Standard Deviation Calculator

What Is Standard Deviation?

Standard deviation tells you how spread out a set of numbers is. A small value means the data clusters tightly around the mean. A large value means it is scattered widely. It is the most common way of putting a single number on the idea of consistency.

The mean on its own can hide a great deal. The sets 49, 50, 51 and 10, 50, 90 share a mean of exactly 50, yet they describe completely different situations. Standard deviation separates them: the first has a spread of about 1, the second about 40. That is the gap the mean cannot see, and it is often the more useful piece of information.

The Standard Deviation Formula

There are two versions of the formula. Which one you use depends on whether your numbers are a sample or an entire population.

The only difference is the divisor: n minus 1 for a sample, n for a population. Everything else is identical.

Step by Step

Take the data set 2, 4, 4, 4, 5, 5, 7, 9. First find the mean: the values total 40 and there are 8 of them, so the mean is 5. Next, subtract the mean from each value to get the deviations: −3, −1, −1, −1, 0, 0, 2, 4. Square each one: 9, 1, 1, 1, 0, 0, 4, 16. Add those squares together to get 32.

Now divide. For a population, 32 divided by 8 gives a variance of 4, and the square root is a standard deviation of exactly 2. For a sample, 32 divided by 7 gives a variance of about 4.571, and the square root is roughly 2.138. Both answers are correct; they simply answer different questions.

Sample or Population?

This is the decision people get wrong most often, so it is worth being precise. Use the population formula when your numbers are the complete group you are describing and you have no intention of generalising beyond it. Use the sample formula when your numbers are a subset and you want them to say something about the larger group they came from.

If you measure the height of all 30 students in one class and only care about that class, that is a population. If you measure 30 students in order to say something about the whole school, the same 30 numbers are a sample. The data has not changed; your intent has.

Why n Minus 1?

A sample almost never captures the most extreme members of its population, so it looks more tightly packed than the real thing. Dividing by n would inherit that bias and understate the spread. Dividing by n minus 1, a step known as Bessel's correction, nudges the answer upward by just enough to compensate. With 5 data points the correction is substantial; with 500 it barely registers.

How to Calculate Standard Deviation on a Calculator

Plenty of people arrive here needing to do this on a physical calculator during an exam, where the tool above is not an option. Both major families handle it in a few keystrokes.

On a Casio fx-Series

Press MODE and select STAT, then choose the single-variable option. Type each value into the table, pressing equals after each one. When the data is in, press AC to leave the editor, then SHIFT followed by 1 to open the STAT menu, and pick Var. You will see sx for the sample standard deviation and σx for the population figure, along with the mean as x̄.

On a TI-83 or TI-84

Press STAT and choose Edit, then enter your values down the L1 column. Press STAT again, move across to the CALC tab, and select 1-Var Stats followed by Enter. The output lists Sx as the sample standard deviation and σx as the population one, with x̄ giving the mean and n confirming your count.

In Excel or Google Sheets

Use STDEV.S for a sample and STDEV.P for a population, passing the range of cells as the argument, for example STDEV.S(A1:A20). The older STDEV function is an alias for the sample version and still works, but the explicit names make your intent clear to anyone reading the spreadsheet later. VAR.S and VAR.P return the corresponding variances.

Variance and Standard Deviation

Variance is the step immediately before standard deviation: it is the average of the squared deviations, and the standard deviation is simply its square root. Because the deviations are squared, variance is expressed in squared units, which is awkward to interpret. A variance of 16 square centimetres does not describe anything you can picture, whereas a standard deviation of 4 centimetres does.

Variance still earns its place in statistical theory, where the fact that variances of independent variables can be added together makes the algebra far cleaner. For reporting a result to a reader, standard deviation almost always communicates better. The calculator shows both.

Reading the Result

Standard deviation carries the same units as your data, so there is no universal cutoff for what counts as large. The usual way to make it comparable is to divide it by the mean, giving the coefficient of variation. Because that ratio has no units, you can use it to compare the consistency of two data sets that measure completely different quantities.

When the data is approximately bell-shaped, the empirical rule offers a quick sense of scale: about 68 per cent of values sit within one standard deviation of the mean, about 95 per cent within two, and about 99.7 per cent within three. The shaded band on the chart above marks that first interval. Do bear in mind the rule assumes a roughly normal distribution and does not hold for heavily skewed data.

Where Standard Deviation Is Used

In finance it is the standard measure of volatility, describing how much an asset's returns swing around their average and feeding directly into risk models. In manufacturing it underpins quality control, where a process that drifts is caught by a rising standard deviation long before any individual part fails inspection.

Researchers report it alongside the mean so readers can judge how representative that mean really is, and it is the raw material for confidence intervals and significance tests. In education it shows whether a class performed uniformly or split into distinct groups, which a single average score would completely conceal.

Common Mistakes

Mixing up the two formulas. Using the population divisor on sample data understates the spread. On small data sets the discrepancy is large enough to change a conclusion, so decide which one applies before you calculate.

Forgetting the square root. Stopping at the variance and calling it the standard deviation is easily done and gives a number in the wrong units entirely.

Comparing across different units. A standard deviation in dollars cannot be meaningfully compared with one in kilograms. Use the coefficient of variation when you need to compare spread across unlike data sets.

Applying the empirical rule to skewed data. The 68-95-99.7 figures describe a normal distribution. Income data, waiting times and many other real-world data sets are strongly skewed, and the rule will mislead you if you apply it without checking the shape first.