WEBVTT 00:00:01.476 --> 00:00:04.996 Hello, in this video, I want to talk about the standard error and this is really 00:00:04.996 --> 00:00:09.777 extending our understanding of sampling distributions and essential limit theorem. 00:00:10.485 --> 00:00:13.222 So, let's talk about what a standard error is. 00:00:14.998 --> 00:00:17.734 First of all, we'll go back to this penguin example and 00:00:17.734 --> 00:00:22.086 you've seen this distribution before as a uniform distribution of data. 00:00:22.958 --> 00:00:26.862 It has, like any distribution, it has-- there's descriptive statistics. 00:00:26.862 --> 00:00:28.524 So, it has a population mean. 00:00:28.524 --> 00:00:30.060 The average is 5.04. 00:00:30.060 --> 00:00:33.876 The average penguin is 5.04 meters from the edge of the ice sheet. 00:00:33.876 --> 00:00:36.747 You can calculate a standard deviation for this. 00:00:36.747 --> 00:00:39.513 So, the deviation is 2.88. 00:00:39.513 --> 00:00:42.311 So, that's the, you know, a measure of the spread. 00:00:42.311 --> 00:00:47.580 And there was 5,000 penguins floating on this ice sheet, that's the n, 00:00:47.580 --> 00:00:49.181 the population size. 00:00:50.105 --> 00:00:54.799 We then discussed about how if you were just to sample either just randomly select 00:00:54.799 --> 00:01:00.189 five penguins at a time or 50 penguins at a time, that each of those samples 00:01:00.189 --> 00:01:04.435 of, let's pick the n equals five for now, each of those five penguins, 00:01:04.435 --> 00:01:07.996 you could calculate how, what the average distance from the front of the edge sheet 00:01:07.996 --> 00:01:13.409 was for each of those individual penguins, sample of five penguins and if you were 00:01:13.409 --> 00:01:17.259 to do that over and over and over again and in this histogram, we did it 00:01:17.259 --> 00:01:22.998 1,000 times, we would be able to generate what's called the sampling distribution. 00:01:24.598 --> 00:01:30.153 And it's the sampling distribution of the sample means, that's what it is 00:01:30.153 --> 00:01:35.257 and I told you that we could calculate from that what the average of 00:01:35.257 --> 00:01:37.867 those sample means across the 1,000 samples was and 00:01:37.867 --> 00:01:43.947 that's this value and the notation that we use for that is this mu and then 00:01:43.947 --> 00:01:53.283 subscript x bar and that's the mean of the sample means and I've forgotten 00:01:53.283 --> 00:01:56.655 what it was, the exact value, but it's pretty much going to approximate 00:01:56.655 --> 00:01:58.208 very, very close. 00:01:58.208 --> 00:02:04.261 So, I just put approximately equal to 5.05, just go back, it's 5.04. 00:02:04.261 --> 00:02:11.111 So, it was-- it's going to approximate the population average and you can 00:02:11.111 --> 00:02:13.494 do that for any sample size. 00:02:13.494 --> 00:02:16.677 So, that was sample size five, let's look at the sample size 50. 00:02:16.677 --> 00:02:24.090 Again, we have the mean of the sampling distribution-- sorry, the mean of 00:02:24.090 --> 00:02:28.577 the sample means and that is also going to be very close to 5.04, it might be 00:02:28.577 --> 00:02:32.167 a little bit closer because our sample size is larger. 00:02:32.881 --> 00:02:35.646 Two other things to notice about these distributions, number one 00:02:35.646 --> 00:02:39.026 they're normally distributed or approx-- sorry, the approximate to normal 00:02:39.026 --> 00:02:43.151 distributions despite the fact for the original distribution of penguins. 00:02:43.151 --> 00:02:46.435 The population distribution was a uniform distribution. 00:02:46.435 --> 00:02:50.962 Second thing to notice, the sample size doesn't really effect where the value 00:02:50.962 --> 00:02:55.979 of the mean, of the sample means, it does effect the standard deviation of 00:02:55.979 --> 00:02:57.428 the sample means. 00:02:57.428 --> 00:03:00.299 So, if this is a normal distribution, or we believe it to approximate, 00:03:00.299 --> 00:03:08.467 and then also this approximates a normal distribution, then, it's clear 00:03:08.467 --> 00:03:14.622 that the distance here, let's just assume that's a standard deviation and I put it 00:03:14.622 --> 00:03:16.771 in the right place. 