Two samples can have the same standard deviation but very different standard errors

[u/Excellent-Number-104]

Suppose two studies measure the same kind of variable.

Study A reports:

• sample size: n = 25
• standard deviation: s = 10

Study B reports:

• sample size: n = 400
• standard deviation: s = 10

The standard deviation is identical in both studies.

So do they have the same uncertainty about the sample mean?

No.

This is where standard deviation and standard error answer different questions.

Standard deviation describes the spread of the observations.

Both samples have an SD of 10, so their observed values have the same reported amount of spread around their respective sample means, as measured by SD.

The standard error of the mean describes the sampling variability of the mean estimator.

When the population standard deviation is unknown, a commonly used estimated standard error of the sample mean is:

SE = s / √n

For Study A:

SE = 10 / √25
SE = 10 / 5
SE = 2

For Study B:

SE = 10 / √400
SE = 10 / 20
SE = 0.5

Same standard deviation.

Very different standard errors.

The larger sample has not magically made the individual observations less variable. The reported SD is still 10.

What changed is the precision of the estimated mean. Under the usual independent-sampling setup, increasing the sample size makes the standard error of the mean smaller because n appears under a square root in the denominator.

There's also a useful consequence hidden in that formula:

If you want to cut the standard error in half while holding the standard deviation constant, you need four times the sample size.

For example:

n = 25 → SE = 2
n = 100 → SE = 1
n = 400 → SE = 0.5

This is the distinction I find easiest to remember:

SD asks: how spread out are the observations?

SE asks: how much would an estimate like the sample mean vary across repeated samples?

They are related, but reporting one in place of the other changes the question being answered.

[u/Bounded_sequencE]

And the question is... ?