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Date: 2016-09-25### Summary

The difference between standard deviation and standard error is demonstrated with a standard normal distribution.### Contents

# Standard Deviation and Standard Error¶

In statistics, there is the standard deviation and the standard error. Their respective estimators are:

So the standard error is by a factor smaller. They have similar but different interpretations:

- Standard Deviation
- This is the width of the population distribution. If you take a single
measurement out of the normal distributed population, it will be within one
standard
*deviation*of the mean in 68% of the cases. - Standard Error
- If you take another measurements and compute the mean of those
measurements, it will be within one standard
*error*of the mean in 68% of the cases. The standard error is therefore the uncertainty of the mean.

Taking more samples from the same distribution will not change the standard
deviation, assuming that you already have sufficient measurements that the
normal distribution is sufficiently apparent. The standard *error* will become
smaller.

I have written a small program, `t_test.py`

, which samples a standard
normal distribution and displays the results in a histogram. You can see that
the shape will come closer to the normal distribution. The green dots and line
show the standard deviation. Those points will always stay close to and
as this is the standard deviation of the underlying distribution.

The standard error is marked by the two red points. There you can see how it shrinks over time as more measurements are drawn from the underlying distribution.