STATS 250 Week 02(a): Chapter 2 Turning Data into Info
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0:01 - 0:03... especially with a large class.
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0:03 - 0:04Alright!
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0:04 - 0:08We haven't seen you since last Thursday. I don't see you quite as often as my Monday, Wednesday, Friday class.
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0:08 - 0:14If you're still having questions about switching a lab because you had something else change in your schedule,
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0:14 - 0:22or still waiting to get into the class, Angie at that statsstudents services email is the best place to contact someone
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0:22 - 0:28or you go right over there. She's in West Hall in the Stat department, and she can actually sometimes help you
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0:28 - 0:34out much better if you go over to see her personally. So [that] is where you can find Angie,
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0:34 - 0:39who is manning my wait-list every morning, issuing permissions for the drops that went through at night,
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0:39 - 0:42so if you still have a question regarding that see Angie.
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0:42 - 0:50The prelabs are working, it wasn't just our site, it wasn't all the 1570 students bringing site-maker down,
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0:50 - 0:55it was some other issue in site-maker certainly, but the prelabs are now available and working fine
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0:55 - 0:59as did their ... fix yesterday and re-load,
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0:59 - 1:05so I hope you can also see them adequately and ... go back to them as needed
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1:05 - 1:09when you start to do your first real homework set and have to make that histogram or boxplot.
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1:09 - 1:16Watch the 2 minute video on how to do that rather than have to go through a whole bunch of other stuff to remember how that works.
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1:16 - 1:21So the videos are there, we still will allow you to finish up your prelab, you know, by tomorrow afternoon, by tomorrow evening,
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1:21 - 1:26the last lab ends at 7, so if you didn't get it done for your lab yesterday because of the difficulties with site-maker
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1:26 - 1:32we will extend that, just email your GSI, work out getting that prelab 1, we just want you to do it, so that
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1:32 - 1:36you are ready to do some of the other things we do in labs and homework, so you get your point if you do it,
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1:36 - 1:41you get no points if you don't ...so complete that please.
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1:41 - 1:48Alright, urh, the other item that's coming up on Thursday is when you go to your lecture book homework tool
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1:48 - 1:52you will start to see a homework there for you on Thursday morning.
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1:52 - 1:55This is my testing student homework side,
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1:55 - 2:01but I have the ability to see anything that's hidden and I have created that first homework, and so, once you click on
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2:01 - 2:06the little pencil thing that say "I'm gonna start writing that homeowork" there's 3 questions ... and you'll be able to click
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2:06 - 2:10through those 3 questions and put in your answers, some of them are multiple choice, some of them are typing up
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2:10 - 2:16a little explanation, and one of them is an upload of a graph to get that practice too.
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2:16 - 2:21The practice homework going up Thursday is practice. I'm putting it under the "required hand in" just so that
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2:21 - 2:28there are points on it, and then it will be submitted by the Wednesday after at 11pm automatically for you.
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2:28 - 2:35The GSIs will get it and they will grade it to go through that process, because we have a few new GSIs this term who haven't done that yet.
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2:35 - 2:40And then you will see what the graded homework looks like back to you, maybe with some feedback too if
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2:40 - 2:46you didn't get something right. It is again just practice so if you're not in the class and haven't subscribed yet that's fine,
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2:46 - 2:50if you don't do it that's fine, it's not gonna affect your grade, except that it does give you a full practice,
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2:50 - 2:55so that when that 1st homework really does come up and you're working on it maybe just the little issues or questions
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2:55 - 2:58you have will be gone ... by then.
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2:58 - 3:03So that will be available, I'll probably give you another highlight on that again on Thursday.
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3:03 - 3:08So, office hours have also started but they're pretty quiet right now, but if you do have an issue,
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3:08 - 3:12if you do want to see how virtual sites works with someone that could help you a little more on that
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3:12 - 3:19come on in, 274 West Hall, Monday through Wednesday, we're there from 9 to 9.
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3:19 - 3:21Thursday 9 o'clock in the morning 'til 5, and Friday just in the morning time, 9 to 12.
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3:21 - 3:27You're welcome to go to any of the office hours even though they may not be your GSI's office hours,
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3:27 - 3:32and instructor office hours are also posted on c-tools, you can go to myself or Dr. ?.
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3:32 - 3:38The first real homework will not be open until January 20th, but the practice one will start this Thursday
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3:38 - 3:41and go from Thursday morning and you got a week basically to do it.
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3:41 - 3:47The following Wednesday night at 11pm, it will be automatically submitted for you.
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3:47 - 3:53We're working through chapter 2. We actually should finish much of that up today, maybe a couple more slides on Thursday.
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3:53 - 3:56For Thursday, I am asking you to read the chapters 3 and 4.
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3:56 - 3:59They're short chapters, they're reading chapters, (um)
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3:59 - 4:03there are not really any formulas in those chapters, maybe just 1 small one.
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4:03 - 4:07I've posted partially complete notes for those 3 and 4.
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4:07 - 4:11You've got the notes in your binder and you can look at a few of the answers.
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4:11 - 4:15I am going to go through the ones that are missing as a recap of those chapters at the beginning on Thursday,
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4:15 - 4:21and then I'll put up a clicker practice quiz, 5 points, I'll actually keep those scores just to show you the histogram
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4:21 - 4:27of them later; but it'll be as if it were a little intermediate practice for you to see if you got those couple ideas
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4:27 - 4:35from those 2 reading chapters, more concepts on different types of bias, an experiment versus an observational study and things like that.
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4:35 - 4:42So try reading through the chapters or at least just look at the notes that are online, they're on c-tools under lecture info,
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4:42 - 4:47just read through them before Thursday and then come ready to recap ideas and try out a little practice quiz.
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4:48 - 4:53So those are my announcements, questions or comments on anything before we recap our histograms that we ended up with,
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4:53 - 4:58question or comment on anything?
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5:03 - 5:05Alright,
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5:05 - 5:10then we're going to recall the 2 types of variables. We're in chapter 2, the 2 types of variables were what?
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5:10 - 5:13categorical or ... quantitative.
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5:13 - 5:20And what was the very first graph we looked at that described a quantitative variable's distribution,
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5:20 - 5:25that gave us the values or categories that your variable can take on and how often they occurred? and how many?
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5:27 - 5:31that first graph we looked at had bars and it was called a
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5:31 - 5:33bar chart or bar graph
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5:33 - 5:36and then we saw the pie graph along with that,
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5:36 - 5:42and the basic thing for a numerical summary is just summarizing the counts or percentages that fall into
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5:42 - 5:45each category, we did that for the sleep deprived status variable,
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5:45 - 5:52and then we went on to our quantitative variable where we had students say how much hours of sleep they typically get per night
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5:52 - 5:57and the quantitative variable to show its distribution, and by distribution we just mean,
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5:57 - 6:00give me an idea of are that that variable can take on
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6:00 - 6:03and how often they tend to occur,
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6:03 - 6:07and the nice picture for the distribution of a quantitative variable was also one with bars but its wasn't
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6:07 - 6:11a bar chart, it was called a ... histogram.
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6:11 - 6:15And the histogram is what we looked at, we gonna recap that on page 8 of your notes, and then
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6:15 - 6:20move on to start summarizing the data that's quantitative with numbers.
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6:20 - 6:24Numbers such as the mean and the median, and we'll get through at least to standard deviation by the end of today.
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6:24 - 6:31So, page 8, we had our histogram of the Amount of Sleep for College Students
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6:31 - 6:37we did a clicker question at the very end where we talked about the shape of that distribution of amount of sleep
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6:37 - 6:44we talked about it being approximately symmetric and another word that would work would be unimodal
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6:44 - 6:49one main mode or peak, and then some of you did want to select the skewness aspect
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6:49 - 6:54it was, it's not perfectly symmetric, a slight skewness to the left, but it's quite slight,
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6:54 - 6:58I would leave that as a secondary and not the primary thing I would be looking for.
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6:58 - 7:01The summary here approximately symmetric, unimodal, centered around 7 hours,
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7:01 - 7:07most of the values between 4 and 10, and I can kind of picture that histogram back with that description.
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7:07 - 7:14So this type of description is what you're asked to write out, even by hand, with your histogram that you make for your prelab.
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7:14 - 7:19Alright ... so lets move to looking at nummerical summaries that would be appropriate.
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7:20 - 7:23Oh, we have one more histogram here for "What if".
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7:23 - 7:29So, you have a response, you measured for a study, it's quantitative, so you make a histogram.
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7:29 - 7:33You look at it graphically first, and this is the picture that you get.
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7:33 - 7:38So this is not unimodal, it's called what? [softly: bimodal]
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7:38 - 7:45Bimodal, more than one mode or peak. A bimodal distribution, what would it tell you?
