0:00:01.486,0:00:04.716
Hello. In this video, I would[br]like to introduce you

0:00:04.716,0:00:08.165
to the concept[br]of sampling distributions.

0:00:08.165,0:00:13.550
Now, sampling distributions are going[br]to be fundamental to understanding

0:00:13.550,0:00:18.598
how we are able to derive at [br]certain knowledge about distributions.

0:00:18.598,0:00:21.227
So what -- Let's, first of all, [br]let's just recap

0:00:21.227,0:00:23.957
what a sample is [br]and what a population is.

0:00:23.957,0:00:26.107
So we've seen this diagram before.

0:00:26.107,0:00:29.767
A population is essentially all [br]the individuals we could possibly

0:00:29.767,0:00:35.343
ever get information on, [br]get data. And populations --

0:00:35.343,0:00:38.161
let's just pick a very, [br]very boring example of height.

0:00:38.161,0:00:43.303
Maybe the population is [br]all UT students, all UT students.

0:00:43.303,0:00:45.643
Now, every single student [br]enrolled at UT,

0:00:45.643,0:00:50.500
if we were able to get all of their [br]heights, there would be some true

0:00:50.500,0:00:53.500
measure of the average height.[br]The average height would be

0:00:53.500,0:00:56.670
a population mean, and there would [br]be some value that that is.

0:00:56.670,0:01:00.530
We'd also be able to get some[br]other truth, we call these truths.

0:01:00.530,0:01:03.280
A standard deviation, if we were[br]able to measure every

0:01:03.280,0:01:05.143
however many thousand[br]UT students (inaudible).

0:01:05.143,0:01:09.387
If we were able to measure those things,[br]we'd be able to get those values,

0:01:09.387,0:01:11.967
and they're the population values,[br]and you may remember

0:01:11.967,0:01:14.663
we call them parameters.

0:01:14.947,0:01:18.417
Now, that's pretty much impossible.[br]We would never actually, in theory,

0:01:18.417,0:01:21.207
ever be able to measure [br]the height of all UT students.

0:01:21.207,0:01:23.670
So what we do, if we were interested,[br]if this was something we were

0:01:23.670,0:01:27.270
interested in for some strange reason,[br]we'd be able to just collect a sample.

0:01:27.270,0:01:30.410
I don't know. Maybe we'd get [br]a sample of five people,

0:01:30.410,0:01:32.730
or maybe we would get [br]a sample of 50 people,

0:01:32.730,0:01:35.460
or maybe we would[br]get a sample of 500.

0:01:35.460,0:01:37.560
In future videos, [br]we'll talk about

0:01:37.560,0:01:39.840
what's an appropriate size [br]of sample to collect

0:01:39.840,0:01:41.520
when we're interested [br]in finding out something.

0:01:41.520,0:01:45.600
But for now, who knows?[br]Maybe we picked ten people,

0:01:45.600,0:01:47.865
but we pick a sample [br]from those individuals.

0:01:47.865,0:01:51.040
Maybe we measure, we measure[br]all their heights and we calculate

0:01:51.040,0:01:55.530
their average height and we call[br]that the sample mean, "x-bar."

0:01:55.702,0:02:00.418
We're also able to calculate their,[br]maybe their sample standard deviation.

0:02:00.418,0:02:03.369
We'd call that "s," [br]something like that.

0:02:03.369,0:02:09.341
These things we call statistics.[br]This is just recap stuff.

0:02:09.341,0:02:12.627
And the purpose of [br]calculating the statistics

0:02:12.627,0:02:16.617
is because we want[br]to estimate these parameters.

0:02:16.617,0:02:20.414
We want to make an estimation[br]of what the true value is.

0:02:20.699,0:02:25.673
Now the issue is, let's say we're[br]just collecting samples of five,

0:02:25.913,0:02:29.533
and then maybe another day,[br]we got a different sample of five,

0:02:29.533,0:02:32.828
and then on another day,[br]we're addicted to measuring

0:02:32.828,0:02:35.638
the heights of people, [br]we get another sample of five.

0:02:35.638,0:02:38.058
Every time we get [br]another sample of five,

0:02:38.058,0:02:40.838
let's, for now, just consider [br]the sample mean.

0:02:40.838,0:02:44.381
We're not that interested in[br]the standard deviation of these.

0:02:44.381,0:02:47.751
We're just gonna get a different [br]average every time we collect a

0:02:47.751,0:02:50.956
sample of five. It'd be [br]very, very unusual

0:02:50.956,0:02:54.596
if we randomly sampled five individuals[br]from all UT students,

0:02:54.596,0:02:58.716
and it was the same five people.[br]It's almost unlikely

0:02:58.716,0:03:01.788
ever to be the same five people,[br]and so the average height

0:03:01.788,0:03:04.698
of the sample mean is going [br]to be different every single time.

0:03:04.698,0:03:07.653
And I don't know, let's say [br]another day we did another five,

0:03:07.653,0:03:08.783
we got another sample mean.

