0:00:01.476,0:00:04.996 Hello, in this video, I want to talk about[br]the standard error and this is really 0:00:04.996,0:00:09.777 extending our understanding of sampling [br]distributions and essential limit theorem. 0:00:10.485,0:00:13.222 So, let's talk about what [br]a standard error is. 0:00:14.998,0:00:17.734 First of all, we'll go back to [br]this penguin example and 0:00:17.734,0:00:22.086 you've seen this distribution before[br]as a uniform distribution of data. 0:00:22.958,0:00:26.862 It has, like any distribution, it has--[br]there's descriptive statistics. 0:00:26.862,0:00:28.524 So, it has a population mean. 0:00:28.524,0:00:30.060 The average is 5.04. 0:00:30.060,0:00:33.876 The average penguin is 5.04 meters [br]from the edge of the ice sheet. 0:00:33.876,0:00:36.747 You can calculate a standard [br]deviation for this. 0:00:36.747,0:00:39.513 So, the deviation is 2.88. 0:00:39.513,0:00:42.311 So, that's the, you know, [br]a measure of the spread. 0:00:42.311,0:00:47.580 And there was 5,000 penguins floating[br]on this ice sheet, that's the n, 0:00:47.580,0:00:49.181 the population size. 0:00:50.105,0:00:54.799 We then discussed about how if you were[br]just to sample either just randomly select 0:00:54.799,0:01:00.189 five penguins at a time or 50 penguins[br]at a time, that each of those samples 0:01:00.189,0:01:04.435 of, let's pick the n equals five for now,[br]each of those five penguins, 0:01:04.435,0:01:07.996 you could calculate how, what the average[br]distance from the front of the edge sheet 0:01:07.996,0:01:13.409 was for each of those individual penguins,[br]sample of five penguins and if you were 0:01:13.409,0:01:17.259 to do that over and over and over again[br]and in this histogram, we did it 0:01:17.259,0:01:22.998 1,000 times, we would be able to generate[br]what's called the sampling distribution. 0:01:24.598,0:01:30.153 And it's the sampling distribution of[br]the sample means, that's what it is 0:01:30.153,0:01:35.257 and I told you that we could calculate[br]from that what the average of 0:01:35.257,0:01:37.867 those sample means across [br]the 1,000 samples was and 0:01:37.867,0:01:43.947 that's this value and the notation that[br]we use for that is this mu and then 0:01:43.947,0:01:53.283 subscript x bar and that's the mean[br]of the sample means and I've forgotten 0:01:53.283,0:01:56.655 what it was, the exact value, but it's[br]pretty much going to approximate 0:01:56.655,0:01:58.208 very, very close. 0:01:58.208,0:02:04.261 So, I just put approximately equal to [br]5.05, just go back, it's 5.04. 0:02:04.261,0:02:11.111 So, it was-- it's going to approximate[br]the population average and you can 0:02:11.111,0:02:13.494 do that for any sample size. 0:02:13.494,0:02:16.677 So, that was sample size five,[br]let's look at the sample size 50. 0:02:16.677,0:02:24.090 Again, we have the mean of the sampling[br]distribution-- sorry, the mean of 0:02:24.090,0:02:28.577 the sample means and that is also going[br]to be very close to 5.04, it might be 0:02:28.577,0:02:32.167 a little bit closer because [br]our sample size is larger. 0:02:32.881,0:02:35.646 Two other things to notice about [br]these distributions, number one 0:02:35.646,0:02:39.026 they're normally distributed or approx--[br]sorry, the approximate to normal 0:02:39.026,0:02:43.151 distributions despite the fact for [br]the original distribution of penguins. 0:02:43.151,0:02:46.435 The population distribution was [br]a uniform distribution. 0:02:46.435,0:02:50.962 Second thing to notice, the sample size[br]doesn't really effect where the value 0:02:50.962,0:02:55.979 of the mean, of the sample means, it does[br]effect the standard deviation of 0:02:55.979,0:02:57.428 the sample means. 0:02:57.428,0:03:00.299 So, if this is a normal distribution,[br]or we believe it to approximate, 0:03:00.299,0:03:08.467 and then also this approximates [br]a normal distribution, then, it's clear 0:03:08.467,0:03:14.622 that the distance here, let's just assume[br]that's a standard deviation and I put it 0:03:14.622,0:03:16.771 in the right place. 