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7:45 - 7:49I certainly see 2 groups, kinna 2 subgroups.
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7:49 - 7:56I see a groups of observations, people that had low scores or low values on the response, and another group that had high scores here.
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7:56 - 7:58So I see a bimodal distribution.
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7:58 - 8:02I see that there seems to be 2 subgroups in my data set.
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8:02 - 8:07And that's one thing I would comment on and want to find, try to figure out why that occurred.
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8:07 - 8:15So there appears to be ... 2 subgroups.
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8:17 - 8:21I would not just note that and then go on and start calculating means and standard deviations because
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8:21 - 8:25I want to first figure out why? what made these 2 groups?
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8:25 - 8:29Investigate why.
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8:29 - 8:35Maybe it turns out ... that the lower observations were for the males in your data set,
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8:35 - 8:37the higher observations for females.
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8:37 - 8:43Maybe it's the old versus the young, or some other aspect that you measured.
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8:43 - 8:48You want to try to investigate why. It might be a gender issue, it might be an age issue ... it could be ... a region issue,
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8:48 - 8:54or something else that affects that response and gives you these 2 subgroups, which is why in a data set
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8:54 - 8:58we don't just record only the outcome of interest, we record a lot of other variables too,
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8:58 - 9:04they may not be the main response variable but they might help to explain features that we see in our response,
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9:04 - 9:12we might have to control for or account for age or gender in our analyses if we see such a picture like this.
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9:12 - 9:16So I would want to try to figure out why, so we look at our data set with other variables that have been measured,
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9:16 - 9:21may be more demographic but they might help us to understand what's going on and
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9:21 - 9:25then of course, I wouldn't just leave the data set group together after this 'cause if you calculate a mean
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9:25 - 9:29the mean's going to be sort of a balancing point and that's not really reflecting of the group very well.
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9:29 - 9:31wouldn't make sense to summarize this data together.
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9:31 - 9:35I'd rather probably lead toward analyzing the data separately by my subgroups from there on out.
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9:35 - 9:45So we might end up analyzing data separately by my subgroups,
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9:47 - 9:56from there on out. Or at least include that variable that just gives us that distinction as a variable to help control for that factor.
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9:56 - 9:59It's important to look at your data first.
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9:59 - 10:02If your histogram turned out to be like this and I asked you "Should you calculate a mean?"
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10:02 - 10:08Well you can, you can always have the calculator do that for you. SPSS gives you the mean by default with the histogram
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10:08 - 10:13so it doesn't even, you know, make you (you know) check first, but it would not make sense to report it.
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10:13 - 10:16It would not be meaningful in this case.
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10:16 - 10:20Alright, couple of histogram comments are laid out there for you.
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10:20 - 10:26The bar chart, we usually have a gap between the bars, 'cause it separates the different categories, whereas
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10:26 - 10:30histograms there's often one right next to the other, unless of course there's a gap and there's no observations there
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10:30 - 10:34and the you have an outlier maybe sticking out at the end.
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10:34 - 10:41How many classes depends ... on a little bit of judgement but the computers and calculators will do a default kinna algorithm to work out
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10:41 - 10:46a reasonable number of categories. You can always go in and change that slightly,
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10:46 - 10:50you don't want too many, because if you had too many categories then you're going to have just a few values
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10:50 - 10:54in each, it's going to be kind of a "flat pancake" kind of histogram.
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10:54 - 10:57If you have too few, then you have everybody in just a couple of classes, and that's not
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10:57 - 11:02looking like a very good picture of a distribution either.
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11:02 - 11:11We like to put "relative frequency" just another name for "proportion" or "percent" on the Y axis whenever you're comparing observations.
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11:11 - 11:15If I wanted to prepare the histogram for male versus female college students, in terms of Amount of Sleep,
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11:15 - 11:19I could do that as long as my axes were matching up.
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11:19 - 11:24Lots of defaults, lots of options, and you get to see that in SPSS a little bit too.
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11:24 - 11:32Alright, urh, last comment here, is just that one of the sections talks about "dot plots" and "stem and leaves plots" in your textbook
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11:32 - 11:37and we're not going to have you do those particular types of graphs but some of the examples in there do comment
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11:37 - 11:41on the shape of the resulting distribution. So it's still good to just kinna glance through,
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11:41 - 11:44but you won't be asked to make those particular types of graphs.
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11:44 - 11:48The histogram's a good choice overall.
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11:48 - 11:57The example that you have on page 10, you can try out on your own ... that was a, one I think on an exam in a spring term.
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11:57 - 12:01It is another histogram of a quantitative variable. See if you can go through and answer those couple questions.
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12:01 - 12:09The solutions are already posted on c-tools. I'll be posting lecture notes filled in up until the end of drop/add
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12:09 - 12:14and so, anything I missed though or also skip in class I will post no matter what.
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12:14 - 12:21So under lecture noted you can find part 1 of chapter 2 notes already filled in.
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12:21 - 12:25Let's turn to the numerical summaries for the rest of today.
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12:25 - 12:30Numerical summaries only appropriate for quantitative data, even if you had categorical data that you coded
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12:30 - 12:34it may not make sense to do any kind of averaging or finding a median there.
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12:34 - 12:42But it certainly makes sense for a quantitative variable where we saw a reasonably unimodal, homogenous set of observations.
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12:42 - 12:46So you've all calculated a mean at some point, and a median perhaps.
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12:46 - 12:48Measures of center.
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12:48 - 12:53We're gonna have a couple formulas along the way, they're going to be based on data looking like those Xs
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12:53 - 12:58X1 is the first observation in your data set, X2 is the second one. That is representing your set of data.
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12:58 - 13:03So n represents the number of items in your data set, its sometimes called the sample size.
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13:03 - 13:06And how do you calculate a mean of a set of data?
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13:06 - 13:10Add them up, divide by the total number, right?
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13:11 - 13:13The formula for that then would be what?
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13:13 - 13:18X1 add X2 add all those observations up,
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13:19 - 13:25up to the last observation and divide by the total number that's represented here by ... n.
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13:25 - 13:33The symbol for that type of mean, when it's the mean of your sample, which is usually what kind of number or data you have.
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13:33 - 13:36Usually you don't have the entire population of values but rather just a sample.
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13:36 - 13:41It's called the sample mean, it's represented by X - bar.
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13:41 - 13:46It's probably a function on one of your calculators where you can put some data in and press that to
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13:46 - 13:48get the mean, and it's the mean of your sample.
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13:48 - 13:55We have a different notation if it is the mean of the population. We've a greek letter "mu" µ that represents a population mean
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13:55 - 14:01but usually what we're looking at when we're summarizing data it's from a sample, part of the larger population,
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14:01 - 14:02not the entire population.
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14:02 - 14:08Anything calculated on a sample, again, is called a statistic ... so that sample mean is easy to find,
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14:08 - 14:14um, a shorthand notation would be to use that summation notation, the summation there, that big "sigma" Σ
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14:14 - 14:21just means add 'em up, add up whatever's after it, and so that's adding up the Xs and dividing by the total.
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14:21 - 14:24The median, how do we find that?
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14:24 - 14:30The median is the bullet value that's in the middle, but you have to first order your data from smallest to largest.
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14:30 - 14:34Let's say you had an odd number of observations, 5 of them,
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14:34 - 14:39then there is, if you order them, a middle value. And that would be your median.
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14:39 - 14:46What if you have only an even number of observations, say 4 ... so the median's gonna be right here.
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14:46 - 14:53Any value that's between those 2 numbers could be the median, it would divide your data set into 50% above and 50% below,
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14:53 - 14:59but in this case we define it to be the average of those 2 middle numbers so we all get the same answer.
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14:59 - 15:04So the median ... when n is an odd number of observations,
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15:04 - 15:11the median or 'm' is going to be the middle observation, 'cause there is a unique one.
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15:11 - 15:17Maybe the same values, some on the either side, but it will be THE middle value,
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15:17 - 15:29whereas if your number of observations is even ... then you're going to define the median to be the average of the 2 middle observations.
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15:33 - 15:35Alright, let's try it out.
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15:35 - 15:38We've got our small set of data from that study where students were looking at
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15:38 - 15:45whether you're getting the same amount of french fries in your small orders, we have the data provided for you there.
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15:45 - 15:48Again the note right here is whenever you have quantitative data, the first step should be,
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15:48 - 15:53not calculate the mean, some standard deviation or something like that but to graph it.
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15:53 - 16:01Proper graph or histogram to show the distribution, to show the values that are possible is a (uh) histogram for quantitative variable.
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16:01 - 16:05I say we have sort of a unimodal, bell-shaped picture again.
