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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:01.48,0:00:04.100,Default,,0000,0000,0000,,Hello, in this video, I want to talk about\Nthe standard error and this is really
Dialogue: 0,0:00:04.100,0:00:09.78,Default,,0000,0000,0000,,extending our understanding of sampling \Ndistributions and essential limit theorem.
Dialogue: 0,0:00:10.48,0:00:13.22,Default,,0000,0000,0000,,So, let's talk about what \Na standard error is.
Dialogue: 0,0:00:14.100,0:00:17.73,Default,,0000,0000,0000,,First of all, we'll go back to \Nthis penguin example and
Dialogue: 0,0:00:17.73,0:00:22.09,Default,,0000,0000,0000,,you've seen this distribution before\Nas a uniform distribution of data.
Dialogue: 0,0:00:22.96,0:00:26.86,Default,,0000,0000,0000,,It has, like any distribution, it has--\Nthere's descriptive statistics.
Dialogue: 0,0:00:26.86,0:00:28.52,Default,,0000,0000,0000,,So, it has a population mean.
Dialogue: 0,0:00:28.52,0:00:30.06,Default,,0000,0000,0000,,The average is 5.04.
Dialogue: 0,0:00:30.06,0:00:33.88,Default,,0000,0000,0000,,The average penguin is 5.04 meters \Nfrom the edge of the ice sheet.
Dialogue: 0,0:00:33.88,0:00:36.75,Default,,0000,0000,0000,,You can calculate a standard \Ndeviation for this.
Dialogue: 0,0:00:36.75,0:00:39.51,Default,,0000,0000,0000,,So, the deviation is 2.88.
Dialogue: 0,0:00:39.51,0:00:42.31,Default,,0000,0000,0000,,So, that's the, you know, \Na measure of the spread.
Dialogue: 0,0:00:42.31,0:00:47.58,Default,,0000,0000,0000,,And there was 5,000 penguins floating\Non this ice sheet, that's the n,
Dialogue: 0,0:00:47.58,0:00:49.18,Default,,0000,0000,0000,,the population size.
Dialogue: 0,0:00:50.10,0:00:54.80,Default,,0000,0000,0000,,We then discussed about how if you were\Njust to sample either just randomly select
Dialogue: 0,0:00:54.80,0:01:00.19,Default,,0000,0000,0000,,five penguins at a time or 50 penguins\Nat a time, that each of those samples
Dialogue: 0,0:01:00.19,0:01:04.44,Default,,0000,0000,0000,,of, let's pick the n equals five for now,\Neach of those five penguins,
Dialogue: 0,0:01:04.44,0:01:07.100,Default,,0000,0000,0000,,you could calculate how, what the average\Ndistance from the front of the edge sheet
Dialogue: 0,0:01:07.100,0:01:13.41,Default,,0000,0000,0000,,was for each of those individual penguins,\Nsample of five penguins and if you were
Dialogue: 0,0:01:13.41,0:01:17.26,Default,,0000,0000,0000,,to do that over and over and over again\Nand in this histogram, we did it
Dialogue: 0,0:01:17.26,0:01:22.100,Default,,0000,0000,0000,,1,000 times, we would be able to generate\Nwhat's called the sampling distribution.
Dialogue: 0,0:01:24.60,0:01:30.15,Default,,0000,0000,0000,,And it's the sampling distribution of\Nthe sample means, that's what it is
Dialogue: 0,0:01:30.15,0:01:35.26,Default,,0000,0000,0000,,and I told you that we could calculate\Nfrom that what the average of
Dialogue: 0,0:01:35.26,0:01:37.87,Default,,0000,0000,0000,,those sample means across \Nthe 1,000 samples was and
Dialogue: 0,0:01:37.87,0:01:43.95,Default,,0000,0000,0000,,that's this value and the notation that\Nwe use for that is this mu and then
Dialogue: 0,0:01:43.95,0:01:53.28,Default,,0000,0000,0000,,subscript x bar and that's the mean\Nof the sample means and I've forgotten
Dialogue: 0,0:01:53.28,0:01:56.66,Default,,0000,0000,0000,,what it was, the exact value, but it's\Npretty much going to approximate
Dialogue: 0,0:01:56.66,0:01:58.21,Default,,0000,0000,0000,,very, very close.
Dialogue: 0,0:01:58.21,0:02:04.26,Default,,0000,0000,0000,,So, I just put approximately equal to \N5.05, just go back, it's 5.04.
Dialogue: 0,0:02:04.26,0:02:11.11,Default,,0000,0000,0000,,So, it was-- it's going to approximate\Nthe population average and you can
Dialogue: 0,0:02:11.11,0:02:13.49,Default,,0000,0000,0000,,do that for any sample size.
