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Standard error

From Wikipedia, the free encyclopedia

For a value that is sampled with an unbiased normally distributed error, the above depicts the proportion of samples that would fall between 0, 1, 2, and 3 standard deviations above and below the actual value.

The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[1] or an estimate of that standard deviation. If the parameter or the statistic is the mean, it is called the standard error of the mean (SEM).

The sampling distribution of a population mean is generated by repeated sampling and recording of the means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

Therefore, the relationship between the standard error and the standard deviation is such that, for a given sample size, the standard error equals the standard deviation divided by the square root of the sample size. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.

In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, e.g., confidence intervals).

YouTube Encyclopedic

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• ✪ Standard error of the mean | Inferential statistics | Probability and Statistics | Khan Academy
• ✪ StatQuickie: Standard Deviation vs Standard Error
• ✪ 📚 How to calculate standard error of the mean and why it's important (Central Limit Theorem)
• ✪ lecture-16 || Standard error || Standard error of mean || Standard error of median
• ✪ Golfer Standard Error and Confidence Interval

Transcription

We've seen in the last several videos, you start off with any crazy distribution. It doesn't have to be crazy. It could be a nice, normal distribution. But to really make the point that you don't have to have a normal distribution, I like to use crazy ones. So let's say you have some kind of crazy distribution that looks something like that. It could look like anything. So we've seen multiple times, you take samples from this crazy distribution. So let's say you were to take samples of n is equal to 10. So we take 10 instances of this random variable, average them out, and then plot our average. We get one instance there. We keep doing that. We do that again. We take 10 samples from this random variable, average them, plot them again. Eventually, you do this a gazillion times-- in theory, infinite number of times-- and you're going to approach the sampling distribution of the sample mean. And n equals 10, it's not going to be a perfect normal distribution, but it's going to be close. It would be perfect only if n was infinity. But let's say we eventually-- all of our samples, we get a lot of averages that are there. That stacks up there. That stacks up there. And eventually, we'll approach something that looks something like that. And we've seen from the last video that, one, if-- let's say we were to do it again. And this time, let's say that n is equal to 20. One, the distribution that we get is going to be more normal. And maybe in future videos, we'll delve even deeper into things like kurtosis and skew. But it's going to be more normal. But even more important here, or I guess even more obviously to us than we saw, then, in the experiment, it's going to have a lower standard deviation. So they're all going to have the same mean. Let's say the mean here is 5. Then the mean here is also going to be 5. The mean of our sampling distribution of the sample mean is going to be 5. It doesn't matter what our n is. If our n is 20, it's still going to be 5. But our standard deviation is going to be less in either of these scenarios. And we saw that just by experimenting. It might look like this. It's going to be more normal, but it's going to have a tighter standard deviation. So maybe it'll look like that. And if we did it with an even larger sample size-- let me do that in a different color. If we do that with an even larger sample size, n is equal to 100, what we're going to get is something that fits the normal distribution even better. We take 100 instances of this random variable, average them, plot it. 100 instances of this random variable, average them, plot it. We just keep doing that. If we keep doing that, what we're going to have is something that's even more normal than either of these. So it's going to be a much closer fit to a true normal distribution, but even more obvious to the human eye, it's going to be even tighter. So it's going to be a very low standard deviation. It's going to look something like that. I'll show you that on the simulation app probably later in this video. So two things happen. As you increase your sample size for every time you do the average, two things are happening. You're becoming more normal, and your standard deviation is getting smaller. So the question might arise, well, is there a formula? So if I know the standard deviation-- so this is my standard deviation of just my original probability density function. This is the mean of my original probability density function. So if I know the standard deviation, and I know n is going to change depending on how many samples I'm taking every time I do a sample mean. If I know my standard deviation, or maybe if I know my variance. The variance is just the standard deviation squared. If you don't remember that, you might want to review those videos. But if I know the variance of my original distribution, and if I know what my n is, how many samples