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16:05 - 16:13Roughly symmetric and a range from the 60s up to the lower 80s, and it's centered around the mid to lower 70s.
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16:13 - 16:19So there's our histograph. It makes sense to calculate a mean to summarize this data, we have some variability there but it is somewhat homogenous.
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16:19 - 16:24So let's calculate that mean first. How'd we do it? and what would it be represented by?
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16:24 - 16:27The symbol would be
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16:27 - 16:28X - bar.
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16:28 - 16:38We'd have to plug into our calculator or ... computer, this data set, add up all 12 observations, and divide by 12.
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16:39 - 16:45Now if you have a calculator. I would, I rarely ask you to calculate a mean on an exam, I know you can do it.
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16:45 - 16:49You can do it with a computer or calculator. I usually give you some basic summary measures.
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16:49 - 16:56If we were to calculate this, would a value like 82 even make sense?
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16:56 - 16:59Right? ... 82 would be way over here.
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16:59 - 17:00Is that the balancing point of this?
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17:00 - 17:02Does that look like the mean?
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17:02 - 17:08So you just want to make sure that any calculation you do do can take it back to the picture you got that it makes somewhat sense with that picture.
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17:08 - 17:1782 would not make sense; 69 may not start to make sense either. But how about more in the middle? A 73.6,
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17:17 - 17:22and the units here are always good to include when you're reporting some numerical summaries and
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17:22 - 17:26the units here are always good to include when you're reporting some numerical summaries,
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17:26 - 17:30and the weight in grams. 73.6 is visually about that balancing point, it makes sense with my histogram.
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17:30 - 17:39We have had pictures of a histogram, where we ask you to pick what you think might be the mean, and maybe which one might be the median
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17:39 - 17:41and you should be able to kind of visualize that but not have to calculate, just from the histogram, and know how would it relates back.
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17:41 - 17:44Alright, how 'bout the median?
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17:45 - 17:49Median you need the data in order, and I provided that for you. We have 12 observations,
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17:50 - 17:59so the median's going to fall in the middle between ... that 72 and that 74. So the median here will be what?
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17:59 - 18:0773, it would be the average of the 2 middle observations... 73 grams.
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18:08 - 18:17Now the mean was 73.6 and the median was 73. The fact that those are kind of similar, does that make sense here too?
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18:17 - 18:24If you had a roughly symmetric distribution, then the mean and the median would be very close to one other, and that supports it also.
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18:24 - 18:28Alright, well we have a couple of "what ifs" there on the bottom.
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18:28 - 18:35What if we had put the data in our calculator quickly and instead of putting the 63 in we actually just put in a 3.
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18:35 - 18:40So our smallest observation was a 63, what if we had accidentally just typed in the 3 only.
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18:40 - 18:45How would that affect my measures of center here that we just calculated? How about the median?
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18:47 - 18:50Would the median change at all? No.
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18:50 - 18:57The median doesn't use all the values, it uses them in terms of its place, in terms of there size or order, but it doesn't use every value in a data set.
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18:57 - 19:03So the median's not going to be affected at all. The median would stay the same.
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19:05 - 19:07How about the mean?
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19:09 - 19:16It will change; the mean uses every value that 3 would enter in instead of a 63 is part of the total on the top.
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19:16 - 19:21The mean is going to do what do you think? It's going to go up? go down? It's gonna go down.
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19:21 - 19:25One smaller value, being smaller yet, is going to drag that mean down.
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19:25 - 19:29That the mean would decrease
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19:30 - 19:33The mean would decrease
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19:34 - 19:36to 68.6
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19:40 - 19:45There would only be a couple observations below the mean, 10 of them above it.
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19:45 - 19:49Much different that what we had before. So the mean would be smaller.
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19:49 - 19:55So we talk about the mean IS affected by extreme observations, the median is not so.
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19:55 - 19:57It is more of a resistant measure.
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19:57 - 20:02Top of the next page, let's fill that idea in.
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20:03 - 20:05The mean IS sensitive
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20:06 - 20:16to extreme observations whereas the median is our more resistant measure of center.
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20:17 - 20:24So which measure of center might be the better one to report if in looking at your histogram it was strongly skewed
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20:24 - 20:27maybe with a couple of outliers, either on the high end or low end?
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20:27 - 20:29The median.
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20:29 - 20:34The median would paint a picture of what really is more of the middle observation, where more of the typical values
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20:34 - 20:39tend to be, rather than the mean. The mean can be affected by those extreme observations.
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20:39 - 20:41Very good!
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20:41 - 20:45Alright, couple pictures to show the relationship between the mean and the median.
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20:45 - 20:48What is the descriptor of this first one again?
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20:48 - 20:54We call this ... bell-shaped, symmetric, unimodal, all of those words would work.
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20:58 - 21:06And, the symmetry is the main idea here. How would the mean and median compare? and where will they be?
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21:06 - 21:15Both right in the middle ... and approximately equal to each other.
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21:16 - 21:20The smoothed out histogram being shown on the right,
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21:20 - 21:23upper right, that is also
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21:23 - 21:26symmetric. What's the other word to describe this one?
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21:27 - 21:33UNIFORM, not unimodal but uniform. It actually doesn't have really any mode, or they're all modes if you will,
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21:33 - 21:37'cause they're all equally likely. Symmetric, uniform, it's still symmetric.
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21:37 - 21:46So it would be quite easy here to also find ... the mean and the median, you just need to find the midpoint.
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21:46 - 21:50So if I gave you the end points, you would be able to find the mean and the median for that one even without
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21:50 - 21:54having more detail of how many observations and all the individual values.
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21:54 - 21:59Alright, you got two skewed distributions, This first one here is again what? skewed to the?
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21:59 - 22:06Skewed to the right, look at where the tail ends up being pulled out. Skewed to the right.
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22:06 - 22:13That is more for income data, sales of home. That type. And skewed to the right means we've got a lot of values that are small
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22:13 - 22:18and then we're throwing in a few really large values into that data set, and averaging.
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22:18 - 22:20So the mean and median are not going to be the same.
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22:20 - 22:24Which one will tend to be higher?
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22:24 - 22:30Got a bunch of ones, twos and threes, and you throw in a couple of really large numbers, it's going to pull the mean
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22:30 - 22:32towards those large values.
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22:32 - 22:40Alright...Skewed to the right, relative to each other, the median would be the smaller one and be less than the mean.
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22:40 - 22:43Exactly how much smaller, and exactly where they're placed.
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22:43 - 22:51Median's going to be the value on the axis so that if you looked at the area to right and left it should be both about 50%
-
22:51 - 22:56'cause the median divides things in half that way. The mean should be more visually the balancing point
-
22:56 - 23:02of that histogram or that smooth curve. But relative to each other that's how they should compare.
-
23:02 - 23:04The mean gets pulled in direction of the tail.
-
23:04 - 23:07Our other distribution here is skewed to the left.
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23:09 - 23:13Very typical of your exam scores in this class.
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23:13 - 23:20The mean and median again will not be equal to each other but it would be the mean that gets pulled down
-
23:20 - 23:25compared to the median ...in this distribution.
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23:26 - 23:31Which is why I often report the 5 number summary or the median ... as far as my measure of center
-
23:31 - 23:35when I report scores on your exams, rather than the mean, the mean can get pulled down by
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23:35 - 23:40that one or two or few low scores that end up happening.
-
23:40 - 23:43And what's the descriptor for our last graph there?
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23:45 - 23:47Bimodal
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23:48 - 23:51Two main modes or peak, it's still a roughly symmetric
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23:51 - 23:55and what would the mean and median be there?
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23:55 - 23:59Kind of in the middle again.
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24:00 - 24:07Does either one of those measures of center really represent what we think of as the center as being that sort of typical value?
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24:07 - 24:19Not very well! ... So neither does a very good job as a summary measure here.
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24:23 - 24:28One of the old exams had a picture that ended up being somewhat bnimodal, showing distinct clusters
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24:28 - 24:34and the question was "should you report the mean here for this quantitative variable? is that an appropriate measure?"
-
24:34 - 24:39If yes go ahead and report it, and it was right in the output summary, if no, explain why not.
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24:39 - 24:42And for that type of picture, I would say no.
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24:42 - 24:47And the reason is that you seem to have two subgroups and you shouldn't be aggregating them, combining them together.
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24:48 - 24:54Alright, there's measures of center. Mean and median are the typical ones, we know when it's appropriate to use either one
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24:54 - 24:57and when it's appropriate to perhaps use one over the other.
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24:57 - 25:01Median preferred when its skewed, or strong outliers in your data set.
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25:02 - 25:06Measures of center are done. Measures of spread come next.