Dialogue: 0,0:02:13.49,0:02:16.68,Default,,0000,0000,0000,,So, that was sample size five,\Nlet's look at the sample size 50.
Dialogue: 0,0:02:16.68,0:02:24.09,Default,,0000,0000,0000,,Again, we have the mean of the sampling\Ndistribution-- sorry, the mean of
Dialogue: 0,0:02:24.09,0:02:28.58,Default,,0000,0000,0000,,the sample means and that is also going\Nto be very close to 5.04, it might be
Dialogue: 0,0:02:28.58,0:02:32.17,Default,,0000,0000,0000,,a little bit closer because \Nour sample size is larger.
Dialogue: 0,0:02:32.88,0:02:35.65,Default,,0000,0000,0000,,Two other things to notice about \Nthese distributions, number one
Dialogue: 0,0:02:35.65,0:02:39.03,Default,,0000,0000,0000,,they're normally distributed or approx--\Nsorry, the approximate to normal
Dialogue: 0,0:02:39.03,0:02:43.15,Default,,0000,0000,0000,,distributions despite the fact for \Nthe original distribution of penguins.
Dialogue: 0,0:02:43.15,0:02:46.44,Default,,0000,0000,0000,,The population distribution was \Na uniform distribution.
Dialogue: 0,0:02:46.44,0:02:50.96,Default,,0000,0000,0000,,Second thing to notice, the sample size\Ndoesn't really effect where the value
Dialogue: 0,0:02:50.96,0:02:55.98,Default,,0000,0000,0000,,of the mean, of the sample means, it does\Neffect the standard deviation of
Dialogue: 0,0:02:55.98,0:02:57.43,Default,,0000,0000,0000,,the sample means.
Dialogue: 0,0:02:57.43,0:03:00.30,Default,,0000,0000,0000,,So, if this is a normal distribution,\Nor we believe it to approximate,
Dialogue: 0,0:03:00.30,0:03:08.47,Default,,0000,0000,0000,,and then also this approximates \Na normal distribution, then, it's clear
Dialogue: 0,0:03:08.47,0:03:14.62,Default,,0000,0000,0000,,that the distance here, let's just assume\Nthat's a standard deviation and I put it
Dialogue: 0,0:03:14.62,0:03:16.77,Default,,0000,0000,0000,,in the right place.
Dialogue: 0,0:03:16.77,0:03:21.49,Default,,0000,0000,0000,,This standard deviation, it's greater than\Nwhatever the corresponding value is
Dialogue: 0,0:03:21.49,0:03:25.12,Default,,0000,0000,0000,,over here, if that's also \Nthe standard deviation.
Dialogue: 0,0:03:25.12,0:03:29.32,Default,,0000,0000,0000,,So, as the sample size gets larger,\Nthe spead of the sample means
Dialogue: 0,0:03:29.32,0:03:33.15,Default,,0000,0000,0000,,gets smaller, so, we can say \Nthe standard deviation gets smaller.
Dialogue: 0,0:03:33.15,0:03:38.42,Default,,0000,0000,0000,,Now, does this standard deviation have any\Nrelationship at all to the original
Dialogue: 0,0:03:38.42,0:03:41.42,Default,,0000,0000,0000,,standard deviation of \Nthe original population.
Dialogue: 0,0:03:41.42,0:03:45.29,Default,,0000,0000,0000,,The original standard deviation was 2.88,\Nso, I'll just say population of
Dialogue: 0,0:03:45.29,0:03:48.72,Default,,0000,0000,0000,,the original-- standard deviation was 2.88
Dialogue: 0,0:03:48.96,0:03:51.82,Default,,0000,0000,0000,,Is there any relationship at all between \Nthese two standard deviations?
Dialogue: 0,0:03:52.20,0:03:55.90,Default,,0000,0000,0000,,Because it's not like the mean of\Nthe sample means, which is pretty
Dialogue: 0,0:03:55.90,0:04:00.75,Default,,0000,0000,0000,,much the same, regardless of the sample \Nsize, I mean it does get better with
Dialogue: 0,0:04:00.75,0:04:04.13,Default,,0000,0000,0000,,larger samples but it approximates,\Nit's close, especially if you have
Dialogue: 0,0:04:04.13,0:04:06.02,Default,,0000,0000,0000,,enough of these samples.
Dialogue: 0,0:04:06.02,0:04:08.98,Default,,0000,0000,0000,,What's the relationship of these standard\Ndeviations because it's clear that when
Dialogue: 0,0:04:08.98,0:04:14.60,Default,,0000,0000,0000,,you change n, this value is going to \Nchange, so is there a relationship?