I'm going to take every time before I average them in order to plot one thing in my sampling distribution of my sample mean, is there a way to predict what the mean of these distributions are? The standard deviation of these distributions. And to make it so you don't get confused between that and that, let me say the variance. If you know the variance, you can figure out the standard deviation because one is just the square root of the other. So this is the variance of our original distribution. Now, to show that this is the variance of our sampling distribution of our sample mean, we'll write it right here. This is the variance of our sample mean. Remember, our true mean is this, that the Greek letter mu is our true mean. This is equal to the mean. While an x with a line over it means sample mean. So here, what we're saying is this is the variance of our sample means. Now, this is going to be a true distribution. This isn't an estimate. If we magically knew the distribution, there's some true variance here. And of course, the mean-- so this has a mean. This, right here-- if we can just get our notation right-- this is the mean of the sampling distribution of the sampling mean. So this is the mean of our means. It just happens to be the same thing. This is the mean of our sample means. It's going to be the same thing as that, especially if we do the trial over and over again. But anyway, the point of this video, is there any way to figure out this variance given the variance of the original distribution and your n? And it turns out, there is. And I'm not going to do a proof here. I really want to give you the intuition of it. And I think you already do have the sense that every trial you take, if you take 100, you're much more likely, when you average those out, to get close to the true mean than if you took an n of 2 or an n of 5. You're just very unlikely to be far away if you took 100 trials as opposed to taking five. So I think you know that, in some way, it should be inversely proportional to n. The larger your n, the smaller a standard deviation. And it actually turns out it's about as simple as possible. It's one of those magical things about mathematics. And I'll prove it to you one day. I want to give you a working knowledge first. With statistics, I'm always struggling whether I should be formal in giving you rigorous proofs, but I've come to the conclusion that it's more important to get the working knowledge first in statistics, and then, later, once you've gotten all of that down, we can get into the real deep math of it and prove it to you. But I think experimental proofs are all you need for right now, using those simulations to show that they're really true. So it turns out that the variance of your sampling distribution of your sample mean is equal to the variance of your original distribution-- that guy right there-- divided by n. That's all it is. So if this up here has a variance of-- let's say this up here has a variance of 20. I'm just making that number up. And then let's say your n is 20. Then the variance of your sampling distribution of your sample mean for an n of 20-- well, you're just going to take the variance up here-- your variance is 20-- divided by your n, 20. So here, your variance is going to be 20 divided by 20, which is equal to 1. This is the variance of your original probability distribution. And this is your n. What's your standard deviation going to be? What's going to be the square root of that? Standard deviation is going to be the square root of 1. Well, that's also going to be 1. So we could also write this. We could take the square root of both sides of this and say, the standard deviation of the sampling distribution of the sample mean is often called the standard deviation of the mean, and it's also called-- I'm going to write this down-- the standard error of the mean. All of these things I just mentioned, these all just mean the standard deviation of the sampling distribution of the sample mean. That's why this is confusing. Because you use the word "mean" and "sample" over and over again. And if it confuses you, let me know. I'll do another video or pause and repeat or whatever. But if we just take the square root of both sides, the standard error of the mean, or the standard deviation of the sampling distribution of the sample mean, is equal to the standard deviation of your original function, of your original probability density function, which could be very non-normal, divided by the square root of n. I just took the square root of both sides of this equation. Personally, I like to remember this, that the variance is just inversely proportional to n, and then I like to go back to this, because this is very simple in my head. You just take the variance divided by n. Oh, and if I want the standard deviation, I just take the square roots of both sides, and I get this formula. So here, when n is 20, the standard deviation of the sampling distribution of the sample mean is going to be 1. Here, when n is 100, our variance-- so our variance of the sampling mean of the sample distribution or our variance of the mean, of the sample mean, we could say, is going to be equal to 20, this guy's variance, divided by n. So it equals-- n is 100-- so it equals one fifth. Now, this guy's standard deviation or the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean, is going to the square