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25:06 - 25:11So what if you had the distribution of scores on an exams that were somewhat
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25:11 - 25:19unimodal, bell-shaped, symmetric, and the mean was reported to be 76
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25:20 - 25:22and you scored an 88, how do you feel?
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25:24 - 25:25Good?
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25:25 - 25:31You're above the mean at least, right? um?. How good should you feel?
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25:31 - 25:39Let's suppose 76 is our mean there, and 88 is right about here ... so there's one model.
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25:39 - 25:46Would you feel better with the distribution of scores looking like that number one, or number two?
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25:47 - 25:52Number two, right? Number two relative to the peers and the scores, you're looking much better
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25:52 - 25:58compared to the first distribution. They both have the same shape, they both have the same center, but
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25:58 - 26:03the differ in their spread, or variation. And it's important to know that aspect of your model too.
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26:03 - 26:13And a score from one distribution being 88 from 76 may look better relative to that idea of how spread out the values are.
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26:13 - 26:16So a couple measures of spread to go along with measures of center.
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26:16 - 26:20The two easiest ones of course, or the easiest one that is, is the range.
-
26:20 - 26:26Just look at the overall range of your data set. The range is defined to be the max. minus the min.
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26:26 - 26:34So if I have you calculate the range I want you to take the largest value and subtract the smallest one off and get that spread of that 100% of your data.
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26:34 - 26:40If I ask you to just comment on spread, one comment you can make is "it goes from 10 to 22"
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26:40 - 26:44and that's kind of giving me the range, but it's not computing it.
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26:44 - 26:48Another way of giving you some idea of the breakdown is to report some percentiles.
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26:48 - 26:53In fact alot of your standardized tests report your score, and what percentile you got.
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26:53 - 26:59In the two pictures we just looked at you would have a higher percentile when there is less spread in you model.
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26:59 - 27:06So percentiles tell you a value so that you know what percent are below it and what percent therefore are above it.
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27:06 - 27:09Common percentiles, we've already done one of 'em.
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27:09 - 27:17The median is actually a percentile of your distribution. It is the 50th percentile 'cause it cuts you data set in half, half below, half above.
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27:17 - 27:24If you were to take the lower half of your data set, below the median, and pretend that's your data set and find the median again
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27:24 - 27:35that would be what is called the first quartile or the 25th percentile, and the first quartile is denoted typically with a Q1.
-
27:35 - 27:41If were to take your data set, find the median and take all the values above the median, find the median of that set again
-
27:41 - 27:50you'd have the upper quartile or Q3 or third quartile ... also known as the 75th percentile
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27:50 - 27:54So it's just dividing each half of the data set in half again,
-
27:54 - 27:58and those two quartiles then give you another little bit of positional idea in your data set.
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27:58 - 28:04If you take those numbers, the median, the quartiles, along with the min. and the max. and put them in a table,
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28:04 - 28:09this is the way your textbook lays out this summary, it's called the 5 Number summary.
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28:09 - 28:14It gives you all in one kind of picture here. A couple measures of spread.
-
28:14 - 28:22One measure of spread would be the max. minus the min. so you get the range by looking at that distance.
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28:22 - 28:28You get the median as your measure of center, and then another measure of spread that sometimes used
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28:29 - 28:34instead of the overall range is called the interquartile range or I Q R.
-
28:37 - 28:40Interquartile range.
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28:41 - 28:47It's the measure of the spread for the middle 50% of your data, 'cause the range depends on your two most extreme values.
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28:47 - 28:51What if one of those values is an outlier, so your range looks distorted to be quite large
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28:51 - 28:55when most of your data is really in this range instead, a smaller range.
-
28:55 - 28:58IQR, interquartile range.
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28:58 - 29:03So nice set of summaries for when you have skewed data or outliers would be a 5 number summary.
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29:03 - 29:06It gives your center and spread in a couple of ways.
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29:07 - 29:10Let's try it out. Our French Fries data set again
-
29:12 - 29:19Why don't you go ahead and try to work out the 5 number summary, we found the median already
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29:19 - 29:22Min. and max. are easy to put in.
-
29:23 - 29:26And then let's find those quartiles.
-
29:26 - 29:28So 12 observations,
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29:31 - 29:32middle,
-
29:32 - 30:04(silence)
-
30:04 - 30:13So I'm finding Q1 and Q3. Not too bad here right? You just take your data that's below and above that median position
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30:14 - 30:19What is going to be Q3? ... Any of these lower observations, there's 6 of them down there,
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30:19 - 30:23so the middle falls right here now, between the 69 and the 70.
-
30:24 - 30:36So Q1, the lower quartile would be what? 69.5 and on the upper side there's another 6 observations in that upper half and
-
30:36 - 30:42finding the middle there would be Q3 78.5.
-
30:43 - 30:50Alright, so the range is easily found, the actual computation of the range would be the maximum value minus the min.
-
30:50 - 30:55The overall range of the data covers ... 20 grams.
-
30:55 - 30:57For a spread.
-
30:57 - 31:03The interquartile range defined again to be Q3 minus Q1
-
31:03 - 31:14Looking at the spread of the middle 50% of your data would be that 78.5 minus the 69 ... 69.5
-
31:14 - 31:18A difference there of only 9 grams instead
-
31:20 - 31:27So the middle 50% cover a range of 9 grams. You can compare from one data set to another with common measures such as IQR
-
31:27 - 31:33to see which one looks like it's more spread out in terms of that middle 50%. ...Now quick "What if" here
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31:33 - 31:39What if that 83 were not there? So instead of having 12 observations, I have 11,
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31:40 - 31:44so now what will be my median of this data set?
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31:45 - 31:47The 72
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31:49 - 31:52and looking at finding Q1 and Q3, what would I do?
-
31:52 - 32:01So now, I'm looking at just the 5 observations below the 72, I'm not going to include the 72 in that lower half.
-
32:01 - 32:07So the definition of Q1 when you have a median being one of the values in your data set is everything strictly below and strictly above
-
32:07 - 32:10that works for both an odd and an even.
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32:10 - 32:20So them my Q1 would have been a 69, and looking at only the 5 observations above, my Q3 would have been the 78
-
32:20 - 32:24just so you know how the calculations the computers are doing, that's typically the method of using it
-
32:24 - 32:33sometimes they specifically take .25 times the sample size to find the position of that 25th observation
-
32:33 - 32:39or 25th percentile, and even interpolate, but this is method that most calculators and computers use.
-
32:39 - 32:44Alright, the test score example is another one that you can try out on your own,
-
32:44 - 32:50it's really not too difficult to work through, just report values from that 5 number summary.
-
32:50 - 32:54It's very similar to one of the examples in chapter one, which I asked you to read initially,
-
32:54 - 32:58and it went through a 5 number summary there and pulling out a few values.
-
32:58 - 33:02So try that one out, that would be posted up on c-tools ... very soon.
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33:02 - 33:08We're gonna look at another graph today ... called boxplots.
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33:08 - 33:14You probably saw that in your prelab if you looked at your prelab video or maybe even ... saw one in lab.
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33:14 - 33:21It is a picture of your 5 number summary ... in graphical form.
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33:21 - 33:26So you take your 5 number summary ... you take your quartiles Q1 and Q3
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33:26 - 33:33and you use those to form your box. The length of the box is your IQR, visually,
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33:33 - 33:43so Q1 and Q3, so this length here is really your interquartile range, it's visually showing you that spread of the middle 50% of your data.
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33:43 - 33:48Inside the box, wherever it occurs you put your median with a line drawn in the middle of the box.
-
33:48 - 33:57Now here it's shown more in the middle, it could be that your Q1 and your median are the same, if you had a lot of repeats in your data set it's possible
-
33:57 - 34:00so there might not even be a line in the middle of the box.
-
34:00 - 34:04And then you do a little check. You check to see if there are any outliers, 'cause if there's outliers,
-
34:04 - 34:08I want to still show them visually in the graph that we're making here.
-
34:08 - 34:14There's a rule for checking for outliers, it's called the 1.5 times IQR rule
-
34:14 - 34:24'cause you calculate 1.5 times IQR, and that quantity's your step, your amount that you go out from the quartiles
-
34:24 - 34:28you take Q1 and you go down one step,
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34:28 - 34:33you take Q3 and you go up ... one step
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34:33 - 34:37and those values are your fences or your boundaries.
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34:37 - 34:44Turns out with a little bit of theory you can show that any values that are OUTSIDE those fences are unusual,
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34:44 - 34:47they would be deemed an outlier using this rule.
-
34:47 - 34:54So you take 1.5 times the IQR, you go out a step from you quartiles, put like a little boundaries there, fences
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34:54 - 34:58and say do I have any observations that are outside those fences?