Dialogue: 0,0:04:14.60,0:04:16.52,Default,,0000,0000,0000,,And it turns out that there is \Na relationship and we're going to
Dialogue: 0,0:04:16.52,0:04:18.17,Default,,0000,0000,0000,,look into that.
Dialogue: 0,0:04:18.17,0:04:23.76,Default,,0000,0000,0000,,This graph here just shows you that \Nthe normal distribution for becomes
Dialogue: 0,0:04:23.76,0:04:26.79,Default,,0000,0000,0000,,better and better the larger \Nthe sample size, so, it's a little
Dialogue: 0,0:04:26.79,0:04:29.85,Default,,0000,0000,0000,,tricky to see but let me, I just want to\Nreally point out one or two things here.
Dialogue: 0,0:04:29.85,0:04:33.86,Default,,0000,0000,0000,,I'm going to pick a color \Nthat represents that.
Dialogue: 0,0:04:33.86,0:04:38.75,Default,,0000,0000,0000,,So, this value here, actually in red, so,\Nif I was just to pick one penguin at
Dialogue: 0,0:04:38.75,0:04:43.51,Default,,0000,0000,0000,,a time, a sample size of one, this is\Nmy estimate of the sample-- I'm going
Dialogue: 0,0:04:43.51,0:04:45.05,Default,,0000,0000,0000,,for the red line here.
Dialogue: 0,0:04:45.05,0:04:49.90,Default,,0000,0000,0000,,That's my estimate of the sample--\Nsorry, let's say that again.
Dialogue: 0,0:04:49.90,0:04:52.16,Default,,0000,0000,0000,,That's the distribution of \Nthe sample means.
Dialogue: 0,0:04:52.16,0:04:53.100,Default,,0000,0000,0000,,It looks like the original population.
Dialogue: 0,0:04:53.100,0:04:57.32,Default,,0000,0000,0000,,So, for a sample size of one, you don't\Nget a normal distribution of the sample
Dialogue: 0,0:04:57.32,0:05:00.62,Default,,0000,0000,0000,,means, you get whatever \Nthe original population was.
Dialogue: 0,0:05:00.62,0:05:06.70,Default,,0000,0000,0000,,Let's look at two and I've got to find it\Non here, so, it's the orange one and
Dialogue: 0,0:05:06.70,0:05:11.63,Default,,0000,0000,0000,,I believe it's this one here.
Dialogue: 0,0:05:11.63,0:05:13.39,Default,,0000,0000,0000,,It is this one here.
Dialogue: 0,0:05:13.39,0:05:14.80,Default,,0000,0000,0000,,This is what it looks like.
Dialogue: 0,0:05:14.80,0:05:17.17,Default,,0000,0000,0000,,This is the n is two.
Dialogue: 0,0:05:17.17,0:05:19.49,Default,,0000,0000,0000,,So, again, not a really \Nnormal distribution.
Dialogue: 0,0:05:19.49,0:05:22.19,Default,,0000,0000,0000,,Now, let's skip to 50.
Dialogue: 0,0:05:22.19,0:05:27.13,Default,,0000,0000,0000,,This is 50 here and you can see it\Nreally, you don't need me to help
Dialogue: 0,0:05:27.13,0:05:28.37,Default,,0000,0000,0000,,you too much.
Dialogue: 0,0:05:28.37,0:05:30.78,Default,,0000,0000,0000,,This is the 50 value, it's very normal.
Dialogue: 0,0:05:30.78,0:05:35.65,Default,,0000,0000,0000,,And then, we got blue at ten--\Nsorry, 25 here.
Dialogue: 0,0:05:35.65,0:05:37.89,Default,,0000,0000,0000,,This is the 25 one and so on.
Dialogue: 0,0:05:37.89,0:05:39.81,Default,,0000,0000,0000,,This is the ten.
Dialogue: 0,0:05:39.81,0:05:41.68,Default,,0000,0000,0000,,This is the five.
Dialogue: 0,0:05:41.68,0:05:44.26,Default,,0000,0000,0000,,I wanted to just show you this graph\Nbecause I wanted to show you that
Dialogue: 0,0:05:44.26,0:05:48.70,Default,,0000,0000,0000,,even with very, very, very small \Nsample sizes of like five, we already
Dialogue: 0,0:05:48.70,0:05:51.26,Default,,0000,0000,0000,,get very close to a normal distribution.