00:03:16.771 --> 00:03:21.487 This standard deviation, it's greater than whatever the corresponding value is 00:03:21.487 --> 00:03:25.124 over here, if that's also the standard deviation. 00:03:25.124 --> 00:03:29.318 So, as the sample size gets larger, the spead of the sample means 00:03:29.318 --> 00:03:33.152 gets smaller, so, we can say the standard deviation gets smaller. 00:03:33.152 --> 00:03:38.419 Now, does this standard deviation have any relationship at all to the original 00:03:38.419 --> 00:03:41.421 standard deviation of the original population. 00:03:41.421 --> 00:03:45.290 The original standard deviation was 2.88, so, I'll just say population of 00:03:45.290 --> 00:03:48.720 the original-- standard deviation was 2.88 00:03:48.955 --> 00:03:51.825 Is there any relationship at all between these two standard deviations? 00:03:52.195 --> 00:03:55.895 Because it's not like the mean of the sample means, which is pretty 00:03:55.895 --> 00:04:00.751 much the same, regardless of the sample size, I mean it does get better with 00:04:00.751 --> 00:04:04.130 larger samples but it approximates, it's close, especially if you have 00:04:04.130 --> 00:04:06.017 enough of these samples. 00:04:06.017 --> 00:04:08.977 What's the relationship of these standard deviations because it's clear that when 00:04:08.977 --> 00:04:14.598 you change n, this value is going to change, so is there a relationship? 00:04:14.598 --> 00:04:16.522 And it turns out that there is a relationship and we're going to 00:04:16.522 --> 00:04:18.166 look into that. 00:04:18.166 --> 00:04:23.755 This graph here just shows you that the normal distribution for becomes 00:04:23.755 --> 00:04:26.789 better and better the larger the sample size, so, it's a little 00:04:26.789 --> 00:04:29.854 tricky to see but let me, I just want to really point out one or two things here. 00:04:29.854 --> 00:04:33.863 I'm going to pick a color that represents that. 00:04:33.863 --> 00:04:38.746 So, this value here, actually in red, so, if I was just to pick one penguin at 00:04:38.746 --> 00:04:43.512 a time, a sample size of one, this is my estimate of the sample-- I'm going 00:04:43.512 --> 00:04:45.049 for the red line here. 00:04:45.049 --> 00:04:49.901 That's my estimate of the sample-- sorry, let's say that again. 00:04:49.901 --> 00:04:52.165 That's the distribution of the sample means. 00:04:52.165 --> 00:04:53.999 It looks like the original population. 00:04:53.999 --> 00:04:57.317 So, for a sample size of one, you don't get a normal distribution of the sample 00:04:57.317 --> 00:05:00.625 means, you get whatever the original population was. 00:05:00.625 --> 00:05:06.698 Let's look at two and I've got to find it on here, so, it's the orange one and 00:05:06.698 --> 00:05:11.632 I believe it's this one here. 00:05:11.632 --> 00:05:13.394 It is this one here. 00:05:13.394 --> 00:05:14.799 This is what it looks like. 00:05:14.799 --> 00:05:17.166 This is the n is two. 00:05:17.166 --> 00:05:19.492 So, again, not a really normal distribution. 00:05:19.492 --> 00:05:22.190 Now, let's skip to 50. 00:05:22.190 --> 00:05:27.129 This is 50 here and you can see it really, you don't need me to help 00:05:27.129 --> 00:05:28.368 you too much. 00:05:28.368 --> 00:05:30.781 This is the 50 value, it's very normal. 00:05:30.781 --> 00:05:35.651 And then, we got blue at ten-- sorry, 25 here. 00:05:35.651 --> 00:05:37.894 This is the 25 one and so on. 00:05:37.894 --> 00:05:39.806 This is the ten. 00:05:39.806 --> 00:05:41.677 This is the five. 00:05:41.677 --> 00:05:44.260 I wanted to just show you this graph because I wanted to show you that 00:05:44.260 --> 00:05:48.701 even with very, very, very small sample sizes of like five, we already 00:05:48.701 --> 00:05:51.258 get very close to a normal distribution. 00:05:51.258 --> 00:05:54.748 It's only with sample sizes of ridiculous sample sizes of like one or two that 00:05:54.748 --> 00:05:56.762 we don't do a very good job, 00:05:56.762 --> 00:05:59.971 So, even with small sample sizes, we get to the normal distribution 00:05:59.971 --> 00:06:02.768 of the normal distribution of the sample means. 00:06:02.768 --> 00:06:07.385 So, back to the problem I just posted a moment ago. 00:06:07.385 --> 00:06:14.501 This is our original standard deviation of a population, this is our population 00:06:14.501 --> 00:06:17.106 and whenever we get a sample, and again, this is just the sample 00:06:17.106 --> 00:06:17.950 size of five. 00:06:17.950 --> 00:06:19.411 This is the distribution of sample means. 00:06:19.411 --> 00:06:26.544 The mean is going to approximate the mean here but what is the relationship of 00:06:26.544 --> 00:06:31.469 the standard deviation to this original population. 00:06:31.469 --> 00:06:32.730 What is the relationship? 00:06:32.730 --> 00:06:38.633 It must be also related to the sample size because it changes with its sample size. 00:06:38.633 --> 00:06:43.024 And it's just a formula and we're not going to talk too much about-- 00:06:43.024 --> 00:06:47.407 we're not going to talk much really at all about how it's derived but this formula 00:06:47.407 --> 00:06:52.141 here, very neatly, just tells us about their relationship and 00:06:52.141 --> 00:06:55.513 so, what we have here is this is our standard deviation of 00:06:55.513 --> 00:06:59.987 the sampling distribution of the sample means. 00:06:59.987 --> 00:07:02.975 So, we call that sigma subscript x bar, 00:07:02.975 --> 00:07:04.401 sigma x bar. 00:07:04.401 --> 00:07:07.863 The standard deviation, so just to really reiterate what we're looking at, this is 00:07:07.863 --> 00:07:12.661 the distribution of sample means, this is-- we're looking for this value 00:07:12.661 --> 00:07:15.812 what's this standard deviation? 00:07:15.812 --> 00:07:22.158 And actually, technically, that's the notation, what is that standard 00:07:22.158 --> 00:07:23.149 deviation? 00:07:23.149 --> 00:07:25.191 So, what we do is, we just take the original population. 00:07:25.191 --> 00:07:30.300 This is the population standard deviation from the original population and we're 00:07:30.300 --> 00:07:35.479 going to divide it by the square root of n and that gives us that this value, 00:07:35.479 --> 00:07:36.752 this standard deviation. 00:07:36.752 --> 00:07:40.715 Its technical name is the standard deviation of the sampling distribution 00:07:40.715 --> 00:07:44.420 of the sample means, which is an awful mouthful but we just call 00:07:44.420 --> 00:07:45.420 it the standard error of 00:07:45.420 --> 00:07:49.447 the mean, which is what we call it the standard error of the mean. 00:07:49.447 --> 00:07:55.089 So, this graph illustrates how the standard error of the mean 00:07:55.089 --> 00:07:56.990 changes by sample size. 00:07:56.990 --> 00:08:06.727 So, if I just go back to-- maybe, I'll just go back to this slide here 00:08:06.727 --> 00:08:11.099 and we were asking the question of, you know, what's this value over 00:08:11.099 --> 00:08:14.841 sample size 50 compared to this value of a sample size of five? 00:08:14.841 --> 00:08:19.229 So, that was the question and I'm going to plot-- maybe here I'll plot it or write it 00:08:19.229 --> 00:08:20.265 sorry. 00:08:20.265 --> 00:08:25.102 So, this is the formula, the standard error of the mean or the standard 00:08:25.102 --> 00:08:27.675 deviation of the sampling distribution of the sample means is equal to 00:08:27.675 --> 00:08:31.601 the original population standard deviation divided by the square root of n. 00:08:31.601 --> 00:08:36.925 So, when we had that sample size of five, which is this one up here, what we're 00:08:36.925 --> 00:08:42.176 really looking at is this, the original standard deviation was 2.88 and 00:08:42.176 --> 00:08:45.774 we're going to divide by the square root of the sample size which is five, so that 00:08:45.774 --> 00:08:47.767 equals 1.3. 00:08:47.767 --> 00:08:53.340 So, the standard deviation here is 1.3 and that standard error we call that is 1.3. 00:08:53.340 --> 00:08:59.429 So, what this is saying is this value here is 1.3 higher that was it, I forget. 00:08:59.429 --> 00:09:04.000 I think it was 5.04 was the mean of the sample means and so this value here 00:09:04.000 --> 00:09:10.068 is going to be a 6.5-- nope, nope, not five. 00:09:10.068 --> 00:09:15.243 It's going to be at 6.34. 00:09:15.243 --> 00:09:22.008 This is one standard deviation above the sample mean but if we have 00:09:22.008 --> 00:09:26.743 a sample size of fifty, then the calculation becomes this. 00:09:26.743 --> 00:09:30.303 Becomes the original standard deviation of the population divided by the square 00:09:30.303 --> 00:09:32.523 root of 50, which is equal to and I've 00:09:32.523 --> 00:09:35.914 written this down so I can check, 0.4. 