## Standard Error of the Mean

The *standard error of the mean*is the average distance of all sample means from the population mean. In other words, if we could take the distance between each sample mean and the population mean, then get the average of all these distances, we would have the standard error of the mean. This is very similar to the standard deviation in a sample, which is the average distance the scores are away from the sample mean. The standard error will be larger or smaller depending on the sample size that is used to form the sampling distribution. This is where the law of large numbers comes in.

The *law of large numbers *refers to the relationship between the sample size and the difference between the sample and population means. The larger the sample size used to make up the sampling distribution of the mean, the smaller will be the difference between sample means and the population mean. Therefore, larger samples will have smaller standard error. In other words, a larger sample will give us a more accurate depiction of the population.

Here’s what I mean. If we make up our sampling distribution of the mean (σ_{M}) from all possible samples of n = 1, and we know that the population standard deviation (σ) is 3, the standard error of our distribution will be:

σ_{M} = σ / √N = 3 / √1 = 3

The standard error, in the case of n = 1, is identical to the population standard deviation. Think about this now, if the *n* for all possible samples is 1, then we are talking about the population itself, one person at a time, and we aren’t really taking samples at all. Therefore, the standard error *should* be exactly the same as the population standard deviation.

What happens if we increase the sample size? What if n = 9?

σ_{M} = σ / √N = 3 / √9 = 1

What if n = 25?

σ_{M} = 3 / √25 = 3/5 = 0.60

What’s happening? The standard error is getting smaller as the sample size gets larger. This shows that with a larger sample size, the average distance between the sample means and the population mean is smaller, which in turn means that any sample we use in our analysis (if sufficiently large) will represent the true population mean much better.

For our purposes, a nice big sample will allow us to more accurately say that whatever is going on in our sample is highly likely to also be going on in the population.

Nice, eh? 🙂

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