How to Calculate Standard Deviation (Sample and Population)

What standard deviation measures

The mean tells you where the centre of a data set is, but not how tightly the values cluster around it. Standard deviation fills that gap. A small standard deviation means the values sit close to the mean; a large one means they are widely scattered. It is measured in the same units as the data, so a standard deviation of 5 centimetres for heights means a typical height is roughly 5 centimetres from the average.

The calculation follows a simple logic. Measure how far each value is from the mean, square those distances so that values above and below the mean do not cancel out, average the squares, and finally take the square root to return to the original units. If you need a refresher on the mean itself, start with mean, median, mode and standard deviation.

The two formulas

There are two versions, depending on whether your data are an entire population or a sample drawn from a larger population:

The population standard deviation $\sigma$ divides by the number of values $N$. The sample standard deviation $s$ divides by $n - 1$. The quantity under the square root is the variance, written $\sigma^2$ or $s^2$. Variance is useful in theory, but its units are squared, such as square centimetres, which is why the standard deviation is the number usually reported.

Five steps, worked by hand

A table with columns for $x$, $x - \bar{x}$ and $(x - \bar{x})^2$ keeps the work organised. A useful check is that the deviations in the second column always add up to zero. If they do not, the mean is wrong, and everything after it will be wrong too.

Why divide by n minus 1 for a sample?

When you work with a sample, you do not know the true population mean, so you measure deviations from the sample mean instead. The sample mean is, by its construction, the point that makes the sum of squared deviations as small as possible for that particular sample. Deviations measured from it are therefore slightly smaller, on average, than deviations from the true mean would be.

Dividing by $n - 1$ instead of $n$ corrects for that, making the sample variance an unbiased estimate of the population variance. This adjustment is called Bessel's correction. Another way to see it is through degrees of freedom: once you know the mean and $n - 1$ of the deviations, the last deviation is forced, because they must add to zero. Only $n - 1$ deviations carry independent information.

In practice, use the sample formula unless your data genuinely include every member of the group you care about, such as the scores of every student in one class when you only care about that class.

A shortcut formula and frequency tables

An algebraically equivalent formula avoids computing each deviation separately:

For the data in Example 1, $\sum x^2 = 36 + 64 + 100 + 144 + 196 = 540$, so $\sigma^2 = \frac{540}{5} - 10^2 = 8$, as before. The shortcut is quick by hand, but computers avoid it with large or very precise numbers, because subtracting two nearly equal large numbers loses accuracy.

Interpreting the result

For data that follow a roughly bell-shaped normal distribution, the empirical rule gives a concrete reading of the standard deviation: about 68 percent of values lie within one standard deviation of the mean, about 95 percent within two and about 99.7 percent within three. If adult heights in a group have mean 170 centimetres and standard deviation 8, roughly 95 percent of the group is between 154 and 186 centimetres tall.

For data of any shape, Chebyshev's inequality still guarantees that at least 75 percent of values lie within two standard deviations of the mean, and at least about 89 percent within three. Expressing a single value's distance from the mean in standard deviations gives its z-score, which lets you compare values measured on different scales.

Standard deviation also responds predictably to changes in the data. Adding the same constant to every value shifts the mean but leaves the standard deviation unchanged, because the spread is the same. Multiplying every value by a constant multiplies the standard deviation by the absolute value of that constant, so converting metres to centimetres multiplies it by 100.

Using a calculator or spreadsheet

Scientific calculators usually show two keys, often labelled $\sigma_x$ or $\sigma_n$ for the population version and $s_x$ or $\sigma_{n-1}$ for the sample version. In spreadsheets such as Excel and Google Sheets, STDEV.P computes the population standard deviation and STDEV.S the sample one; the older STDEV function also gives the sample value. Choosing the wrong one is the most common source of mismatched answers.

The standard deviation calculator shows both versions together with the full deviation table, which is ideal for checking homework done by hand. The mean, median and mode calculator adds the other summary statistics, and the z-score calculator uses the result to standardise individual values.

Comparing spread across different scales

Standard deviations are hard to compare when the means are very different. A standard deviation of 2 kilograms is large for newborn babies but tiny for adult elephants. The coefficient of variation solves this by dividing the standard deviation by the mean, often expressed as a percentage. Two investments with average yearly returns of 6 and 9 percent and standard deviations of 3 and 9 percentage points have coefficients of variation of 50 and 100 percent, so the second is twice as volatile relative to its typical return.

Common mistakes

• Forgetting to square the deviations, so that positive and negative values cancel to zero.
• Forgetting the final square root and reporting the variance instead.
• Mixing the sample and population formulas, or pressing the wrong calculator key.
• Rounding the mean heavily before computing deviations, which compounds rounding error.
• Treating a frequency table as if each value appeared once.

Frequently asked questions