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34:58 - 35:02If I do, I want to plot them separately and draw attention to them 'cause they're sticking out from the rest,
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35:02 - 35:11if I don't ...then just extend the lines out to the actual min. and max. 'cause there were no outliers in that data set
-
35:11 - 35:16so you plot them individually if you have any, if not just extend them out to the smallest and largest observation
-
35:16 - 35:22if there are outliers you say apart from those outliers what are the min. and max. and draw your boxplot accordingly.
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35:22 - 35:26So this boxplot right here is for your 12 orders of french fries, the weights.
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35:26 - 35:32I do not see any points plotted outside the length of those boxes, those whiskers going out individually,
-
35:32 - 35:37I didn't have any outliers, in this example. If you remember the histogram that we looked at a little bit ago
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35:37 - 35:41it didn't have any outliers either. With this set of 12 observations.
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35:41 - 35:48So I see no outliers on either end here, they would be plotted separately,
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35:48 - 35:51I don't see any outliers, so what we're going to do is confirm that there aren't any outliers
-
35:51 - 35:59trying out this rule once ... and then pretend that we do have one larger value and see how that affects the boxplot overall.
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35:59 - 36:04Boxplots are gonna be made for you typically, and I just want you to know how they're constructed
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36:04 - 36:08so let's do one little check on how that rule works.
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36:08 - 36:11What we're doing basically is verifying that there are no outliers in our data set.
-
36:11 - 36:17We didn't see it visually in the histogram, the boxplot we just saw there didn't show it , so let's see
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36:17 - 36:21what that rule is that shows us there aren't any outliers.
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36:21 - 36:29The interquartile range is the first thing to compute, and that was a diference of 9 grams, we just did that together ... a minute ago.
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36:29 - 36:32Calculate that thing called a step
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36:32 - 36:351.5 times the IQR.
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36:35 - 36:40So this is my step amount ...13.5.
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36:40 - 36:4613.5 is not put on my actual boxplot anywhere, it's just the amount I'm going to go out from the quartiles
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36:46 - 36:50to figure out where these fences can be drawn or thought through.
-
36:50 - 36:58So the lower boundary or fence is taking Q1 and subtracting one step. So go from 69.5 which is Q1
-
36:58 - 37:07subtract off one step amount and come up with your lower boundary which is a 56.
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37:07 - 37:11Now that's not necessarily gonna be a value in your data set, but you're going to ask yourself,
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37:11 - 37:16"in my data set did I have any values that were even smaller than that lower boundary?"
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37:16 - 37:22Any observation that fell below ... 56. What was our lowest value?
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37:22 - 37:2663. So we have no outliers on the low side.
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37:26 - 37:33Any observation that fall below this lower boundary? no, so there are no low outliers.
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37:35 - 37:41If my lowest value was a 55 or a 54 that would start to be deemed an outlier, and I would be plotting that separately
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37:41 - 37:48but there is no outlier on the low end. The 56 didn't appear on my boxplot that you had at the top of that page
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37:48 - 37:52but that is the boundary that I'm using to determine if there would be a point I would plot separately.
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37:52 - 38:01What about on the upper end? The upper boundary or fence is 78.5 going up one step
-
38:02 - 38:05This is a boundary of 92.
-
38:06 - 38:09Again, you may or may not even have that value in your data set, but you're gonna be asking
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38:09 - 38:16do I have any values that go beyond that number? Any values on the high end, above 92.
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38:16 - 38:20Our largest value was? ... 83
-
38:20 - 38:28So the answer here is also no. So there are no ... high outliers or large outliers.
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38:29 - 38:34So we've just confirmed the graph that we just made was the graph using that rule, just that it didn't have any outliers
-
38:34 - 38:37so we just drew the lines out to the min. and max.
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38:37 - 38:41What would happen if there were an outlier? How does the boxplot change?
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38:41 - 38:49So we're going to pretend on the next page that that largest value of 83 is really a 93 so we will have an outlier
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38:49 - 38:52and see how the graph changes.
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38:53 - 39:01So if we change from 83 to 93, just makes the one large value larger a bit larger, does the median change?
-
39:01 - 39:06is the median affected by those extreme kind of values? no, still 73.
-
39:06 - 39:11In fact the quartiles are going to be the same too. We're just shifting that one large value out there further
-
39:11 - 39:17so it's still going to be the same Q1 and Q3. Everything's the same except for the largest values which is a 93 now.
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39:18 - 39:27The boundaries that we computed on the previous slide. That 56 lower boundary and that 92 still stand as the boundaries to determine if we have outliers
-
39:27 - 39:32but now on the high side, 92's out there and I have a 93.
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39:32 - 39:37It goes beyond ... so now I do have one high outlier, the value of my 93.
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39:37 - 39:44That's the point that's plotted separately here.
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39:46 - 39:53A different symbol's used depending on the package, it might be dots or asterisks, but any point that's plotted separately is an outlier by this rule.
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39:53 - 40:01There actually could have been, without knowing the data set, at least one outlier here, 'cause there could be two values at that same high value
-
40:01 - 40:07But we plot that separately and then ... the lower box whisker goes out to still the 63 'cause we didn't have any low outliers
-
40:07 - 40:17That upper whisker or line goes up to an 80. Notice that this doesn't go up to the 92, which was our boundary, 'cause we didn't have any values at 92,
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40:17 - 40:21that was just used to determine if we have an outlier or not.
-
40:21 - 40:29The upper end here should go up to the largest value in our data set that's NOT an outlier which we would have plotted separately
-
40:29 - 40:33and our largest value that is not an outlier is?
-
40:33 - 40:3480
-
40:34 - 40:35Why does the line extend out to 80?
-
40:37 - 40:38It's the largest value
-
40:42 - 40:44that is not an outlier.
-
40:52 - 40:58The 50, the lower boundary of 56 and the upper boundary of 92 don't appear on the graph.
-
40:58 - 41:03They are sort of your invisible fences, one kind of fence right there, and one way down over here just to
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41:03 - 41:08determine which values, if there are any, would be plotted separately.
-
41:08 - 41:13So now I can see truly where most of the values are. I see the outlier plotted separately, I see the gap
-
41:13 - 41:15which I would have seen in the histogram too.
-
41:15 - 41:20You have the histogram and just that one larger outcome was moved way out larger you would have
-
41:20 - 41:26had a gap of values and seen that outlier through the histogram, but also drawing to your attention the boxplot.
-
41:26 - 41:32Alright ... so the boxplots are going to be made for you, I don't like to have you work our this IQR rule by hand
-
41:32 - 41:38and try to draw them out, but know what these values are that apply separately and how they came to be plotted that way.
-
41:38 - 41:40Couple notes then to fill in.
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41:41 - 41:44Boxplots are very helpful when you want to make comparisons.
-
41:44 - 41:51You can make side by side boxplots, one boxplot for males, one for females, to compare the two distributions quite easily.
-
41:51 - 42:00So side by side boxplots are good for ... comparing ... two or more ... sets of observations.
-
42:06 - 42:14It automatically puts them on the same axes, the same scaling so that you don't have to go in and click the axes and make the two histograms have the same
-
42:14 - 42:18X and Y axis scaling, so you can compare them, puts 'em right next to each other.
-
42:18 - 42:24We do that quite often for a visual check when we look at multiple comparisons for different groups,
-
42:24 - 42:27watching out for those points that are plotted separately.
-
42:27 - 42:30They're called what again? ... They're outliers.
-
42:30 - 42:35And my point here is that they are still part of the data set.
-
42:41 - 42:47I don't want you to ignore them, in fact they were plotted there separately so you would be drawn to see them.
-
42:48 - 42:52Maybe that outlier is the most interesting observation in your data set.
-
42:53 - 43:00Here's all the data, that observation's really good. How did you get to be THAT good for that combination of factors that I put together?
-
43:00 - 43:03I might wanna focus on that one rather than the rest.
-
43:03 - 43:08Alright, so they're still part of the data set, don't ignore them, they're there to show you that they did stand out
-
43:08 - 43:11from the general rest of the data set.
-
43:11 - 43:17One thing that boxplots do not do very well, that we can't see from a boxplot alone very well is shape.
-
43:17 - 43:22You can't confirm shape from a boxplot only.
-
43:28 - 43:38What graph does a much better job of showing us the shape of the distribution of our quantitative variable? ... A histogram.
-
43:45 - 43:52Your boxplot can look ... beautifully symmetric but your data set underlying it may not be.
-
43:52 - 44:00Just because your median is right in the middle to Q1 and Q3, you don't know what happens between Q1 and the median and Q3.
-
44:00 - 44:05How are the data distributed that way?! They could be all clumped to one end, and then clumped to the middle on the other side,
-
44:05 - 44:10so it may not ... have that same pattern that you're thinking that the boxplot tends to show.