0:03:16.771,0:03:21.487 This standard deviation, it's greater than[br]whatever the corresponding value is 0:03:21.487,0:03:25.124 over here, if that's also [br]the standard deviation. 0:03:25.124,0:03:29.318 So, as the sample size gets larger,[br]the spead of the sample means 0:03:29.318,0:03:33.152 gets smaller, so, we can say [br]the standard deviation gets smaller. 0:03:33.152,0:03:38.419 Now, does this standard deviation have any[br]relationship at all to the original 0:03:38.419,0:03:41.421 standard deviation of [br]the original population. 0:03:41.421,0:03:45.290 The original standard deviation was 2.88,[br]so, I'll just say population of 0:03:45.290,0:03:48.720 the original-- standard deviation was 2.88 0:03:48.955,0:03:51.825 Is there any relationship at all between [br]these two standard deviations? 0:03:52.195,0:03:55.895 Because it's not like the mean of[br]the sample means, which is pretty 0:03:55.895,0:04:00.751 much the same, regardless of the sample [br]size, I mean it does get better with 0:04:00.751,0:04:04.130 larger samples but it approximates,[br]it's close, especially if you have 0:04:04.130,0:04:06.017 enough of these samples. 0:04:06.017,0:04:08.977 What's the relationship of these standard[br]deviations because it's clear that when 0:04:08.977,0:04:14.598 you change n, this value is going to [br]change, so is there a relationship? 0:04:14.598,0:04:16.522 And it turns out that there is [br]a relationship and we're going to 0:04:16.522,0:04:18.166 look into that. 0:04:18.166,0:04:23.755 This graph here just shows you that [br]the normal distribution for becomes 0:04:23.755,0:04:26.789 better and better the larger [br]the sample size, so, it's a little 0:04:26.789,0:04:29.854 tricky to see but let me, I just want to[br]really point out one or two things here. 0:04:29.854,0:04:33.863 I'm going to pick a color [br]that represents that. 0:04:33.863,0:04:38.746 So, this value here, actually in red, so,[br]if I was just to pick one penguin at 0:04:38.746,0:04:43.512 a time, a sample size of one, this is[br]my estimate of the sample-- I'm going 0:04:43.512,0:04:45.049 for the red line here. 0:04:45.049,0:04:49.901 That's my estimate of the sample--[br]sorry, let's say that again. 0:04:49.901,0:04:52.165 That's the distribution of [br]the sample means. 0:04:52.165,0:04:53.999 It looks like the original population. 0:04:53.999,0:04:57.317 So, for a sample size of one, you don't[br]get a normal distribution of the sample 0:04:57.317,0:05:00.625 means, you get whatever [br]the original population was. 0:05:00.625,0:05:06.698 Let's look at two and I've got to find it[br]on here, so, it's the orange one and 0:05:06.698,0:05:11.632 I believe it's this one here. 0:05:11.632,0:05:13.394 It is this one here. 0:05:13.394,0:05:14.799 This is what it looks like. 0:05:14.799,0:05:17.166 This is the n is two. 0:05:17.166,0:05:19.492 So, again, not a really [br]normal distribution. 0:05:19.492,0:05:22.190 Now, let's skip to 50. 0:05:22.190,0:05:27.129 This is 50 here and you can see it[br]really, you don't need me to help 0:05:27.129,0:05:28.368 you too much. 0:05:28.368,0:05:30.781 This is the 50 value, it's very normal. 0:05:30.781,0:05:35.651 And then, we got blue at ten--[br]sorry, 25 here. 0:05:35.651,0:05:37.894 This is the 25 one and so on. 0:05:37.894,0:05:39.806 This is the ten. 0:05:39.806,0:05:41.677 This is the five. 0:05:41.677,0:05:44.260 I wanted to just show you this graph[br]because I wanted to show you that 0:05:44.260,0:05:48.701 even with very, very, very small [br]sample sizes of like five, we already 0:05:48.701,0:05:51.258 get very close to a normal distribution. 0:05:51.258,0:05:54.748 It's only with sample sizes of ridiculous[br]sample sizes of like one or two that 0:05:54.748,0:05:56.762 we don't do a very good job, 0:05:56.762,0:05:59.971 So, even with small sample sizes, [br]we get to the normal distribution 0:05:59.971,0:06:02.768 of the normal distribution of [br]the sample means. 0:06:02.768,0:06:07.385 So, back to the problem [br]I just posted a moment ago. 0:06:07.385,0:06:14.501 This is our original standard deviation[br]of a population, this is our population 0:06:14.501,0:06:17.106 and whenever we get a sample,[br]and again, this is just the sample 0:06:17.106,0:06:17.950 size of five. 0:06:17.950,0:06:19.411 This is the distribution of sample means. 