Dialogue: 0,0:05:51.26,0:05:54.75,Default,,0000,0000,0000,,It's only with sample sizes of ridiculous\Nsample sizes of like one or two that
Dialogue: 0,0:05:54.75,0:05:56.76,Default,,0000,0000,0000,,we don't do a very good job,
Dialogue: 0,0:05:56.76,0:05:59.97,Default,,0000,0000,0000,,So, even with small sample sizes, \Nwe get to the normal distribution
Dialogue: 0,0:05:59.97,0:06:02.77,Default,,0000,0000,0000,,of the normal distribution of \Nthe sample means.
Dialogue: 0,0:06:02.77,0:06:07.38,Default,,0000,0000,0000,,So, back to the problem \NI just posted a moment ago.
Dialogue: 0,0:06:07.38,0:06:14.50,Default,,0000,0000,0000,,This is our original standard deviation\Nof a population, this is our population
Dialogue: 0,0:06:14.50,0:06:17.11,Default,,0000,0000,0000,,and whenever we get a sample,\Nand again, this is just the sample
Dialogue: 0,0:06:17.11,0:06:17.95,Default,,0000,0000,0000,,size of five.
Dialogue: 0,0:06:17.95,0:06:19.41,Default,,0000,0000,0000,,This is the distribution of sample means.
Dialogue: 0,0:06:19.41,0:06:26.54,Default,,0000,0000,0000,,The mean is going to approximate the mean\Nhere but what is the relationship of
Dialogue: 0,0:06:26.54,0:06:31.47,Default,,0000,0000,0000,,the standard deviation to \Nthis original population.
Dialogue: 0,0:06:31.47,0:06:32.73,Default,,0000,0000,0000,,What is the relationship?
Dialogue: 0,0:06:32.73,0:06:38.63,Default,,0000,0000,0000,,It must be also related to the sample size\Nbecause it changes with its sample size.
Dialogue: 0,0:06:38.63,0:06:43.02,Default,,0000,0000,0000,,And it's just a formula and we're not\Ngoing to talk too much about--
Dialogue: 0,0:06:43.02,0:06:47.41,Default,,0000,0000,0000,,we're not going to talk much really at all\Nabout how it's derived but this formula
Dialogue: 0,0:06:47.41,0:06:52.14,Default,,0000,0000,0000,,here, very neatly, just tells us \Nabout their relationship and
Dialogue: 0,0:06:52.14,0:06:55.51,Default,,0000,0000,0000,,so, what we have here is this is \Nour standard deviation of
Dialogue: 0,0:06:55.51,0:06:59.99,Default,,0000,0000,0000,,the sampling distribution of\Nthe sample means.
Dialogue: 0,0:06:59.99,0:07:02.98,Default,,0000,0000,0000,,So, we call that sigma subscript x bar,
Dialogue: 0,0:07:02.98,0:07:04.40,Default,,0000,0000,0000,,sigma x bar.
Dialogue: 0,0:07:04.40,0:07:07.86,Default,,0000,0000,0000,,The standard deviation, so just to really\Nreiterate what we're looking at, this is
Dialogue: 0,0:07:07.86,0:07:12.66,Default,,0000,0000,0000,,the distribution of sample means,\Nthis is-- we're looking for this value
Dialogue: 0,0:07:12.66,0:07:15.81,Default,,0000,0000,0000,,what's this standard deviation?
Dialogue: 0,0:07:15.81,0:07:22.16,Default,,0000,0000,0000,,And actually, technically, that's \Nthe notation, what is that standard
Dialogue: 0,0:07:22.16,0:07:23.15,Default,,0000,0000,0000,,deviation?
Dialogue: 0,0:07:23.15,0:07:25.19,Default,,0000,0000,0000,,So, what we do is, we just take \Nthe original population.
Dialogue: 0,0:07:25.19,0:07:30.30,Default,,0000,0000,0000,,This is the population standard deviation\Nfrom the original population and we're
Dialogue: 0,0:07:30.30,0:07:35.48,Default,,0000,0000,0000,,going to divide it by the square root of n\Nand that gives us that this value,
Dialogue: 0,0:07:35.48,0:07:36.75,Default,,0000,0000,0000,,this standard deviation.
Dialogue: 0,0:07:36.75,0:07:40.72,Default,,0000,0000,0000,,Its technical name is the standard\Ndeviation of the sampling distribution
Dialogue: 0,0:07:40.72,0:07:44.42,Default,,0000,0000,0000,,of the sample means, which is an awful\Nmouthful but we just call
Dialogue: 0,0:07:44.42,0:07:45.42,Default,,0000,0000,0000,,it the standard error of
Dialogue: 0,0:07:45.42,0:07:49.45,Default,,0000,0000,0000,,the mean, which is what we call it\Nthe standard error of the mean.