-
44:10 - 44:15So it doesn't confirm shape, you can start to see skewness. A boxplot showing some skewness would be
-
44:15 - 44:20a long tail going out and a bunch of outliers on one end, so you can start to see or visualize that,
-
44:20 - 44:23BUT it still doesn't confirm the shape, we would wanna use a histogram instead.
-
44:23 - 44:28And then one of the graph you're gonna look at in labs #2
-
44:28 - 44:33um, no labs next week, remember?! cause there's no Monday classes, so we don't have labs at all for next week.
-
44:33 - 44:40But the 2nd lab we're gonna look at Q-Q plots... That'll show you whether you have a bell-shaped model in a better way.
-
44:40 - 44:45We're gonna look at time plots 'cause a lot of data we gather over time. So a couple more graphs that we'll look at.
-
44:45 - 44:50And then my last comment really helps you on an exam ... or a quiz if we were to give one
-
44:50 - 44:54and that is that "Show me what you're doing when you're reading values off the graph!"
-
44:54 - 44:58If you're looking all over and reading that value that was the outlier a 92 instead of 93
-
44:58 - 45:05well, I might give you credit for something that's reasonable, based on where the axes are and how fine they are for you to read values from.
-
45:05 - 45:11Show me what you're doing so I can see your approach ... and still give you credit for the right process.
-
45:11 - 45:15Alright, so on the next ... Oh! I got my comic of the day
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45:15 - 45:20Recycling, go green (sings softly: aluminum cans, bottles and) (soft laughter) Recycle those boxploh ... uh yeah, Ok!
-
45:24 - 45:31What I have next for your is to try out a question that was on an exam, in the past. We have here scores
-
45:31 - 45:40on a standardized test, for children, and some of those children ate breakfast, some of them did not.
-
45:40 - 45:45SO we have their scores compared side by side. Side by side boxplots. So I'm gonna ask you to review
-
45:45 - 45:49the notes on that page 16 that we just went through, or keep them in mind anyway, work with a neighbor.
-
45:49 - 45:56Try out these 3 questions, and then we'll click them in. I'm gonna give you some choices then for the actual answers
-
45:56 - 45:58but try them out , and we'll see how you do.
-
45:58 - 46:04This is a chance for you to talk with your neighbor and work out a problem.
-
46:04 - 49:32(silence)
-
49:32 - 49:36(inaudible talking in background)
-
49:36 - 49:42So this question that you have, was on an exam before but not with multiple choice on the exam.
-
49:42 - 49:51If your answer's not exactly there ... it's probably close to one of those answers perhaps ... pick the closest one.
-
49:52 - 49:57Do you notice in the questions there are sometimes words that are bold? ... or italicized?
-
49:57 - 50:01That usually means they're important ...to help you guide to the right boxplot.
-
50:09 - 50:19We're asking here for the approximate lowest score or grade... for a child who... "does" have breakfast.
-
50:21 - 50:24So I want "Do you have breakfast? YES!"
-
50:25 - 50:27That's the boxplot I wanna work with.
-
50:28 - 50:35Looking pretty good! Here's your distribution, most of you indeed are picking the right answer. Which is ... B) ... 4.5
-
50:35 - 50:40Plus 5 points minus 10 over here
-
50:41 - 50:43Alright!
-
50:46 - 50:49The... lowest value 4.5 is this value right here. What is that called? That's an ... outlier. (Outlier)
-
50:52 - 50:56But is it still part of the data set? (Yes) (It is)
-
50:56 - 51:01It's the lowest value. There could've even been two children that had 4.5. But it is the minimum.
-
51:01 - 51:06If you ignored the outliers, yes, then the next smallest observation is 6.
-
51:06 - 51:10But 6 is not the minimum value here, for that data set.
-
51:10 - 51:16And of course the other, um, answer of 4 was for the other group in case you went to the wrong boxplot.
-
51:16 - 51:18Highlighting that.
-
51:18 - 51:23Now if someone wrote, and I saw someone in my other classes wrote 4.6, I would have given credit for that
-
51:23 - 51:24'cause that's close to what I have there.
-
51:24 - 51:30You know it's pretty much right in the middle, but if you read it off as 4.4 or 4.6 you'd still credit,
-
51:30 - 51:35especially if you were to circle it or bring it over and kinda tell me what you think that number is, on the axis.
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51:35 - 51:43Alright, a little more curious about this ... next question ...and go ahead!
-
51:46 - 51:56Among children who did NOT eat breakfast ... 25% has a score of at least how many points?
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51:59 - 52:14(inaudible taking in background)
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52:15 - 52:17We're a little more split on this one. (Uh oh)
-
52:18 - 52:21For our question 2.
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52:22 - 52:26Alright, let's take a look.
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52:27 - 52:30Wanna change your answer? Um?
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52:31 - 52:41Question 2, I see a 25%, so I'm sort of thinking a quartile rather than the median. So the choice over these three.
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52:41 - 52:48Let's take a look. I gonna tell you right up front that the right answer is ... A)
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52:49 - 52:51Let's see WHY?
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52:52 - 52:58Alright we're looking at which boxplot? The one with the children that did ... not eat breakfast. So the "NO" group.
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52:58 - 53:04I do see what? what is the median for this one? About 6...and a half.
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53:04 - 53:12That would be my median... and my quartiles are about this 5.5 ... and then about 7.5
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53:13 - 53:18And I see a 25%, but 25% does not necessarily mean it's gonna be a Q1.
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53:19 - 53:26When you take Q1, you got 25% percent below and what percent above Q1? ... 75
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53:26 - 53:34When you take Q3, you got another 75, 25 split. There's 25% above it and 75% below it.
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53:34 - 53:40So depending on which quartile you use depends on which direction you were looking to go for that 25%.
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53:41 - 53:47So often in these kind of questions it's almost easier to just say "I know it's a Q1 or Q3", 'cause it could be kinda tricky that way.
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53:47 - 53:51Put each one in and see if it makes the sentence correct.
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53:51 - 53:57So let's try the one that was the most common answer. Most of you took 25% and said "oh it's Q1, 5.5"
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53:57 - 54:09So what if we put 5.5 in our sentence here? It says ... you're looking at the percent of students that had a score of ...at least ... 5.5 points.
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54:09 - 54:15So at least 5.5. points means 5.5 points or ... more.
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54:15 - 54:20And what percent of the students had a score of 5.5 points or more? or at least 5 and a half points?
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54:20 - 54:2475% ... NOT 25%.
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54:24 - 54:31There are 75% ... that had a score of at least 5 and a half points.
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54:31 - 54:40Let's put in the other Q3 of 7.5. We put in 7.5 and say what percent of the students had a score of at least 7 and a half points.
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54:40 - 54:48At least means that many points or? ... more. And what percent are there?... 25%
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54:48 - 54:52The correct answer A).
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54:52 - 55:00If it were "at most" instead of "at least" ... if I had at most there then you would be calculating that many points or?... less,
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55:00 - 55:03and then you do want 25%, so that's Q1.
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55:03 - 55:11So is it "at least" or "at most"? is it more than or less than? and you want the 25% in that spot , or the 75% tells you which one to use.
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55:11 - 55:15You just have to think it through a little bit, or kinda try out both numbers and see which one ends up
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55:15 - 55:22making it correct... noting the direction you're going and what percent you want there... in that direction.
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55:23 - 55:31Alright, so that one I put there specifically 'cause that was one on a quiz, I think for spring term and I did have a number get it wrong.
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55:31 - 55:35I'm doing it now so you won't get it wrong if it were asked somewhere on your exam.
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55:35 - 55:39Our last one is a true/false. Let's see what you choose there.
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55:40 - 55:46We have a very nice symmetric boxplot for our "not eating breakfast" group.
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55:46 - 55:54The statement is that this implies the distribution for scores, is set nice bell-shaped symmetric distribution.
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55:55 - 55:57No outliers in that one.
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55:58 - 56:04And ... most of you paid attention to the notes on the previous page.
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56:05 - 56:08Notes on the previous page would lead you to say "This is false".
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56:09 - 56:16It is false. ... It does not imply the distribution is bell-shaped, just because the distribution is symmetric.
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56:19 - 56:26A symmetric boxplot ... does not necessarily guarantee that your data when you plot it would be symmetric.
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56:26 - 56:31It tells you that you median and your Qs, your Q1 and Q3, are nice and symmetric from each other,
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56:31 - 56:34and even if you went up to a max. and min. it would be the same in distance.
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56:34 - 56:39But what happens in between, to that 25% chunks you've got?
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56:39 - 56:42They could be distributed in different ways that are not mirrored or symmetric.