0:06:19.411,0:06:26.544 The mean is going to approximate the mean[br]here but what is the relationship of 0:06:26.544,0:06:31.469 the standard deviation to [br]this original population. 0:06:31.469,0:06:32.730 What is the relationship? 0:06:32.730,0:06:38.633 It must be also related to the sample size[br]because it changes with its sample size. 0:06:38.633,0:06:43.024 And it's just a formula and we're not[br]going to talk too much about-- 0:06:43.024,0:06:47.407 we're not going to talk much really at all[br]about how it's derived but this formula 0:06:47.407,0:06:52.141 here, very neatly, just tells us [br]about their relationship and 0:06:52.141,0:06:55.513 so, what we have here is this is [br]our standard deviation of 0:06:55.513,0:06:59.987 the sampling distribution of[br]the sample means. 0:06:59.987,0:07:02.975 So, we call that sigma subscript x bar, 0:07:02.975,0:07:04.401 sigma x bar. 0:07:04.401,0:07:07.863 The standard deviation, so just to really[br]reiterate what we're looking at, this is 0:07:07.863,0:07:12.661 the distribution of sample means,[br]this is-- we're looking for this value 0:07:12.661,0:07:15.812 what's this standard deviation? 0:07:15.812,0:07:22.158 And actually, technically, that's [br]the notation, what is that standard 0:07:22.158,0:07:23.149 deviation? 0:07:23.149,0:07:25.191 So, what we do is, we just take [br]the original population. 0:07:25.191,0:07:30.300 This is the population standard deviation[br]from the original population and we're 0:07:30.300,0:07:35.479 going to divide it by the square root of n[br]and that gives us that this value, 0:07:35.479,0:07:36.752 this standard deviation. 0:07:36.752,0:07:40.715 Its technical name is the standard[br]deviation of the sampling distribution 0:07:40.715,0:07:44.420 of the sample means, which is an awful[br]mouthful but we just call 0:07:44.420,0:07:45.420 it the standard error of 0:07:45.420,0:07:49.447 the mean, which is what we call it[br]the standard error of the mean. 0:07:49.447,0:07:55.089 So, this graph illustrates how [br]the standard error of the mean 0:07:55.089,0:07:56.990 changes by sample size. 0:07:56.990,0:08:06.727 So, if I just go back to-- maybe, [br]I'll just go back to this slide here 0:08:06.727,0:08:11.099 and we were asking the question of, [br]you know, what's this value over 0:08:11.099,0:08:14.841 sample size 50 compared to this [br]value of a sample size of five? 0:08:14.841,0:08:19.229 So, that was the question and I'm going to[br]plot-- maybe here I'll plot it or write it 0:08:19.229,0:08:20.265 sorry. 0:08:20.265,0:08:25.102 So, this is the formula, the standard[br]error of the mean or the standard 0:08:25.102,0:08:27.675 deviation of the sampling distribution[br]of the sample means is equal to 0:08:27.675,0:08:31.601 the original population standard deviation[br]divided by the square root of n. 0:08:31.601,0:08:36.925 So, when we had that sample size of five,[br]which is this one up here, what we're 0:08:36.925,0:08:42.176 really looking at is this, the original[br]standard deviation was 2.88 and 0:08:42.176,0:08:45.774 we're going to divide by the square root[br]of the sample size which is five, so that 0:08:45.774,0:08:47.767 equals 1.3. 0:08:47.767,0:08:53.340 So, the standard deviation here is 1.3 and[br]that standard error we call that is 1.3. 0:08:53.340,0:08:59.429 So, what this is saying is this value here[br]is 1.3 higher that was it, I forget. 0:08:59.429,0:09:04.000 I think it was 5.04 was the mean of[br]the sample means and so this value here 0:09:04.000,0:09:10.068 is going to be a 6.5-- nope, nope, not five. 0:09:10.068,0:09:15.243 It's going to be at 6.34. 0:09:15.243,0:09:22.008 This is one standard deviation above [br]the sample mean but if we have 0:09:22.008,0:09:26.743 a sample size of fifty, then [br]the calculation becomes this. 0:09:26.743,0:09:30.303 Becomes the original standard deviation[br]of the population divided by the square 0:09:30.303,0:09:32.523 root of 50, which is equal to and I've 0:09:32.523,0:09:35.914 written this down so I can check, 0.4. 0:09:35.914,0:09:40.234 So, back to this graph, [br]this value is 0.4, 0:09:40.234,0:09:43.002 and this value is 1.3. 0:09:43.002,0:09:46.554 And so, it gets smaller the bigger the[br]sample size. 0:09:46.554,0:09:52.456 This graph here that I got to previously[br]is actually showing us 0:09:52.456,0:09:56.063 how the standard error changes by[br]the sample size. 0:09:56.063,0:09:59.661 So we just had a sample size of 50,[br]which is approximately here. 0:09:59.661,0:10:06.839 If we go across to this value on this[br]axis, it tells us that's about 0.4, 0:10:06.839,0:10:11.837 sample size of 50, and if we had [br]a sample size of 5, 0:10:11.837,0:10:15.991 which is approximately here --[br]I'm doing a line, not very well, 0:10:15.991,0:10:20.372 but it goes to about there.[br]This was about 1.3. 0:10:20.372,0:10:23.829 And I just want you to -- there's nothing[br]really too much for you to take home 0:10:23.829,0:10:27.337 from this graph other than showing you[br]that as the sample size increases, 0:10:27.337,0:10:32.416 that the -- any population [br]standard deviation that we have, 0:10:32.416,0:10:36.584 the standard error is going to get[br]much smaller very rapidly. 0:10:36.584,0:10:41.044 A sample size of 5 is still quite high up[br]on this curve, 0:10:41.044,0:10:44.432 but once you come down to sample sizes[br]of 20 or 30 or more, 0:10:44.432,0:10:49.222 then we get a very, very small [br]standard error. 0:10:50.578,0:10:56.786 This is just to reiterate that point so[br]you can see what these are on this graph. 0:10:56.786,0:11:00.394 So let's put together what we've [br]just learned about the standard error 0:11:00.394,0:11:04.525 with what we have learned previously about[br]the Central Limit Theorem. 0:11:04.525,0:11:08.797 So what we have just been discussing is [br]that we just know that we have 0:11:08.797,0:11:10.929 an original population, [br]it could be any distribution, 0:11:10.929,0:11:13.369 here's our uniform distribution. 0:11:13.369,0:11:16.762 If we take many samples from it,[br]we get our sampling distribution. 0:11:16.762,0:11:24.601 In this case, of the sample means,[br]is normally distributed 0:11:24.601,0:11:27.102 or approximately normally distributed. 0:11:27.102,0:11:34.696 And we know that the sampling distribution[br]has a mean that is approximately equal to 0:11:34.696,0:11:40.404 the population mean and we've just learned[br]that we just know now that 0:11:40.404,0:11:44.118 the standard deviation of this [br]approximately normal distribution, 0:11:44.118,0:11:46.826 this is the standard error. 0:11:46.826,0:11:49.653 I'll write here, "standard error." 0:11:50.319,0:11:53.645 So we can actually write this in[br]notation form, 0:11:53.645,0:11:56.844 and we say that this sampling distribution[br]is approximately normal, 0:11:56.844,0:11:59.943 this is what this tilde squiggle means, [br]is approximately normal, 0:11:59.943,0:12:07.615 approximately normal and it has a mean[br]of the population mean, 0:12:07.615,0:12:11.153 so I'll just write here, [br]the mean is the population mean. 0:12:11.153,0:12:13.251 And the standard deviation of that [br]distribution, 0:12:13.251,0:12:15.767 and we're talking about this distribution[br]down here, 0:12:15.767,0:12:18.941 the standard deviation of that [br]distribution is the standard error, 0:12:18.941,0:12:20.604 that's what we call it. 0:12:20.604,0:12:22.572 And it's approximately equal to the [br]standard deviation of the 0:12:22.572,0:12:26.758 original population divided by the[br]square root of the sample size n. 0:12:26.758,0:12:34.628 So, this is a key thing that we know.[br]If we have at a population of any -- 0:12:34.628,0:12:37.634 I'll just write "uniform" in here, [br]of any type, it could bimodal, 0:12:37.634,0:12:40.490 it could be uniform, it could be skewed,[br]we know that if we were to take 0:12:40.490,0:12:43.704 thousands and thousands of samples[br]or just one thousand -- or just a few, 0:12:43.704,0:12:47.370 hundred samples, the sample means that[br]we get from all those samples 0:12:47.370,0:12:50.035 are going to approximate [br]a normal distribution 0:12:50.035,0:12:52.651 if our sample size is larger, [br]it's going to approximate 0:12:52.651,0:12:57.088 a normal distribution even more.