Dialogue: 0,0:07:49.45,0:07:55.09,Default,,0000,0000,0000,,So, this graph illustrates how \Nthe standard error of the mean
Dialogue: 0,0:07:55.09,0:07:56.99,Default,,0000,0000,0000,,changes by sample size.
Dialogue: 0,0:07:56.99,0:08:06.73,Default,,0000,0000,0000,,So, if I just go back to-- maybe, \NI'll just go back to this slide here
Dialogue: 0,0:08:06.73,0:08:11.10,Default,,0000,0000,0000,,and we were asking the question of, \Nyou know, what's this value over
Dialogue: 0,0:08:11.10,0:08:14.84,Default,,0000,0000,0000,,sample size 50 compared to this \Nvalue of a sample size of five?
Dialogue: 0,0:08:14.84,0:08:19.23,Default,,0000,0000,0000,,So, that was the question and I'm going to\Nplot-- maybe here I'll plot it or write it
Dialogue: 0,0:08:19.23,0:08:20.26,Default,,0000,0000,0000,,sorry.
Dialogue: 0,0:08:20.26,0:08:25.10,Default,,0000,0000,0000,,So, this is the formula, the standard\Nerror of the mean or the standard
Dialogue: 0,0:08:25.10,0:08:27.68,Default,,0000,0000,0000,,deviation of the sampling distribution\Nof the sample means is equal to
Dialogue: 0,0:08:27.68,0:08:31.60,Default,,0000,0000,0000,,the original population standard deviation\Ndivided by the square root of n.
Dialogue: 0,0:08:31.60,0:08:36.92,Default,,0000,0000,0000,,So, when we had that sample size of five,\Nwhich is this one up here, what we're
Dialogue: 0,0:08:36.92,0:08:42.18,Default,,0000,0000,0000,,really looking at is this, the original\Nstandard deviation was 2.88 and
Dialogue: 0,0:08:42.18,0:08:45.77,Default,,0000,0000,0000,,we're going to divide by the square root\Nof the sample size which is five, so that
Dialogue: 0,0:08:45.77,0:08:47.77,Default,,0000,0000,0000,,equals 1.3.
Dialogue: 0,0:08:47.77,0:08:53.34,Default,,0000,0000,0000,,So, the standard deviation here is 1.3 and\Nthat standard error we call that is 1.3.
Dialogue: 0,0:08:53.34,0:08:59.43,Default,,0000,0000,0000,,So, what this is saying is this value here\Nis 1.3 higher that was it, I forget.
Dialogue: 0,0:08:59.43,0:09:04.00,Default,,0000,0000,0000,,I think it was 5.04 was the mean of\Nthe sample means and so this value here
Dialogue: 0,0:09:04.00,0:09:10.07,Default,,0000,0000,0000,,is going to be a 6.5-- nope, nope, not five.
Dialogue: 0,0:09:10.07,0:09:15.24,Default,,0000,0000,0000,,It's going to be at 6.34.
Dialogue: 0,0:09:15.24,0:09:22.01,Default,,0000,0000,0000,,This is one standard deviation above \Nthe sample mean but if we have
Dialogue: 0,0:09:22.01,0:09:26.74,Default,,0000,0000,0000,,a sample size of fifty, then \Nthe calculation becomes this.
Dialogue: 0,0:09:26.74,0:09:30.30,Default,,0000,0000,0000,,Becomes the original standard deviation\Nof the population divided by the square
Dialogue: 0,0:09:30.30,0:09:32.52,Default,,0000,0000,0000,,root of 50, which is equal to and I've
Dialogue: 0,0:09:32.52,0:09:35.91,Default,,0000,0000,0000,,written this down so I can check, 0.4.
Dialogue: 0,0:09:35.91,0:09:40.23,Default,,0000,0000,0000,,So, back to this graph, \Nthis value is 0.4,
Dialogue: 0,0:09:40.23,0:09:43.00,Default,,0000,0000,0000,,and this value is 1.3.
Dialogue: 0,0:09:43.00,0:09:46.55,Default,,0000,0000,0000,,And so, it gets smaller the bigger the\Nsample size.
Dialogue: 0,0:09:46.55,0:09:52.46,Default,,0000,0000,0000,,This graph here that I got to previously\Nis actually showing us
Dialogue: 0,0:09:52.46,0:09:56.06,Default,,0000,0000,0000,,how the standard error changes by\Nthe sample size.
Dialogue: 0,0:09:56.06,0:09:59.66,Default,,0000,0000,0000,,So we just had a sample size of 50,\Nwhich is approximately here.
