Standard error vs standard deviation

Question

A colleague measures daily returns, finds a sample standard deviation of $2\%$ , and concludes: "my estimate of the mean daily return is accurate to about $2\%$ ."

What is the difference between a standard deviation and a standard error, and what did the colleague get wrong?

Accepted Answer

They confused two different quantities. Standard deviation (\sigma) measures the spread of individual observations around their mean. It is a property of the distribution/data and does not shrink as you collect more data, with more days you estimate the same 2\% more precisely, but it stays 2\%. Standard error (SE) measures the spread of a statistic computed from the sample, usually the sample mean. For the mean, \operatorname{SE}(\bar X) = \frac{\sigma}{\sqrt{n}}. This does shrink, at rate 1/\sqrt{n}. With n = 100 days, \operatorname{SE}(\bar X) = \frac{2\%}{\sqrt{100}} = 0.2\%, not 2\%. Wit…

Standard error vs standard deviation

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