[br]And we can already determine what the 0:12:57.088,0:13:01.234 shape of that distribution is going to be[br]because we know that the population mean 0:13:01.234,0:13:04.340 is approximately equal to the mean [br]of the sample means, 0:13:04.340,0:13:09.130 and we know that the standard deviation,[br]this is the standard error, 0:13:09.130,0:13:15.096 we know that that, the standard error,[br]is the standard deviation of the 0:13:15.096,0:13:17.330 sampling distribution. 0:13:17.330,0:13:19.712 Okay, so we can work that out. 0:13:19.712,0:13:22.833 But the thing is, what you're probably [br]already thinking is, 0:13:22.833,0:13:25.780 "why do you care?" And you may not care,[br]and that's fine. 0:13:25.780,0:13:29.255 There's no reason to particularly. 0:13:29.255,0:13:34.286 But, it can be very, very helpful.[br]I'm just going to just float this idea 0:13:34.286,0:13:38.121 and we'll return to it in future videos. 0:13:38.121,0:13:42.588 Hopefully it's gone through your head [br]that why is this strange person 0:13:42.588,0:13:45.715 taking thousands of samples all the time? 0:13:45.715,0:13:47.446 You know, you're not going to go to this[br]penguin ice sheet and just keep 0:13:47.446,0:13:50.951 randomly picking 5 penguins at random[br]1,000 times. 0:13:50.951,0:13:53.627 Science and other types of time --[br]when we collect data, 0:13:53.627,0:13:56.998 it doesn't work like that.[br]We pretty much usually only just collect 0:13:56.998,0:13:59.349 one sample of data. 0:13:59.349,0:14:03.331 And so, when we collect one sample of data[br]and this here -- I've got 0:14:03.331,0:14:07.558 sampling distribution of n = 5 penguins. 0:14:07.558,0:14:09.768 This is when we did do it 1,000 times. 0:14:09.768,0:14:14.352 But let's just say that we did it one time[br]and we got a value around about here, 0:14:14.352,0:14:17.504 around about 7 meters, [br]that was our sample. 0:14:17.504,0:14:20.123 We just got one sample. 0:14:20.123,0:14:24.697 If we just got one sample, [br]we don't know anything really about that 0:14:24.697,0:14:30.402 in terms of how certain or how uncertain [br]are we that this truly is the sample mean. 0:14:30.402,0:14:34.368 We knew if we did this many, many times[br]the average of al the sample means 0:14:34.368,0:14:36.618 would converge on the true [br]population mean. 0:14:36.618,0:14:38.455 And that's our ultimate goal, [br]we're trying to est-- 0:14:38.455,0:14:42.139 normally we don't know the population mean[br]we're trying to estimate it. 0:14:42.139,0:14:46.211 So in our one sample, we just got this [br]value of 7, say. 0:14:46.211,0:14:50.854 How confident are we that that is [br]the population mean? 0:14:50.854,0:14:55.793 And so, what we're able to do by having[br]this belief that we're able to know 0:14:55.793,0:15:01.224 that this value of 7 does come from [br]in theory, 0:15:01.224,0:15:03.710 a sampling distribution that exists. 0:15:03.710,0:15:07.569 And in theory, this sampling distribution[br]exists with a standard deviation 0:15:07.569,0:15:11.300 that we call the standard error.[br]We're able to understand how far 0:15:11.300,0:15:16.439 this value of 7, or any value that [br]we collected, it could be some other value 0:15:16.439,0:15:19.850 but our one sample was 7 meters,[br]we get a sense of how far away 0:15:19.850,0:15:23.890 from the mean that is in the units[br]of standard deviations 0:15:23.890,0:15:26.063 or technically, [br]with a sampling distribution, 0:15:26.063,0:15:27.410 standard errors. 0:15:27.410,0:15:30.264 So we're going to come back to this topic,[br]but really the value of the standard error 0:15:30.264,0:15:33.996 is that enables us to determine [br]when we collect one sample, 0:15:33.996,0:15:39.677 we're able to work out how far away[br]or how confident we are in our value, 0:15:39.677,0:15:41.717 is how far away is it from the [br]population mean, 0:15:41.717,0:15:45.652 how confident we are that this is a true[br]representation of the population mean. 0:15:45.652,0:15:48.456 We're going to come back to this[br]in future videos.