00:09:35.914 --> 00:09:40.234 So, back to this graph, this value is 0.4, 00:09:40.234 --> 00:09:43.002 and this value is 1.3. 00:09:43.002 --> 00:09:46.554 And so, it gets smaller the bigger the sample size. 00:09:46.554 --> 00:09:52.456 This graph here that I got to previously is actually showing us 00:09:52.456 --> 00:09:56.063 how the standard error changes by the sample size. 00:09:56.063 --> 00:09:59.661 So we just had a sample size of 50, which is approximately here. 00:09:59.661 --> 00:10:06.839 If we go across to this value on this axis, it tells us that's about 0.4, 00:10:06.839 --> 00:10:11.837 sample size of 50, and if we had a sample size of 5, 00:10:11.837 --> 00:10:15.991 which is approximately here -- I'm doing a line, not very well, 00:10:15.991 --> 00:10:20.372 but it goes to about there. This was about 1.3. 00:10:20.372 --> 00:10:23.829 And I just want you to -- there's nothing really too much for you to take home 00:10:23.829 --> 00:10:27.337 from this graph other than showing you that as the sample size increases, 00:10:27.337 --> 00:10:32.416 that the -- any population standard deviation that we have, 00:10:32.416 --> 00:10:36.584 the standard error is going to get much smaller very rapidly. 00:10:36.584 --> 00:10:41.044 A sample size of 5 is still quite high up on this curve, 00:10:41.044 --> 00:10:44.432 but once you come down to sample sizes of 20 or 30 or more, 00:10:44.432 --> 00:10:49.222 then we get a very, very small standard error. 00:10:50.578 --> 00:10:56.786 This is just to reiterate that point so you can see what these are on this graph. 00:10:56.786 --> 00:11:00.394 So let's put together what we've just learned about the standard error 00:11:00.394 --> 00:11:04.525 with what we have learned previously about the Central Limit Theorem. 00:11:04.525 --> 00:11:08.797 So what we have just been discussing is that we just know that we have 00:11:08.797 --> 00:11:10.929 an original population, it could be any distribution, 00:11:10.929 --> 00:11:13.369 here's our uniform distribution. 00:11:13.369 --> 00:11:16.762 If we take many samples from it, we get our sampling distribution. 00:11:16.762 --> 00:11:24.601 In this case, of the sample means, is normally distributed 00:11:24.601 --> 00:11:27.102 or approximately normally distributed. 00:11:27.102 --> 00:11:34.696 And we know that the sampling distribution has a mean that is approximately equal to 00:11:34.696 --> 00:11:40.404 the population mean and we've just learned that we just know now that 00:11:40.404 --> 00:11:44.118 the standard deviation of this approximately normal distribution, 00:11:44.118 --> 00:11:46.826 this is the standard error. 00:11:46.826 --> 00:11:49.653 I'll write here, "standard error." 00:11:50.319 --> 00:11:53.645 So we can actually write this in notation form, 00:11:53.645 --> 00:11:56.844 and we say that this sampling distribution is approximately normal, 00:11:56.844 --> 00:11:59.943 this is what this tilde squiggle means, is approximately normal, 00:11:59.943 --> 00:12:07.615 approximately normal and it has a mean of the population mean, 00:12:07.615 --> 00:12:11.153 so I'll just write here, the mean is the population mean. 00:12:11.153 --> 00:12:13.251 And the standard deviation of that distribution, 00:12:13.251 --> 00:12:15.767 and we're talking about this distribution down here, 00:12:15.767 --> 00:12:18.941 the standard deviation of that distribution is the standard error, 00:12:18.941 --> 00:12:20.604 that's what we call it. 00:12:20.604 --> 00:12:22.572 And it's approximately equal to the standard deviation of the 00:12:22.572 --> 00:12:26.758 original population divided by the square root of the sample size n. 00:12:26.758 --> 00:12:34.628 So, this is a key thing that we know. If we have at a population of any -- 00:12:34.628 --> 00:12:37.634 I'll just write "uniform" in here, of any type, it could bimodal, 00:12:37.634 --> 00:12:40.490 it could be uniform, it could be skewed, we know that if we were to take 00:12:40.490 --> 00:12:43.704 thousands and thousands of samples or just one thousand -- or just a few, 00:12:43.704 --> 00:12:47.370 hundred samples, the sample means that we get from all those samples 00:12:47.370 --> 00:12:50.035 are going to approximate a normal distribution 00:12:50.035 --> 00:12:52.651 if our sample size is larger, it's going to approximate 00:12:52.651 --> 00:12:57.088 a normal distribution even more. And we can already determine what the 00:12:57.088 --> 00:13:01.234 shape of that distribution is going to be because we know that the population mean 00:13:01.234 --> 00:13:04.340 is approximately equal to the mean of the sample means, 00:13:04.340 --> 00:13:09.130 and we know that the standard deviation, this is the standard error, 00:13:09.130 --> 00:13:15.096 we know that that, the standard error, is the standard deviation of the 00:13:15.096 --> 00:13:17.330 sampling distribution. 