The Standard Error of the Mean (SE(M)) is analogous to the Standard Deviation (SD), in that it is an estimate of variability. It is different, however, in that while the Standard Deviation gives one a sense of how much variability there is in the individual values that make up one single sample, the Standard Error of the Mean gives one a sense of how much variability there is in the means of small samples (of n individual values) of a larger population.
As an example, assume that you measured the height of a population of 1000 people. The SD is 3.0 cm. This tells you how much individual variability there is among individuals. If you only measured 500 people, your standard deviation would still be very close to 3.0 cm. Same thing if you measured 250 people. With reasonably large sample sizes, SD will always be the same.
Now, imagine you measured the average height of ten random people. Then, imagine you measured the height of another ten random people. These mean heights would be different. HOW different they were would be a function of both the SD (i.e. how much individual variability there is), and the sample size – as you choose larger and larger groups of people, the means will deviate less. Groups of 5 are likely to have a lot of variability in the means, because if one person is very tall or very short, the mean will be thrown off. Groups of 100, by contrast, are almost always going to have similar means, because variability is averaged out. This is, in essence, what we are referring to when we talk about the Standard Error of the Mean. When there is a lot of individual variability (SD is high) and n (number per sample) is small, SE(M) will be large. When there is very little individual variability and n is large, SE(M) will be small.
This becomes important when comparing groups. If the expected variability between groups is large (i.e. SE(M) is large), then the actual difference between groups will have to be very large if it is to be considered statistically significant. This is exemplified in the calculation of the Z Test.
Standard Error of the Mean:
