Last updated: 2026-07-01

What This Calculator Does

This calculator finds the standard error of the mean (SEM) from a sample's standard deviation and size, and builds the confidence interval around the sample mean. Enter the standard deviation, the sample size, the confidence level, and the observed sample mean.

What the standard error of the mean is

When you estimate an average from a sample, that estimate is never exact — a different random sample would produce a slightly different mean. The standard error of the mean quantifies this: it is the standard deviation of the sampling distribution of the sample mean, or how much the sample mean is expected to vary from one sample to the next.

A small standard error means your sample mean pins down the population mean tightly; a large one means the estimate is loose. It is the key ingredient in confidence intervals and hypothesis tests about a mean.

The formula

• SEM — the standard error of the mean
• s — the sample standard deviation
• n — the sample size

The standard deviation measures the spread of the raw data; dividing by $\sqrt{n}$ converts that into the spread of the average. Averaging many observations cancels out individual noise, so the mean is more stable than any single data point — and the more you average, the more stable it gets.

Confidence interval

Once you have the standard error, a confidence interval for the population mean is the sample mean plus or minus a critical value times the standard error:

The critical value $z$ comes from the standard normal distribution and is set by the confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%). At 95% confidence this means that if the study were repeated many times, about 95% of the intervals built this way would contain the true population mean.

Standard error versus standard deviation

These two are easy to confuse because both measure "spread," but they describe different things:

Collecting more data does not make individual values less variable, so the standard deviation is stable. But it does make the average more reliable, so the standard error keeps falling.

Why the standard error shrinks with √n

Because the sample size enters under a square root, the standard error falls in proportion to $1/\sqrt{n}$ rather than in inverse proportion to $n$. This produces diminishing returns:

To halve the standard error you must quadruple the sample. Each additional unit of precision becomes progressively more expensive to buy.

Worked example

Suppose a sample of size 36 has a standard deviation of 12 and a mean of 100, and a 95% confidence interval is wanted. The critical value is 1.96:

The standard error of the mean is 2, and the 95% confidence interval runs from 96.08 to 103.92. Based on this sample, the population mean is plausibly anywhere in that range.

When the standard deviation is estimated from a small sample

The interval above uses the normal critical value $z$, which is appropriate when the population standard deviation is known or the sample is large enough that $s$ is a dependable substitute. A common rule of thumb is $n \geq 30$.

When $s$ is estimated from a small sample, replace $z$ with a value from the t-distribution, which has heavier tails to account for the extra uncertainty:

The t-value depends on the degrees of freedom, one fewer than the sample size, and is always slightly larger than the corresponding z-value, so it widens the interval. As n grows, the t-value converges back to z.

Related calculators

The Confidence Interval Calculator builds the full interval around your estimate for both means and proportions, and the Margin of Error Calculator reports the half-width $z \cdot \text{SEM}$ directly. To find the standard deviation from raw data first, use the Variance and Standard Deviation Calculator.