root of that. So 1 over the square root of 5. And so this guy will have to be a little bit under one half the standard deviation, while this guy had a standard deviation of 1. So you see it's definitely thinner. Now, I know what you're saying. Well, Sal, you just gave a formula. I don't necessarily believe you. Well, let's see if we can prove it to ourselves using the simulation. So just for fun, I'll just mess with this distribution a little bit. So that's my new distribution. And let me take an n-- let me take two things it's easy to take the square root of, because we're looking at standard deviations. So let's say we take an n of 16 and n of 25. And let's do 10,000 trials. So in this case, every one of the trials, we're going to take 16 samples from here, average them, plot it here, and then do a frequency plot. Here, we're going to do a 25 at a time and then average them. I'll do it once animated just to remember. So I'm taking 16 samples, plot it there. I take 16 samples, as described by this probability density function, or 25 now. Plot it down here. Now, if I do that 10,000 times, what do I get? What do I get? All right. So here, just visually, you can tell just when n was larger, the standard deviation here is smaller. This is more squeezed together. But actually, let's write this stuff down. Let's see if I can remember it here. Here, n is 6. So in this random distribution I made, my standard deviation was 9.3. I'm going to remember these. Our standard deviation for the original thing was 9.3. And so standard deviation here was 2.3, and the standard deviation here is 1.87. Let's see if it conforms to our formula. So I'm going to take this off screen for a second, and I'm going to go back and do some mathematics. So I have this on my other screen so I can remember those numbers. So, in the trial we just did, my wacky distribution had a standard deviation of 9.3. When n was equal to 16-- just doing the experiment, doing a bunch of trials and averaging and doing all the thing-- we got the standard deviation of the sampling distribution of the sample mean, or the standard error of the mean. We experimentally determined it to be 2.33. And then when n is equal to 25, we got the standard error of the mean being equal to 1.87. Let's see if it conforms to our formulas. So we know that the variance-- or we could almost say the variance of the mean or the standard error-- the variance of the sampling distribution of the sample mean is equal to the variance of our original distribution divided by n. Take the square roots of both sides. Then you get standard error of the mean is equal to standard deviation of your original distribution, divided by the square root of n. So let's see if this works out for these two things. So if I were to take 9.3-- so let me do this case. So 9.3 divided by the square root of 16-- n is 16-- so divided by the square root of 16, which is 4. What do I get? So 9.3 divided by 4. Let me get a little calculator out here. Let's see. We want to divide 9.3 divided by 4. 9.3 divided by our square root of n-- n was 16, so divided by 4-- is equal to 2.32. So this is equal to 2.32, which is pretty darn close to 2.33. This was after 10,000 trials. Maybe right after this I'll see what happens if we did 20,000 or 30,000 trials where we take samples of 16 and average them. Now let's look at this. Here, we would take 9.3. So let me draw a little line here. Maybe scroll over. That might be better. So we take our standard deviation of our original distribution-- so just that formula that we've derived right here would tell us that our standard error should be equal to the standard deviation of our original distribution, 9.3, divided by the square root of n, divided by square root of 25. 4 was just the square root of 16. So this is equal to 9.3 divided by 5. And let's see if it's 1.87. So let me get my calculator back. So if I take 9.3 divided by 5, what do I get? 1.86, which is very close to 1.87. So we got in this case 1.86. So as you can see, what we got experimentally was almost exactly-- and this is after 10,000 trials-- of what you would expect. Let's do another 10,000. So you got another 10,000 trials. Well, we're still in the ballpark. We're not going to-- maybe I can't hope to get the exact number rounded or whatever. But, as you can see, hopefully that'll be pretty satisfying to you, that the variance of the sampling distribution of the sample mean is just going to be equal to the variance of your original distribution, no matter how wacky that distribution might be, divided by your sample size, by the number of samples you take for every basket that you average, I guess is the best way to think about it. And sometimes this can get confusing, because you are taking samples of averages based on samples. So when someone says sample size, you're like, is sample size the number of times I took averages or the number of things I'm taking averages of each time? And it doesn't hurt to clarify that. Normally when they talk about sample size, they're talking about n. And, at least in my head, when I think of the trials as you take a sample of size of 16, you average it, that's one trial. And you plot it. Then you do it again, and you do another trial. And you do it over and over again. But anyway, hopefully this makes everything clear. And then you now also understand how to get to the standard error of the mean.

