Stats what is standard error

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 , 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 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 I'm just making that number up.

And then let's say your n is Then the variance of your sampling distribution of your sample mean for an n of well, you're just going to take the variance up here-- your variance is divided by your n, 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 , 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 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. The standard error is a measure of the variability of the sampling distribution. Just as the standard deviation is a measure of the dispersion of values in the sample, the standard error is a measure of the dispersion of values in the sampling distribution.

That is, of the dispersion of means of samples if a large number of different samples had been drawn from the population. The standard error of a sample mean is represented by the following formula:.

That is, the standard error is equal to the standard deviation divided by the square root of the sample size, n. This shows that the larger the sample size, the smaller the standard error. Given that the larger the divisor, the smaller the result and the smaller the divisor, the larger the result. The symbol for standard error of the mean is s M or when symbols are difficult to produce, it may be represented as, S.

The standard error of the mean can provide a rough estimate of the interval in which the population mean is likely to fall. The SEM, like the standard deviation, is multiplied by 1. Then subtract the result from the sample mean to obtain the lower limit of the interval. The resulting interval will provide an estimate of the range of values within which the population mean is likely to fall.

This interval is a crude estimate of the confidence interval within which the population mean is likely to fall. A more precise confidence interval should be calculated by means of percentiles derived from the t-distribution. Another use of the value, 1. Consider, for example, a researcher studying bedsores in a population of patients who have had open heart surgery that lasted more than 4 hours.

Suppose the mean number of bedsores was 0. If the standard error of the mean is 0. This is interpreted as follows: The population mean is somewhere between zero bedsores and 20 bedsores. Given that the population mean may be zero, the researcher might conclude that the 10 patients who developed bedsores are outliers. That in turn should lead the researcher to question whether the bedsores were developed as a function of some other condition rather than as a function of having heart surgery that lasted longer than 4 hours.

The standard error of the estimate S. Specifically, it is calculated using the following formula:. Therefore, the standard error of the estimate is a measure of the dispersion or variability in the predicted scores in a regression. In a scatterplot in which the S. When the S. Figure 1. Low S. Figure 2. Large S. Every inferential statistic has an associated standard error. Although not always reported, the standard error is an important statistic because it provides information on the accuracy of the statistic 4.

As discussed previously, the larger the standard error, the wider the confidence interval about the statistic. In fact, the confidence interval can be so large that it is as large as the full range of values, or even larger. In that case, the statistic provides no information about the location of the population parameter.

And that means that the statistic has little accuracy because it is not a good estimate of the population parameter. In this way, the standard error of a statistic is related to the significance level of the finding. When the standard error is large relative to the statistic, the statistic will typically be non-significant. However, if the sample size is very large, for example, sample sizes greater than 1,, then virtually any statistical result calculated on that sample will be statistically significant.

For example, a correlation of 0. However, a correlation that small is not clinically or scientifically significant. I was not able to understand standard error. The procedures for calculating is given but i cant understand the process of calculation. Standard Deviation is the square root of variance, so its kind of trivial to state the conclusion about the increasing standard error with respect to standard error.

Also please look into the symbol of sigma mentioned in the explanation of standard error. Thank you for flagging the symbol errors on the page Rohit. These have been updated now. Many thanks, Emma. Hi, Thank you! The denominator should be n Hi Wesley. Thank you for the comment. There is indeed a different formula, which uses n — 1 rather than N, when calculating the standard deviation of a sample.

The resource here provides a really good explanation too. Hope that proves useful. Conducting successful research requires choosing the appropriate study design. This article describes the most common types of designs conducted by researchers. What are the key steps in EBM?