00:13:17.330 --> 00:13:19.712 Okay, so we can work that out. 00:13:19.712 --> 00:13:22.833 But the thing is, what you're probably already thinking is, 00:13:22.833 --> 00:13:25.780 "why do you care?" And you may not care, and that's fine. 00:13:25.780 --> 00:13:29.255 There's no reason to particularly. 00:13:29.255 --> 00:13:34.286 But, it can be very, very helpful. I'm just going to just float this idea 00:13:34.286 --> 00:13:38.121 and we'll return to it in future videos. 00:13:38.121 --> 00:13:42.588 Hopefully it's gone through your head that why is this strange person 00:13:42.588 --> 00:13:45.715 taking thousands of samples all the time? 00:13:45.715 --> 00:13:47.446 You know, you're not going to go to this penguin ice sheet and just keep 00:13:47.446 --> 00:13:50.951 randomly picking 5 penguins at random 1,000 times. 00:13:50.951 --> 00:13:53.627 Science and other types of time -- when we collect data, 00:13:53.627 --> 00:13:56.998 it doesn't work like that. We pretty much usually only just collect 00:13:56.998 --> 00:13:59.349 one sample of data. 00:13:59.349 --> 00:14:03.331 And so, when we collect one sample of data and this here -- I've got 00:14:03.331 --> 00:14:07.558 sampling distribution of n = 5 penguins. 00:14:07.558 --> 00:14:09.768 This is when we did do it 1,000 times. 00:14:09.768 --> 00:14:14.352 But let's just say that we did it one time and we got a value around about here, 00:14:14.352 --> 00:14:17.504 around about 7 meters, that was our sample. 00:14:17.504 --> 00:14:20.123 We just got one sample. 00:14:20.123 --> 00:14:24.697 If we just got one sample, we don't know anything really about that 00:14:24.697 --> 00:14:30.402 in terms of how certain or how uncertain are we that this truly is the sample mean. 00:14:30.402 --> 00:14:34.368 We knew if we did this many, many times the average of al the sample means 00:14:34.368 --> 00:14:36.618 would converge on the true population mean. 00:14:36.618 --> 00:14:38.455 And that's our ultimate goal, we're trying to est-- 00:14:38.455 --> 00:14:42.139 normally we don't know the population mean we're trying to estimate it. 00:14:42.139 --> 00:14:46.211 So in our one sample, we just got this value of 7, say. 00:14:46.211 --> 00:14:50.854 How confident are we that that is the population mean? 00:14:50.854 --> 00:14:55.793 And so, what we're able to do by having this belief that we're able to know 00:14:55.793 --> 00:15:01.224 that this value of 7 does come from in theory, 00:15:01.224 --> 00:15:03.710 a sampling distribution that exists. 00:15:03.710 --> 00:15:07.569 And in theory, this sampling distribution exists with a standard deviation 00:15:07.569 --> 00:15:11.300 that we call the standard error. We're able to understand how far 00:15:11.300 --> 00:15:16.439 this value of 7, or any value that we collected, it could be some other value 00:15:16.439 --> 00:15:19.850 but our one sample was 7 meters, we get a sense of how far away 00:15:19.850 --> 00:15:23.890 from the mean that is in the units of standard deviations 00:15:23.890 --> 00:15:26.063 or technically, with a sampling distribution, 00:15:26.063 --> 00:15:27.410 standard errors. 00:15:27.410 --> 00:15:30.264 So we're going to come back to this topic, but really the value of the standard error 00:15:30.264 --> 00:15:33.996 is that enables us to determine when we collect one sample, 00:15:33.996 --> 00:15:39.677 we're able to work out how far away or how confident we are in our value, 00:15:39.677 --> 00:15:41.717 is how far away is it from the population mean, 00:15:41.717 --> 00:15:45.652 how confident we are that this is a true representation of the population mean. 00:15:45.652 --> 00:15:48.456 We're going to come back to this in future videos.