Standard error of the mean

Population

The standard error of the mean (SEM) can be expressed as:

${\displaystyle {\sigma }_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}}$

where

σ is the standard deviation of the population.
n is the size (number of observations) of the sample.

Estimate

Since the population standard deviation is seldom known, the standard error of the mean is usually estimated as the sample standard deviation divided by the square root of the sample size (assuming statistical independence of the values in the sample).

${\displaystyle {\sigma }_{\bar {x}}\ \approx {\frac {s}{\sqrt {n}}}}$

where

s is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
n is the size (number of observations) of the sample.

Sample

In those contexts where standard error of the mean is defined not as the standard deviation of the sample mean, but as its estimate, this is the estimate typically given as its value. Thus, it is common to see standard deviation of the mean alternatively defined as:

${\displaystyle {\text{s}}_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}}$

The standard deviation of the sample mean is equivalent to the standard deviation of the error in the sample mean with respect to the true mean, since the sample mean is an unbiased estimator. Therefore, the standard error of the mean can also be understood as the standard deviation of the error in the sample mean with respect to the true mean (or an estimate of that statistic).

Note: the standard error and the standard deviation of small samples tend to systematically underestimate the population standard error and standard deviation: the standard error of the mean is a biased estimator of the population standard error. With n = 2 the underestimate is about 25%, but for n = 6 the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect.[2] Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20.[3] See unbiased estimation of standard deviation for further discussion.

A practical result: Decreasing the uncertainty in a mean value estimate by a factor of two requires acquiring four times as many observations in the sample. Or decreasing the standard error by a factor of ten requires a hundred times as many observations.

Derivations

The formula may be derived from the variance of a sum of independent random variables.[4]

• If ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ are ${\displaystyle n}$ independent observations from a population that has a mean ${\displaystyle \mu }$ and standard deviation ${\displaystyle \sigma }$, then the variance of the total ${\displaystyle T=(x_{1}+x_{2}+\cdots +x_{n})}$ is ${\displaystyle n\sigma ^{2}.}$
• The variance of ${\displaystyle T/n}$ (the mean ${\displaystyle {\bar {x}}}$) must be ${\displaystyle n\left({\frac {\sigma ^{2}}{n^{2}}}\right)={\frac {\sigma ^{2}}{n}}.}$ Alternatively, ${\displaystyle {\text{var}}({\frac {T}{n}})={\frac {1}{n^{2}}}{\text{var}}(T)={\frac {1}{n^{2}}}n\sigma ^{2}={\frac {\sigma ^{2}}{n}}.}$
• And the standard deviation of ${\displaystyle T/n}$ must be ${\displaystyle \sigma /{\sqrt {n}}}$

Student approximation when σ value is unknown

In many practical applications, the true value of σ is unknown. As a result, we need to use a distribution that takes into account that spread of possible σ's. When the true underlying distribution is known to be Gaussian, although with unknown σ, then the resulting estimated distribution follows the Student t-distribution. The standard error is the standard deviation of the Student t-distribution. T-distributions are slightly different from Gaussian, and vary depending on the size of the sample. Small samples are somewhat more likely to underestimate the population standard deviation and have a mean that differs from the true population mean, and the Student t-distribution accounts for the probability of these events with somewhat heavier tails compared to a Gaussian. To estimate the standard error of a Student t-distribution it is sufficient to use the sample standard deviation "s" instead of σ, and we could use this value to calculate confidence intervals.

Note: The Student's probability distribution is approximated well by the Gaussian distribution when the sample size is over 100. For such samples one can use the latter distribution, which is much simpler.

Assumptions and usage

An example of how SE is used, is to make confidence intervals of the unknown population mean. If the sampling distribution is normally distributed, the sample mean, the standard error, and the quantiles of the normal distribution can be used to calculate confidence intervals for the true population mean. The following expressions can be used to calculate the upper and lower 95% confidence limits, where ${\displaystyle {\bar {x}}}$ is equal to the sample mean, ${\displaystyle SE}$ is equal to the standard error for the sample mean, and 1.96 is the 0.975 quantile of the normal distribution:

Upper 95% limit ${\displaystyle ={\bar {x}}+({\text{SE}}\times 1.96),}$ and
Lower 95% limit ${\displaystyle ={\bar {x}}-({\text{SE}}\times 1.96).}$

In particular, the standard error of a sample statistic (such as sample mean) is the actual or estimated standard deviation of the error in the process by which it was generated. In other words, it is the actual or estimated standard deviation of the sampling distribution of the sample statistic. The notation for standard error can be any one of SE, SEM (for standard error of measurement or mean), or SE.

Standard errors provide simple measures of uncertainty in a value and are often used because:

Standard error of mean versus standard deviation

In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[5]

Put simply, the standard error of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the standard deviation of the sample is the degree to which individuals within the sample differ from the sample mean.[6] If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases.