0:03:08.783,0:03:11.713
So we're trying [br]to estimate these things,

0:03:11.713,0:03:15.253
but our values are going [br]to generally be different,

0:03:15.253,0:03:17.391
and then they're never going [br]to be exactly -- well they could

0:03:17.391,0:03:20.358
actually be exactly the population [br]mean, but it's unlikely.

0:03:20.358,0:03:23.640
So what we've just done, by the way,[br]is a sampling distribution.

0:03:23.640,0:03:26.812
I kind of, by accident, have [br]introduced you to it.

0:03:26.812,0:03:30.162
A sampling distribution is these things:

0:03:30.162,0:03:33.742
It's when you collect lots, [br]and lots, and lots of samples.

0:03:33.742,0:03:36.242
You collect lots and lots[br]of samples, and then you

0:03:36.242,0:03:38.072
just look at all [br]the values that you got.

0:03:38.072,0:03:41.332
And then you tend to plot [br]them as a histogram.

0:03:41.332,0:03:44.022
So in the next couple of slides,[br]I'll just more formally go through this,

0:03:44.022,0:03:48.032
because it gets very slightly [br]more exciting than this.

0:03:48.032,0:03:50.037
Well this distribution, actually,

0:03:50.037,0:03:52.717
looks quite nice [br]when you collect the data.

0:03:52.717,0:03:56.938
Okay, so in this case, I don't have --[br]my population is not all UT students.

0:03:56.938,0:04:00.528
My population is four people.[br]There's just four people.

0:04:00.528,0:04:05.038
Maybe my population is four people[br]that are roommates together,

0:04:05.331,0:04:10.111
and I ask them, "How many people[br]do you phone in one week?"

0:04:10.111,0:04:13.891
And one person said they phoned[br]four people, one person said seven,

0:04:13.891,0:04:16.231
one person said five, [br]one person said eight.

0:04:16.231,0:04:19.367
So, if you look at [br]our population mean, it's six.

0:04:19.367,0:04:23.796
That's just 4+5+7+8 divided by [br]the number of individuals,

0:04:23.796,0:04:26.956
which is four. Our mean is six.

0:04:26.956,0:04:30.189
So, I was really interested [br]in knowing how many,

0:04:30.189,0:04:33.315
this population of this one house,

0:04:33.315,0:04:36.577
I wanted to know what's[br]the average number of people

0:04:36.577,0:04:37.917
that they phone in a week.

0:04:37.917,0:04:41.037
But I thought collecting data on [br]four people was too much work,

0:04:41.037,0:04:44.253
so I decided, I'm just going [br]to collect samples of three people.

0:04:44.253,0:04:47.093
And so, I got a five, [br]a seven, and an eight.

0:04:47.093,0:04:49.111
They were the first three people [br]that I sampled.

0:04:49.111,0:04:51.840
And so my sample mean [br]here is 6.67,

0:04:51.840,0:04:56.167
which is 5+7+8 divided by [br]my sample size of three.

0:04:56.339,0:04:59.979
So I got one sample mean,[br]but this is not a sampling distribution

0:04:59.979,0:05:01.549
because it's just one sample.

0:05:01.549,0:05:04.563
For a sampling distribution,[br]I need to do this over and over

0:05:04.563,0:05:06.263
and over again.

0:05:06.263,0:05:09.077
So I can do something like this,[br]where I do, let's say,

0:05:09.894,0:05:13.771
one sample, two samples, three samples,[br]four samples, five samples, six samples,

0:05:13.771,0:05:17.841
and here are my values:[br]5,7,8; 4,5,8; 4,7,8; 4, 8, 8;

0:05:17.841,0:05:20.812
and for each of these, [br]I calculate the sample mean.

0:05:21.281,0:05:24.212
This is a sampling distribution.[br]I've now got many samples,

0:05:24.212,0:05:27.314
and I've collected [br]the sample mean for each of them.

0:05:28.031,0:05:30.601
One thing -- there's two things[br]you may have noticed about this:

0:05:30.601,0:05:34.341
one thing I want you to notice is that [br]I've actually sampled with replacement.

0:05:34.341,0:05:36.660
So in the UT example, I said[br]it would be really unlikely

0:05:36.660,0:05:39.447
to ever get [br]the same five people.

0:05:39.447,0:05:41.835
More than that, if you randomly[br]selected five people,

0:05:41.835,0:05:45.255
it's unlikely to get the same [br]individual twice in one sample.

0:05:45.255,0:05:47.785
But for sampling distributions,[br]we tend to actually say

0:05:47.785,0:05:49.563
that we will sample [br]with replacement.

0:05:49.563,0:05:51.907
That's just a -- it's not something[br]we actually need to worry

0:05:51.907,0:05:54.277
too much about right now.[br]I just want you to be aware of --

0:05:54.277,0:05:57.297
in this particular example,[br]I sampled with replacement,

0:05:57.297,0:06:00.207
which means you could sample [br]the same individual twice,

0:06:00.207,0:06:03.170
or even three times, because [br]you'll notice the 64th sample

0:06:03.170,0:06:05.830
had the same individual[br]three times and I got 8,8,8,

0:06:05.830,0:06:08.805
and the mean of [br]that sample mean is 8.

