Recall that in the sampling distribution of the mean the Central Limit Theorem stated that no matter what the shape of the parent population distribution, the sampling distribution of the mean would have a normal shape. The shape becomes more normal with increasing numbers of samples.
No matter how the actual values in a population are distributed, we know how the sample means will be distributed. Since our interest in inferential statistics is in making an inference about a population from a sample, it is important to know what kind of variability to expect in the samples we take. If there is a large amount of variability from sample to sample, we will be less certain in our use of a single sample value to estimate the population than if there is little variability from sample to sample.
The standard deviation, just as with other distributions we have worked with, will be the measure of variability we use; this time for the sampling distribution. When we measure the average deviation of a set of sample means, we are measuring how much variability there is from sample to sample. This measure of the standard deviation of a distribution of sample means it is called the
and is the standard deviation of the sampling distribution of the mean. Just as we calculated how much variability existed in a distribution of values sampled from the population, the standard error is how much variability exists in a distribution of sample means.
In this demonstration we start with the same samples we used in the sampling distribution demonstration. Recall that the range of values for the pop quiz scores is 0-9 and the mean for the population was equal to 3.90.
Since the sampling distribution is theoretical we don't actually need to calculate the standard error every time we want to conduct an inferential test. Instead we estimate it from the population or the sample. Rule 3 of the Central Limit Theorem states: To estimate the standard error for the population: