The Standard Error of the Mean

Watch "Sample Size" in Action

Each frame of this animation shows the mean of 8 samples. Each frame shows a new set of 8 samples.

The black triangles show the mean of each of the samples. Of course, the sample of size 1 just shows the value of that one observation.
The green bars above each sample mean, indicate the range of values within 2 standard deviations or 2 standard errors of the population mean.
"Freq." counts how many times the mean of the sample was within 2 standard deviations/standard errors of the population mean. "Reps." counts how many replications--the total number of samples.
"%" indicates the proportion of means that are within 2 standard deviations/standard errors of the population mean.
Any time a mean is outside the range of "green bar" values, the triangle turns "red" and the "Freq" is not incremented.
Notice: Each of the percentages ("%") is close to the theoretical value of 95.453 (the percentage of observations in a normal distribution within 2 standard deviations of the mean.
Ripple Software's "Teaching Statistics" application allows users to set either the theoretical central percentage or z-score surrounding the population mean to demonstrate the validity of the Central Limit Theorem.

[Text continues below animation.]

Here is the IMPORTANT bit of information.

The Green Bar for n = 1 is (4 X the population standard deviations wide) and centered at the value of the population mean.

The Green Bars for all the other sample sizes divide the population standard deviation by the square root of sample size. They are not based on the means of the samples drawn. There's a example table given below

The Standard Error of the Mean (aka "Standard Error") tells how much the sample mean will "jump around" from 1 sample to the next just as the "Standard Deviation" tells how much a single observation will "jump around" from 1 observation to the next.

The Standard Error of the Mean is the Standard Deviation of the Sampling Distribution of the Mean.

**Standard Deviation and Standard Errror of the Mean (SEM)**
Sample Size	Standard Deviation/Standard Error	Width of Green Bar (arbitrary units)
1	Population Standard Deviation =100	400
2	SEM = 100/ (sqrt(2) = 70.71	283
4	SEM = 100 / sqrt(4) = 50.00	200
8	SEM = 100 / sqrt(8) = 35.36	141
16	SEM = 100 / sqrtt(16) = 25.00	100
32	SEM = 100 / sqrt(32) = 17.68	72
64	SEM = 100 / sqrt(64) = 12.5	50
128	SEM = 100 / sqrt(128) = 8.84	35

For the hardcore

Click here to see an animation for the center 50% instead of the center 95%.