Dialogue: 0,0:09:59.66,0:10:06.84,Default,,0000,0000,0000,,If we go across to this value on this\Naxis, it tells us that's about 0.4,
Dialogue: 0,0:10:06.84,0:10:11.84,Default,,0000,0000,0000,,sample size of 50, and if we had \Na sample size of 5,
Dialogue: 0,0:10:11.84,0:10:15.99,Default,,0000,0000,0000,,which is approximately here --\NI'm doing a line, not very well,
Dialogue: 0,0:10:15.99,0:10:20.37,Default,,0000,0000,0000,,but it goes to about there.\NThis was about 1.3.
Dialogue: 0,0:10:20.37,0:10:23.83,Default,,0000,0000,0000,,And I just want you to -- there's nothing\Nreally too much for you to take home
Dialogue: 0,0:10:23.83,0:10:27.34,Default,,0000,0000,0000,,from this graph other than showing you\Nthat as the sample size increases,
Dialogue: 0,0:10:27.34,0:10:32.42,Default,,0000,0000,0000,,that the -- any population \Nstandard deviation that we have,
Dialogue: 0,0:10:32.42,0:10:36.58,Default,,0000,0000,0000,,the standard error is going to get\Nmuch smaller very rapidly.
Dialogue: 0,0:10:36.58,0:10:41.04,Default,,0000,0000,0000,,A sample size of 5 is still quite high up\Non this curve,
Dialogue: 0,0:10:41.04,0:10:44.43,Default,,0000,0000,0000,,but once you come down to sample sizes\Nof 20 or 30 or more,
Dialogue: 0,0:10:44.43,0:10:49.22,Default,,0000,0000,0000,,then we get a very, very small \Nstandard error.
Dialogue: 0,0:10:50.58,0:10:56.79,Default,,0000,0000,0000,,This is just to reiterate that point so\Nyou can see what these are on this graph.
Dialogue: 0,0:10:56.79,0:11:00.39,Default,,0000,0000,0000,,So let's put together what we've \Njust learned about the standard error
Dialogue: 0,0:11:00.39,0:11:04.52,Default,,0000,0000,0000,,with what we have learned previously about\Nthe Central Limit Theorem.
Dialogue: 0,0:11:04.52,0:11:08.80,Default,,0000,0000,0000,,So what we have just been discussing is \Nthat we just know that we have
Dialogue: 0,0:11:08.80,0:11:10.93,Default,,0000,0000,0000,,an original population, \Nit could be any distribution,
Dialogue: 0,0:11:10.93,0:11:13.37,Default,,0000,0000,0000,,here's our uniform distribution.
Dialogue: 0,0:11:13.37,0:11:16.76,Default,,0000,0000,0000,,If we take many samples from it,\Nwe get our sampling distribution.
Dialogue: 0,0:11:16.76,0:11:24.60,Default,,0000,0000,0000,,In this case, of the sample means,\Nis normally distributed
Dialogue: 0,0:11:24.60,0:11:27.10,Default,,0000,0000,0000,,or approximately normally distributed.
Dialogue: 0,0:11:27.10,0:11:34.70,Default,,0000,0000,0000,,And we know that the sampling distribution\Nhas a mean that is approximately equal to
Dialogue: 0,0:11:34.70,0:11:40.40,Default,,0000,0000,0000,,the population mean and we've just learned\Nthat we just know now that
Dialogue: 0,0:11:40.40,0:11:44.12,Default,,0000,0000,0000,,the standard deviation of this \Napproximately normal distribution,
Dialogue: 0,0:11:44.12,0:11:46.83,Default,,0000,0000,0000,,this is the standard error.
Dialogue: 0,0:11:46.83,0:11:49.65,Default,,0000,0000,0000,,I'll write here, "standard error."
Dialogue: 0,0:11:50.32,0:11:53.64,Default,,0000,0000,0000,,So we can actually write this in\Nnotation form,
Dialogue: 0,0:11:53.64,0:11:56.84,Default,,0000,0000,0000,,and we say that this sampling distribution\Nis approximately normal,
Dialogue: 0,0:11:56.84,0:11:59.94,Default,,0000,0000,0000,,this is what this tilde squiggle means, \Nis approximately normal,
Dialogue: 0,0:11:59.94,0:12:07.62,Default,,0000,0000,0000,,approximately normal and it has a mean\Nof the population mean,
Dialogue: 0,0:12:07.62,0:12:11.15,Default,,0000,0000,0000,,so I'll just write here, \Nthe mean is the population mean.
Dialogue: 0,0:12:11.15,0:12:13.25,Default,,0000,0000,0000,,And the standard deviation of that \Ndistribution,
Dialogue: 0,0:12:13.25,0:12:15.77,Default,,0000,0000,0000,,and we're talking about this distribution\Ndown here,
Dialogue: 0,0:12:15.77,0:12:18.94,Default,,0000,0000,0000,,the standard deviation of that \Ndistribution is the standard error,
Dialogue: 0,0:12:18.94,0:12:20.60,Default,,0000,0000,0000,,that's what we call it.
Dialogue: 0,0:12:20.60,0:12:22.57,Default,,0000,0000,0000,,And it's approximately equal to the \Nstandard deviation of the
Dialogue: 0,0:12:22.57,0:12:26.76,Default,,0000,0000,0000,,original population divided by the\Nsquare root of the sample size n.
Dialogue: 0,0:12:26.76,0:12:34.63,Default,,0000,0000,0000,,So, this is a key thing that we know.\NIf we have at a population of any --
Dialogue: 0,0:12:34.63,0:12:37.63,Default,,0000,0000,0000,,I'll just write "uniform" in here, \Nof any type, it could bimodal,
Dialogue: 0,0:12:37.63,0:12:40.49,Default,,0000,0000,0000,,it could be uniform, it could be skewed,\Nwe know that if we were to take
Dialogue: 0,0:12:40.49,0:12:43.70,Default,,0000,0000,0000,,thousands and thousands of samples\Nor just one thousand -- or just a few,
Dialogue: 0,0:12:43.70,0:12:47.37,Default,,0000,0000,0000,,hundred samples, the sample means that\Nwe get from all those samples
Dialogue: 0,0:12:47.37,0:12:50.04,Default,,0000,0000,0000,,are going to approximate \Na normal distribution
Dialogue: 0,0:12:50.04,0:12:52.65,Default,,0000,0000,0000,,if our sample size is larger, \Nit's going to approximate
Dialogue: 0,0:12:52.65,0:12:57.09,Default,,0000,0000,0000,,a normal distribution even more.\NAnd we can already determine what the
Dialogue: 0,0:12:57.09,0:13:01.23,Default,,0000,0000,0000,,shape of that distribution is going to be\Nbecause we know that the population mean
Dialogue: 0,0:13:01.23,0:13:04.34,Default,,0000,0000,0000,,is approximately equal to the mean \Nof the sample means,
Dialogue: 0,0:13:04.34,0:13:09.13,Default,,0000,0000,0000,,and we know that the standard deviation,\Nthis is the standard error,
Dialogue: 0,0:13:09.13,0:13:15.10,Default,,0000,0000,0000,,we know that that, the standard error,\Nis the standard deviation of the
Dialogue: 0,0:13:15.10,0:13:17.33,Default,,0000,0000,0000,,sampling distribution.
Dialogue: 0,0:13:17.33,0:13:19.71,Default,,0000,0000,0000,,Okay, so we can work that out.
Dialogue: 0,0:13:19.71,0:13:22.83,Default,,0000,0000,0000,,But the thing is, what you're probably \Nalready thinking is,
Dialogue: 0,0:13:22.83,0:13:25.78,Default,,0000,0000,0000,,"why do you care?" And you may not care,\Nand that's fine.
Dialogue: 0,0:13:25.78,0:13:29.26,Default,,0000,0000,0000,,There's no reason to particularly.
Dialogue: 0,0:13:29.26,0:13:34.29,Default,,0000,0000,0000,,But, it can be very, very helpful.\NI'm just going to just float this idea
Dialogue: 0,0:13:34.29,0:13:38.12,Default,,0000,0000,0000,,and we'll return to it in future videos.
Dialogue: 0,0:13:38.12,0:13:42.59,Default,,0000,0000,0000,,Hopefully it's gone through your head \Nthat why is this strange person
Dialogue: 0,0:13:42.59,0:13:45.72,Default,,0000,0000,0000,,taking thousands of samples all the time?
Dialogue: 0,0:13:45.72,0:13:47.45,Default,,0000,0000,0000,,You know, you're not going to go to this\Npenguin ice sheet and just keep
Dialogue: 0,0:13:47.45,0:13:50.95,Default,,0000,0000,0000,,randomly picking 5 penguins at random\N1,000 times.
Dialogue: 0,0:13:50.95,0:13:53.63,Default,,0000,0000,0000,,Science and other types of time --\Nwhen we collect data,
Dialogue: 0,0:13:53.63,0:13:56.100,Default,,0000,0000,0000,,it doesn't work like that.\NWe pretty much usually only just collect
Dialogue: 0,0:13:56.100,0:13:59.35,Default,,0000,0000,0000,,one sample of data.
Dialogue: 0,0:13:59.35,0:14:03.33,Default,,0000,0000,0000,,And so, when we collect one sample of data\Nand this here -- I've got
Dialogue: 0,0:14:03.33,0:14:07.56,Default,,0000,0000,0000,,sampling distribution of n = 5 penguins.
Dialogue: 0,0:14:07.56,0:14:09.77,Default,,0000,0000,0000,,This is when we did do it 1,000 times.
Dialogue: 0,0:14:09.77,0:14:14.35,Default,,0000,0000,0000,,But let's just say that we did it one time\Nand we got a value around about here,
Dialogue: 0,0:14:14.35,0:14:17.50,Default,,0000,0000,0000,,around about 7 meters, \Nthat was our sample.
Dialogue: 0,0:14:17.50,0:14:20.12,Default,,0000,0000,0000,,We just got one sample.
Dialogue: 0,0:14:20.12,0:14:24.70,Default,,0000,0000,0000,,If we just got one sample, \Nwe don't know anything really about that
Dialogue: 0,0:14:24.70,0:14:30.40,Default,,0000,0000,0000,,in terms of how certain or how uncertain \Nare we that this truly is the sample mean.
Dialogue: 0,0:14:30.40,0:14:34.37,Default,,0000,0000,0000,,We knew if we did this many, many times\Nthe average of al the sample means
Dialogue: 0,0:14:34.37,0:14:36.62,Default,,0000,0000,0000,,would converge on the true \Npopulation mean.
Dialogue: 0,0:14:36.62,0:14:38.46,Default,,0000,0000,0000,,And that's our ultimate goal, \Nwe're trying to est--
Dialogue: 0,0:14:38.46,0:14:42.14,Default,,0000,0000,0000,,normally we don't know the population mean\Nwe're trying to estimate it.
Dialogue: 0,0:14:42.14,0:14:46.21,Default,,0000,0000,0000,,So in our one sample, we just got this \Nvalue of 7, say.
Dialogue: 0,0:14:46.21,0:14:50.85,Default,,0000,0000,0000,,How confident are we that that is \Nthe population mean?
Dialogue: 0,0:14:50.85,0:14:55.79,Default,,0000,0000,0000,,And so, what we're able to do by having\Nthis belief that we're able to know
Dialogue: 0,0:14:55.79,0:15:01.22,Default,,0000,0000,0000,,that this value of 7 does come from \Nin theory,
Dialogue: 0,0:15:01.22,0:15:03.71,Default,,0000,0000,0000,,a sampling distribution that exists.
Dialogue: 0,0:15:03.71,0:15:07.57,Default,,0000,0000,0000,,And in theory, this sampling distribution\Nexists with a standard deviation
Dialogue: 0,0:15:07.57,0:15:11.30,Default,,0000,0000,0000,,that we call the standard error.\NWe're able to understand how far
Dialogue: 0,0:15:11.30,0:15:16.44,Default,,0000,0000,0000,,this value of 7, or any value that \Nwe collected, it could be some other value
Dialogue: 0,0:15:16.44,0:15:19.85,Default,,0000,0000,0000,,but our one sample was 7 meters,\Nwe get a sense of how far away
Dialogue: 0,0:15:19.85,0:15:23.89,Default,,0000,0000,0000,,from the mean that is in the units\Nof standard deviations
Dialogue: 0,0:15:23.89,0:15:26.06,Default,,0000,0000,0000,,or technically, \Nwith a sampling distribution,
Dialogue: 0,0:15:26.06,0:15:27.41,Default,,0000,0000,0000,,standard errors.
Dialogue: 0,0:15:27.41,0:15:30.26,Default,,0000,0000,0000,,So we're going to come back to this topic,\Nbut really the value of the standard error
Dialogue: 0,0:15:30.26,0:15:33.100,Default,,0000,0000,0000,,is that enables us to determine \Nwhen we collect one sample,
Dialogue: 0,0:15:33.100,0:15:39.68,Default,,0000,0000,0000,,we're able to work out how far away\Nor how confident we are in our value,
Dialogue: 0,0:15:39.68,0:15:41.72,Default,,0000,0000,0000,,is how far away is it from the \Npopulation mean,
Dialogue: 0,0:15:41.72,0:15:45.65,Default,,0000,0000,0000,,how confident we are that this is a true\Nrepresentation of the population mean.
Dialogue: 0,0:15:45.65,0:15:48.46,Default,,0000,0000,0000,,We're going to come back to this\Nin future videos.