0:06:08.805,0:06:11.385
The second thing you may [br]have noticed is that

0:06:11.385,0:06:14.605
the sample number [br]only goes up to 64 here.

0:06:14.605,0:06:16.605
I've done 64 samples, [br]and that's because

0:06:16.605,0:06:20.855
when you collect three numbers[br]from four potential numbers,

0:06:21.213,0:06:25.233
there's only 64 combinations of the data,[br]so I just stopped at 64.

0:06:25.233,0:06:29.448
Okay, but what I want you to realize[br]is that I have collected 64 samples.

0:06:29.448,0:06:31.638
These dots just refer to --[br]I didn't put all the data

0:06:31.638,0:06:35.764
between 6 and 64.[br]I have 64 sample means.

0:06:35.764,0:06:40.269
So I have 64 of these things,[br]so what should I do with them?

0:06:40.269,0:06:42.699
Well, I could plot them on a histogram.

0:06:42.699,0:06:47.272
So, here's my histogram [br]of the 64 sample means,

0:06:47.609,0:06:50.999
and I wonder what you think about it.

0:06:50.999,0:06:55.179
Well, one thing is I've used a nice color,[br]I think, for the bars, but another thing

0:06:55.179,0:06:59.216
is maybe you see that there's[br]potentially a shape in this data.

0:06:59.365,0:07:04.639
So if I, if you allow me to be[br]kind of, draw a curve over it,

0:07:04.639,0:07:07.211
I haven't done a great job,[br]but there's a curve,

0:07:07.211,0:07:09.624
and it looks normal-ish.

0:07:09.624,0:07:11.946
I'm using the word, "ish," because[br]it's obviously not normal.

0:07:11.946,0:07:15.184
It's kind of getting there to be [br]normal distributed.

0:07:15.184,0:07:18.367
So this is just a histogram of all[br]the possible sample means.

0:07:18.367,0:07:23.252
And here is the one where we had 8,8,8,[br]and actually there's one down here

0:07:23.252,0:07:27.689
where we got 4,4,4, but there's all[br]the ones in between as well.

0:07:30.486,0:07:33.310
Now what we --[br]And I'll just put the curve back on it.

0:07:33.310,0:07:35.739
Maybe I'll actually this time [br]do a better job.

0:07:35.739,0:07:37.674
I'm not sur --[br]No, I didn't.

0:07:37.674,0:07:39.437
I'm gonna undo that because that[br]was a terrible job.

0:07:39.437,0:07:41.211
Let's see if I can --

0:07:41.211,0:07:44.501
I think going slow...[br]no, it will do.

0:07:44.501,0:07:45.899
Mmm... no it won't.

0:07:45.899,0:07:47.010
Let's do it again.

0:07:47.010,0:07:49.369
This time, third time's lucky.

0:07:50.478,0:07:51.983
Okay I'm happy with that.

0:07:51.983,0:07:55.000
This is the sampling distribution.

0:07:55.000,0:07:57.338
I just told you that,[br]but I want you to formally know

0:07:57.338,0:07:58.404
what it means.

0:07:58.404,0:08:03.836
It's the sampling distribution,[br]dot dot dot, for the sample mean,

0:08:03.836,0:08:05.111
that's what we collected.

0:08:05.111,0:08:08.201
We collected the sample mean,[br]so it's a sampling distribution of the

0:08:08.201,0:08:12.162
sample mean, and then we say[br]for n=3.

0:08:12.162,0:08:15.938
Because our sample [br]could have been 2, n=2m

0:08:15.938,0:08:17.272
in which case we would have [br]had a different distribution.

0:08:17.272,0:08:19.782
It could've been 1. We could've [br]just been really lazy and said,

0:08:19.782,0:08:22.110
"I only want to just check [br]one person and ask them,"

0:08:22.110,0:08:24.948
and calculated [br]these sample mean for 1.

0:08:24.948,0:08:27.509
But is the sample -- [br]we did three people,

0:08:27.509,0:08:30.317
and so this is technically [br]the sampling distribution

0:08:30.317,0:08:33.085
for the sample mean for n=3.

0:08:34.833,0:08:39.363
Okay, so that's probably enough for now [br]on introducing sampling distribution.

0:08:39.363,0:08:42.585
I hope you understand[br]about what it is.

0:08:42.585,0:08:44.152
It is the -- [br]you collect many samples,

0:08:44.152,0:08:46.358
and you collect some information [br]about each of those samples --

0:08:46.358,0:08:50.606
in this case, it was the sample mean --[br]and then you plot them as a histogram,

0:08:50.606,0:08:53.177
and you are able to look at [br]the shape of the distribution.

0:08:53.415,0:08:55.441
And that's what we call [br]a sampling distribution.

0:08:55.441,0:08:58.013
And we're going to extend [br]this idea in future videos.