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56:42 - 56:47And even if it were symmetric, does it mean it's gonna be this type of symmetry? bell-shaped?
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56:47 - 56:58Isn't there other symmetry models? Bimodal will give you a symmetric boxplot. Just the same as a unimodal bell-shaped curve will give you a symmetric boxplot.
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56:58 - 57:01So you can't see the clusters.
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57:01 - 57:05You can't see bimodality from your boxplot. It hides that aspect of shape.
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57:05 - 57:13Boxplots hide clusters and gaps, you don't see them as readily. So shape is not best to imply from any boxplot.
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57:13 - 57:17Look at the histogram along with it. So false!.
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57:17 - 57:25Alright ... now we did have one outlier, at least one outlier in our data set, there could actually be two children or more there, at that low value.
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57:25 - 57:29What do you do with outliers?
-
57:29 - 57:33A small section in your text. Section 2.6 talks about how to handle them.
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57:33 - 57:37Gives you a couple good examples for you to look at.
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57:37 - 57:44The primary idea is that you just can't throw them out, and say "oh they don't match, let's just not use them".
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57:44 - 57:50You have to take a look at your data, it might be a legitimate value, it might represent something that's been gathered under different conditions.
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57:51 - 58:00Joel did the measuring of these parts, knew how to operate this machine. Jim came in while he went out, to the bathroom, and did a measurement and it was way wrong,
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58:00 - 58:04and that would indicate that you shouldn't use it 'cause it wasn't calibrated correctly or whatever.
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58:04 - 58:08It could just be natural variability, and you occasionally get a value that stands out,
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58:08 - 58:14but more than likely it could be that someone else made the measurement. The measurement was entered incorrectly,
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58:14 - 58:17you can go back and check your records, just a switching around of the values or something,
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58:17 - 58:23but you can't throw it out unless there's a legitimate reason to say it doesn't fit with the rest of the data for this situation,
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58:23 - 58:28it's measured by different person or on a different machine ... or under a different condition.
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58:28 - 58:34It could be that it's an interesting value to look at, and that might be your focus.
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58:34 - 58:38It could be that that particular person or item belongs to a different group.
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58:38 - 58:42That was the only male in your data set and everybody else that you asked the question of were all females.
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58:42 - 58:46So that could give you a different rating or response.
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58:46 - 58:52Alright, so, some good examples there. We have techniques that are more resistant to outliers than others,
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58:52 - 58:54we have a median that's better to report than a mean ...if there were an outlier.
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58:54 - 59:00Question in the back? ... or just stretching?
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59:00 - 59:08Alright, one last measure of spread ....Standard Deviation... and we end with our pictures of the day.
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59:09 - 59:15So how many of you have heard of standard deviation too as another nummerical summary?
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59:15 - 59:18How it's calculated, how you interpret it, would be our last focus.
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59:19 - 59:28So if you have the median as your measure of center, then you're most likely gonna report the range or the IQR, interquartile range, to go along with it.
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59:28 - 59:36If you had the mean, as a reasonable measure of center, then often you see next to that is the standard deviation as a measure of spread.
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59:36 - 59:43'Cause what does the standard deviation do? It measures the spread of your observations from, the mean.
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59:43 - 59:51The standard deviations takes every value of a new data set, and looks at how far away every observation is from the mean.
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59:51 - 59:56We're going to interpret the standard deviation in the following way.
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59:56 - 60:04We want to interpret it as being roughly the average distance, that your values are ... from the mean.
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60:04 - 60:09It's that typical distance, that average distance, of the values from the mean.
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60:09 - 60:09Now it's not exactly the average.
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60:10 - 60:13If you were going to calculate the average, you'd have a little different formula than what we're gonna write up here.
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60:13 - 60:19But it does take distances from the mean, and we want to interpret it or view it as being roughly that average distance,
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60:19 - 60:23so it can be kind of a yardstick for us ... to see how far away we are from the mean.
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60:23 - 60:26So here's how it's computed. Let's work that out.
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60:26 - 60:30We're gonna take every observation and look at how far away it is from the mean.
-
60:30 - 60:32So the first observation would be X1
-
60:32 - 60:41and what was the mean of a sample again? We represent that by X - bar, that represents the sample mean. The mean of your set of data.
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60:41 - 60:47So there's the first distance, and I'm gonna calculate the next distance, my next observation from the mean.
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60:47 - 60:50And I'm gonna do that for all of the observations.
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60:52 - 60:58Every single value's used in calculating the standard deviation. Just as it is for calculating the mean.
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60:58 - 61:03And if I were to average these, I would be starting to add them up then, right?
-
61:03 - 61:11Average these distances. And if you did that calculation right there, that would always come out to be zero.
-
61:11 - 61:16'Cause some of those distances are positive, some values were above the mean and some were below the mean.
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61:16 - 61:23And so every negative and positive ends up, when you work them all out as a total, to add up to zero. That's a property of the mean.
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61:23 - 61:26The total of the distances of every value from the mean is zero.
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61:26 - 61:29So you can't really just average that 'cause you always get zero.
-
61:29 - 61:35So what's one way of getting rid of the negatives versus positives is to take ... absolute value, and that would work
-
61:35 - 61:41that would be called "the mean absolute deviation", which is a measure of spread,
-
61:41 - 61:47which really is more of that average distance idea, but it doesn't work very well mathematically.
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61:47 - 61:54Trying to integrate or show properties of this kind of measurement here, this statistic when there's absolute values to work with ... not so fun.
-
61:54 - 61:58So another way to get rid of it is to ... square everything.
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61:58 - 62:06Then a value that's 2 below the mean contributes 4, just as a value that sits 2 above the mean contributes the same distance of 4.
-
62:06 - 62:09But it's in squared units now.
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62:09 - 62:14So if I were to use squares, I would average those squares by doing what?
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62:14 - 62:22Add all those squared distances up and divide by ... 'n', we're gonna divide by n minus 1.
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62:24 - 62:29n minus one, sometimes gonna be called, later for us, the degrees of freedom.
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62:30 - 62:34Those quantities that were in the top numerator there,
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62:34 - 62:39and before we put the squares there, we said they added up to zero.
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62:39 - 62:46So if you has 10 numbers and you know that the distances add up to zero, 9 of those those numbers could be anything
-
62:46 - 62:54giving any of the distance at all, that last one though, has to be fixed, because it has to make that total add up to zero.
-
62:54 - 62:58So there are 10 observations but only 9 that are free to vary, and be whatever they want to be.
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62:58 - 63:03That last one's fixed by this constraint, because we're looking at that distance from the mean.
-
63:03 - 63:10It also turns out, if you do a "n-1" on the bottom ... you're gonna get a better estimate out from your data
-
63:10 - 63:18for what the true standard deviation would be for the population, than if you didn't correct for that, losing of one degree of freedom.
-
63:18 - 63:21So later it's gonna be called
-
63:21 - 63:24degrees of freedom
-
63:25 - 63:30when we use some inference using this as our measure of spread.
-
63:30 - 63:37Right now, we have, looking at roughly the average of the squared deviation of the values from the mean,
-
63:37 - 63:42and I don't want to have a measurement in squared units when it comes out, so we take the square root of that whole thing.
-
63:44 - 63:52And you got the standard deviation ... represented by that little letter 's' for our sample standard deviation.
-
63:52 - 63:56's' goes along with X - bar... those are the symbols for a sample standard deviation.
-
63:56 - 64:01The shorthand notation that's on your yellow card, that has your second formula there, I think the mean is listed there too.
-
64:01 - 64:12would be that you take the sum of the X minus the X - bars, and square them, dividing that total or sum by that 'n-1'.
-
64:12 - 64:15So that's just that more compact notation.
-
64:15 - 64:18You've already likely have on your calculator an 's' button too.
-
64:18 - 64:23Or when you have it do some basic summaries it will give you that standard deviation, which is nice.
-
64:23 - 64:29What we calculated before we took a square root ... is called s² (s squared), that's your variance.
-
64:29 - 64:36Variance is not in original units. Variance is in squared units so we wanna bring it back to the original units to be my yardstick.
-
64:36 - 64:40Here's the average and my give or take, my standard deviation each way.
-
64:40 - 64:43So that's how it's computed. Let's try it out once.
-
64:43 - 64:46There's an example in the bottom that you can look at, on your own.
-
64:46 - 64:54We're gonna calculate it once but mostly focus on the interpretations that we have in the middle of that next page. Page 19.
-
64:58 - 65:03This is probably the only time you're gonna calculate the standard deviation by hand, working it out.
-
65:03 - 65:07The rest of time I want you to use the calculator or I provide it on the exam for you to interpret.
-
65:07 - 65:14These are our observations, the average was 73.6, that was our X - bar.
