Standard deviation calculator

What is standard deviation?

Standard deviation measures how far values usually sit from their mean. A small standard deviation means the numbers are bunched close to the average. A large one means they are spread out.

Two classes can both average 80 on a test. If one class has a standard deviation of 3, nearly everyone scored between about 74 and 86. If the other has a standard deviation of 15, the scores run from the 50s to the high 90s. The mean alone hides that difference.

It is reported in the same units as your data, so a standard deviation of 9.45 points on a test means 9.45 points. That is the main reason people report it instead of the variance, which is in squared units.

Standard deviation formula

There are two versions. Use the sample formula when your numbers are a subset of a larger group, and the population formula when you have every member of the group.

Sample: s = √( Σ(x − x̄)² / (n − 1) )

Population: σ = √( Σ(x − μ)² / N )

Here x is each value, x̄ or μ is the mean, and n or N is how many values there are. The only difference is the divisor. The National Library of Medicine’s statistics course works through the population version using the heights of nine students.

How to find standard deviation

To find the standard deviation of a data set, you need the mean first. Every step after that measures how far the values sit from the mean. This is how to calculate it by hand:

1. Find the mean: add the values and divide by how many there are.
2. Subtract the mean from each value.
3. Square each of those differences so negatives do not cancel positives.
4. Add the squares. This total is the sum of squares.
5. Divide by n − 1 for a sample or by N for a population. The result is the variance.
6. Take the square root of the variance.

Worked example

Take the eight values 2, 4, 4, 4, 5, 5, 7 and 9.

• Mean: 40 / 8 = 5
• Differences from the mean: −3, −1, −1, −1, 0, 0, 2, 4
• Squared: 9, 1, 1, 1, 0, 0, 4, 16
• Sum of squares: 32
• Population variance: 32 / 8 = 4, so σ = √4 = 2
• Sample variance: 32 / 7 = 4.5714, so s = √4.5714 = 2.1381

Click “Small example” in the calculator to see the same working laid out step by step.

Shortcut formula

Many textbooks also give a version that skips the individual differences and works from the sum of the values and the sum of their squares:

s² = ( Σx² − (Σx)² / n ) / (n − 1)

For the same eight values, Σx = 40 and Σx² = 4 + 16 + 16 + 16 + 25 + 25 + 49 + 81 = 232. That gives s² = (232 − 1,600 / 8) / 7 = 32 / 7 = 4.5714, the same answer as before. It is faster on paper, but with large values that sit close together it can lose precision, so the calculator uses the two-step method.

Sample vs population standard deviation

Most real data is a sample. Ten test scores from one class, 30 customer orders from last week or eight months of fund returns are all samples if you want to say something about the wider group they came from. Use s.

Use σ only when the data is the whole group you care about and you are not generalizing past it, such as the final grades of every student in a class of 25 when you only want to describe that class.

Dividing by n − 1 is called Bessel’s correction, and that n − 1 is the degrees of freedom of the sample. A sample’s values sit a little closer to their own mean than to the true population mean, so dividing by n would understate the spread. With large data sets the gap is tiny: for 1,000 values the two results differ by about 0.05%. With 5 values the sample figure is about 12% larger.

If you are not sure, pick sample. The “Data set is” dropdown switches the tool between a sample standard deviation calculator and a population standard deviation calculator, and the results show the other value too, so you can compare.

Standard deviation of a frequency table

When a value repeats many times, it is quicker to enter each value once with its count. The formula weights each squared difference by its frequency f:

s = √( Σf(x − x̄)² / (n − 1) ), where n = Σf

For 60 dice rolls where 1 came up 8 times, 2 came up 11 times, 3 came up 9 times, 4 came up 12 times, 5 came up 10 times and 6 came up 10 times, the mean is 215 / 60 = 3.5833. The sum of squares is 164.5833, the sample variance is 164.5833 / 59 = 2.7895 and s = 1.6702. Switch to the Frequency table tab and click “Dice rolls” to check it.

The mean step is the same idea as a weighted average, with the counts acting as the weights. The same counts also go straight into the histogram maker if you want to see the spread rather than read it off a single number.

How to interpret standard deviation

What a standard deviation tells you depends on the shape of your data. For data that follows a bell-shaped normal distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. Drawn out, that picture is the standard deviation graph you have seen in textbooks, and the exact areas are 68.27%, 95.45% and 99.73%. The test scores sample has a mean of 80 and s = 9.45, so a typical score sits somewhere between about 70.5 and 89.5. Six of the ten scores do.

The z score in the chart under the calculator tells you how many standard deviations each value is from the mean. A score of 95 has z = 1.59. Values beyond about 2 or 3 are unusual in normal data and worth a second look. Those 68, 95 and 99.7 figures are probabilities, and the probability calculator will combine one of them with a second event if you need the chance of both, or of either.

Standard deviation is pulled up by outliers because each difference is squared. One extreme value can double it, so it is worth checking whether that value counts as an outlier before you report anything. If your data is skewed, report the median alongside the mean, as covered in our guide to mean vs median.

To compare spread between data sets with different scales, use the coefficient of variation, which is the standard deviation divided by the mean. The calculator shows it as a percentage.

Standard deviation, variance and standard error

Variance is the standard deviation squared, and our variance calculator shows it as the main result. It is what you get at the step before the square root, and it is in squared units, such as points squared. Statistical tests often work with variance, while reports usually quote the standard deviation.

Standard error is the standard deviation divided by the square root of n, and some books call it the standard deviation of the mean. It describes how precise your sample mean is, not how spread out the individual values are. For the test scores, s = 9.45 and the standard error is 9.45 / √10 = 2.99. Collect more data and the standard error shrinks, while the standard deviation stays about the same.

Standard deviation in Excel and Google Sheets

Both programs have a separate function for each version:

=STDEV.S(A2:A11) sample

=STDEV.P(A2:A11) population

Microsoft’s pages for STDEV.S and STDEV.P explain that both skip text and empty cells in a range. Google Sheets also accepts the older names STDEV and STDEVP, described in its STDEV help page. In our test of 20 data sets, Excel and Google Sheets returned matching results on every case we could compare, STDEV.S included.

If your rows are a sample, use STDEV.S. For 2, 4, 4, 4, 5, 5, 7 and 9, STDEV.P returns 2 and STDEV.S returns 2.1381, so the choice changes the answer even on a short list.