Correction for finite population

The formula given above for the standard error assumes that the sample size is much smaller than the population size, so that the population can be considered to be effectively infinite in size. This is usually the case even with finite populations, because most of the time, people are primarily interested in managing the processes that created the existing finite population; this is called an analytic study, following W. Edwards Deming. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative study.

When the sampling fraction is large (approximately at 5% or more) in an enumerative study, the estimate of the standard error must be corrected by multiplying by a "finite population correction":[7] [8]

${\displaystyle {\text{FPC}}={\sqrt {\frac {N-n}{N-1}}}}$

which, for large N:

${\displaystyle {\text{FPC}}\approx {\sqrt {1-{\frac {n}{N}}}}}$

to account for the added precision gained by sampling close to a larger percentage of the population. The effect of the FPC is that the error becomes zero when the sample size n is equal to the population size N.

Correction for correlation in the sample

Expected error in the mean of A for a sample of n data points with sample bias coefficient ρ. The unbiased standard error plots as the ρ=0 diagonal line with log-log slope -½.

If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of the mean (actually a correction on the standard deviation part) may be obtained by multiplying the calculated standard error of the sample by the factor f:

${\displaystyle f={\sqrt {\frac {1+\rho }{1-\rho }}},}$

where the sample bias coefficient ρ is the widely used Prais–Winsten estimate of the autocorrelation-coefficient (a quantity between −1 and +1) for all sample point pairs. This approximate formula is for moderate to large sample sizes; the reference gives the exact formulas for any sample size, and can be applied to heavily autocorrelated time series like Wall Street stock quotes. Moreover, this formula works for positive and negative ρ alike.[9] See also unbiased estimation of standard deviation for more discussion.

Relative standard error

The relative standard error of a sample mean is the standard error divided by the mean and expressed as a percentage. It can only be calculated if the mean is a non-zero value.

As an example of the use of the relative standard error, consider two surveys of household income that both result in a sample mean of $50,000. If one survey has a standard error of$10,000 and the other has a standard error of \$5,000, then the relative standard errors are 20% and 10% respectively. The survey with the lower relative standard error can be said to have a more precise measurement, since it has proportionately less sampling variation around the mean. In fact, data organizations often set reliability standards that their data must reach before publication. For example, the U.S. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more – for an estimate to be reported.)[10]

References

1. ^ Everitt, B. S. (2003). The Cambridge Dictionary of Statistics. CUP. ISBN 978-0-521-81099-9.
2. ^ Gurland, J; Tripathi RC (1971). "A simple approximation for unbiased estimation of the standard deviation". American Statistician. 25 (4): 30–32. doi:10.2307/2682923. JSTOR 2682923.
3. ^ Sokal; Rohlf (1981). Biometry: Principles and Practice of Statistics in Biological Research (2nd ed.). p. 53. ISBN 978-0-7167-1254-1.
4. ^ Hutchinson, T. P. Essentials of Statistical Methods, in 41 pages. Adelaide: Rumsby. ISBN 978-0-646-12621-0.
5. ^ Barde, M. (2012). "What to use to express the variability of data: Standard deviation or standard error of mean?". Perspect. Clin. Res. 3 (3): 113–116. doi:10.4103/2229-3485.100662. PMC 3487226. PMID 23125963.
6. ^ Wassertheil-Smoller, Sylvia (1995). Biostatistics and Epidemiology : A Primer for Health Professionals (Second ed.). New York: Springer. pp. 40–43. ISBN 0-387-94388-9.
7. ^ Isserlis, L. (1918). "On the value of a mean as calculated from a sample". Journal of the Royal Statistical Society. 81 (1): 75–81. doi:10.2307/2340569. JSTOR 2340569. (Equation 1)
8. ^ Bondy, Warren; Zlot, William (1976). "The Standard Error of the Mean and the Difference Between Means for Finite Populations". The American Statistician. 30 (2): 96–97. JSTOR 2683803. (Equation 2)
9. ^ Bence, James R. (1995). "Analysis of Short Time Series: Correcting for Autocorrelation". Ecology. 76 (2): 628–639. doi:10.2307/1941218. JSTOR 1941218.
10. ^ Klein, RJ. "Healthy People 2010 criteria for data suppression" (PDF). Statistical Notes (24). Retrieved 17 July 2014.
This page was last edited on 3 June 2019, at 19:21
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