-
65:14 - 65:18The standard deviation or 's' for our sample.
-
65:18 - 65:22We're gonna need a very large square root.
-
65:22 - 65:34We're gonna have to start calculating distances of each of the values from their mean, so the first one's a 77 ... minus the 73.6 ... but I do have to square that.
-
65:35 - 65:43We're kind of cheating a little bit here, and putting in the first and the last observation...
-
65:43 - 65:46to show would it would be that you would have to work out and do.
-
65:47 - 65:54With only a very basic calculator this would take a bit of time, especially with a mean of 73.6, in decimal form like that,
-
65:54 - 65:57but I would do all of those squared terms and add them up.
-
65:57 - 65:59So, an observation that's further away from the mean contributes more.
-
65:59 - 66:06Leading to a higher spread or more variation than if you had a lot of observations that are close to the mean.
-
66:06 - 66:12What will we divide by here? ...12 is our observation number, so 12 minus ...1
-
66:12 - 66:18Under the square root you would get a thirty...five point four
-
66:20 - 66:2435.4 has what name again?
-
66:25 - 66:29It's called the variance. It's call the variance.
-
66:29 - 66:31So sometimes you'll see reports that show the variance and the standard deviation.
-
66:31 - 66:38Standard deviation's preferred because we bring it back to original units, and that would be about 5.9
-
66:38 - 66:44and so we can say the standard deviation's 5.9 grams versus a variance of 35.4 grams squared.
-
66:44 - 66:47Hard to think of squared units.
-
66:47 - 66:53So there's out standard deviation. If you looked at the histogram, the values did vary from the mean
-
66:53 - 67:00some of them are more than 5.9 away, some of them are closer ... than 5.9 away.
-
67:00 - 67:04But the average distance can be thought of as being ABOUT 5.9 grams.
-
67:04 - 67:10Probably the more important thing is to know what this number tells you in terms of an interpretation or viewing it,
-
67:10 - 67:18and then also when you have certain shapes of distributions it's very useful for a yardstick as we will see ...not today, on Thursday.
-
67:18 - 67:23So interpretation's coming first. This is gonna be an important page.
-
67:23 - 67:29We're definitely gonna have on real homework number 1, somewhere interpreting a standard deviation.
-
67:29 - 67:33I think in one of your modules, it might even be in module one, there's some good examples to look at.
-
67:33 - 67:38If you didn't get to them in lab that were wrong interpretations and right ones, in picking out the right ones.
-
67:38 - 67:45The weights of our small orders of french fries. They weren't all the same ... they did vary.
-
67:45 - 67:51They vary, they are about how far way from the mean? Roughly how far on average?
-
67:51 - 68:02So these weights of small order of french fries are roughly about ... 5.9 grams away from their mean, which happened to be this 73.6.
-
68:02 - 68:09And then we've got that important clarifier there ... "on average".
-
68:09 - 68:13I'm not saying that every order of french fries have the weight that is exactly 5.9 grams away from the mean.
-
68:13 - 68:21There were some that were further away, and some that were closer; and if you look at the average of those distances, it would be ABOUT 5.9.
-
68:21 - 68:25So come, some keys parts here is that we are clarifying it 'cause we didn't calculate the absolute values,
-
68:25 - 68:28we did square things and then took the square root.
-
68:28 - 68:34We are talking about an average distance... and I give you the frame of reference from which you're looking at
-
68:34 - 68:37the values from what? from their mean.
-
68:37 - 68:40Another way to write it correctly is to start out with the "On average" so you ...
-
68:40 - 68:47"On average, these weights did vary ... by ABOUT how much... from their mean?"
-
68:47 - 68:55by ABOUT 5.9 grams .. from their mean, which happens to be known here as 73.6.
-
68:56 - 69:02So there's a couple examples of where you have all the right parts ... for an interpretation.
-
69:02 - 69:09A standard deviation's thought of as being roughly an average distance. The average distance that the values vary from ...
-
69:09 - 69:11what frame of reference from? The mean.
-
69:12 - 69:19A data set that had a standard deviation of 5.9 compared to a data set that had a standard deviation of 59.
-
69:19 - 69:23Much more spread from the mean on average, compared to the other one.
-
69:23 - 69:26Alright, standard deviation. Couple of notes
-
69:26 - 69:30Well, how would we get a standard deviation of 0?
-
69:30 - 69:32What would would it mean?
-
69:32 - 69:39There's no variation, no spread. Has to occur when all the values are ... the same.
-
69:39 - 69:46So 's' could be zero, that's the smallest it could be. It represents that there is no spread at all, no variation,
-
69:46 - 69:51and it would occur of course when all the observations ...are the same.
-
69:53 - 70:00Otherwise, what kind of values do you get for an 's', a standard deviation? It'd have to be positive.
-
70:00 - 70:03You can't get an 's' of -2.8. It would be marked really wrong.
-
70:04 - 70:13The larger the value, the more spread. The closer to zero, the more consistent the values are, and close to the mean on average.
-
70:13 - 70:21Now, the mean was our measure of spread that we, or measure of center that we talked about was somewhat sensitive to those extreme observation,
-
70:21 - 70:28an extreme observation would pull the mean quite down or up; and the standard deviation looks at every value from that mean
-
70:28 - 70:34so the standard deviation is also ... SENSITIVE to extreme observations.
-
70:39 - 70:48So when do we prefer to use the mean and the standard deviation, to go along with a graph, to show or summarize a quantitative variable?
-
70:48 - 70:55We do that when our distribution looks to be reasonably symmetric, and bell-shaped.
-
70:59 - 71:06If our distribution overall looks reasonably symmetric, unimodal, bell-shaped ... it's even more ideal.
-
71:06 - 71:11Then the standard deviation and a mean work very well, in fact we're gonna see in the beginning of next class
-
71:11 - 71:16this empirical rules that allows us to really see how it works with a bell-shaped model.
-
71:16 - 71:215 number summary ... uses a median, uses the interquartile range to measure spread,
-
71:21 - 71:26which are more resistant and better to use when you have what kind of distributions?
-
71:26 - 71:33Strongly skewed distributions, not just slightly skewed, slightly is okay, to use a mean and standard deviation, but
-
71:33 - 71:37strongly skewed distributions, OR if you have outliers.
-
71:42 - 71:48Strongly skewed distributions or if outliers, 'cause they will affect the mean and the standard deviation.
-
71:48 - 71:51The last bullet ... and then a picture of the day.
-
71:51 - 71:56The last bullet is: your calculator will often have lots of these numbers that can be summarized for you,
-
71:56 - 71:59and calculated for you, but sometimes it even gives you more.
-
72:01 - 72:04Do you remember the first sample mean?
-
72:04 - 72:14X - bar, that's the mean of a sample. That's called a statistic. And 'mu' ... is the notation for a population mean.
-
72:15 - 72:24For standard deviation we have a parallel, of sigma versus 's'. Most of the time, I know on TI calculators
-
72:24 - 72:28when you get your summary measures, sometimes on regular calculators, the little buttons,
-
72:28 - 72:36they have a sigma button or an 's' button. The sigma is if you had a population of values and you had all those numbers
-
72:36 - 72:41put into your calculator, then you want the population standard deviation, which is computed a little differently
-
72:41 - 72:49than the sample one ... and you need to use ... sigma. But most of the time the data that we have is from a sample.
-
72:49 - 72:54Most of the time, we're computing sample statistics, and NOT true population values,
-
72:54 - 73:00so we're gonna often pull off the 's' and not the sigma when that comes out from your calculator, so you know that difference.
-
73:00 - 73:04And the you get to see pictures of the day before you leave.
-
73:04 - 73:09I have two dogs! First Lily ... and there's Molly.
-
73:09 - 73:14Lily is a 16 year old Beagle mix that can't hear, can't see but she's still around.
-
73:14 - 73:18(inaudible talking in background) Molly is a lot of fun! She's kind of a cat-dog. (laugh in background)
-
73:18 - 73:22She keeps us (inaudible) busy, with walks... I hope you have a good day. We'll see you on Thursday!
- Title:
- STATS 250 Week 02(a): Chapter 2 Turning Data into Info
- Description:
-
A lecture from Statistics 250 - Introduction to Statistics and Data Analysis.
Instructor: Brenda Gunderson
View the course materials:
https://open.umich.edu/education/lsa/statistics250/fall2011Creative Commons Attribution-Non Commercial-Share Alike 3.0 License
http://creativecommons.org/licenses/by-nc-sa/3.0/ - Video Language:
- English
- Duration:
- 01:13:24
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
openmichigan.video edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info | ||
Kristy edited English subtitles for STATS 250 Week 02(a): Chapter 2 Turning Data into Info |