Standard Error of the Mean

Measures how much a sample mean moves from sample to sample, as the sample standard deviation ÷ √n. Where the standard deviation describes the spread of the data, the standard error describes the precision of the mean as an estimate.

Draw a fresh sample from the same population and the sample mean comes out a little different each time. The standard error measures the size of that wobble.

SE = \dfrac{s}{\sqrt{n}}

$SE$ is the standard error
$s$ is the sample standard deviation, with the squared deviations divided by $n - 1$ before taking the root
$n$ is the number of values

Despite the similar names, the standard deviation and the standard error describe different things. The standard deviation is the spread of the data itself, and collecting more data does not shrink it; it merely converges on the population's spread. The standard error is the precision of the mean as an estimate, and it does shrink as data accumulates.

The $\sqrt{n}$ in the denominator is the crucial part. Halving the standard error takes four times as much data, not twice as much. Going from 100 observations to 400 buys exactly one halving. That square root is why extra survey effort yields so little extra precision.

Example

Take the default data 12, 15, 18, 20, 25. There are 5 values and the mean is 18.

The deviations from the mean are −6, −3, 0, 2 and 7, and their squares sum to 98. The sample standard deviation divides that by $5 - 1 = 4$ and takes the root: $\sqrt{24.5}$ , about 4.9497.

Dividing by $\sqrt{5}$ gives a standard error of about 2.2136. Read it as: the individual values scatter by around 4.9497, but their mean only wobbles by around 2.2136.

Where it is used

Confidence intervals are built from it. A 95% interval for the population mean stretches from the sample mean by a t critical value times the standard error in each direction, so a smaller standard error means a tighter interval.

Error bars on charts often show it too. What the bar represents — standard deviation, standard error, or a confidence interval — completely changes its meaning, which is why captions are expected to say. Standard error bars are always shorter, so using them without saying so makes the data look tidier than it is.

Points to watch

The sample standard deviation divides by $n - 1$ . Using the population version, which divides by $n$ , understates the standard error slightly.

A single observation gives nothing. Measuring spread needs at least two values.