Standard Deviation Calculator

What standard deviation tells you

The standard deviation (SD, often written σ for a population and s for a sample) measures how far values typically sit from their mean. A small SD means the numbers cluster tightly around the average. A large SD means they are spread out. Because it is expressed in the same units as the data, it is easier to read than the variance, which is in squared units.

Two classes with the same average mark can be very different: in one, nearly everyone scored close to 70, in the other half scored 40 and half scored 100. The mean is identical, the standard deviation is not, and the SD is the number that tells the two classes apart.

How to use the calculator

For continuous data (measurements such as time, revenue, weight or scores), type or paste the values into the Numbers box, separated by commas, spaces, tabs or new lines. Then choose the type:

• Sample if your numbers are a sample drawn from a larger population and you want to estimate that population's spread. It divides by n − 1.
• Population if your numbers are every value that exists. It divides by n.

You get the standard deviation, variance, mean, the standard error of the mean (SEM), the range and the count. At least two values are required.

For proportions (rates), such as a conversion or survival rate, switch the data type, enter the sample size and the observed outcome as a proportion (0.05), a percentage rate (5) or a number of events (10). The result is the standard deviation, variance and standard error of that binary outcome.

How to calculate it by hand

1. Add the values and divide by how many there are to get the mean.
2. Subtract the mean from every value to get the deviations.
3. Square each deviation, so negatives and positives cannot cancel out.
4. Add the squares.
5. Divide the total by n (population) or n − 1 (sample). This is the variance.
6. Take the square root of the variance. This is the standard deviation.

Example with the values 3, 7, 7 and 19. The mean is 36 ÷ 4 = 9. The deviations are −6, −2, −2 and 10, whose squares are 36, 4, 4 and 100, adding up to 144.

For the sample version the standard error of the mean is 6.9282 ÷ √4 = 3.4641, and the range is 3 to 19. Those are the figures the calculator prints for this list.

Standard deviation formulas

Population: σ = √( Σ (xᵢ − μ)² ÷ N )

Sample: s = √( Σ (xᵢ − x̄)² ÷ (n − 1) )

Standard error of the mean: SEM = SD ÷ √n

For a yes/no outcome with probability p, each observation has a variance of p(1 − p), so the standard deviation does not depend on the sample size:

SD = √( p (1 − p) ), SEM = √( p (1 − p) ÷ n )

With p = 0.2 and n = 400 the SD is √0.16 = 0.40, the variance is 0.16 and the SEM is 0.40 ÷ 20 = 0.02. A proportion of 0.5 gives the largest possible SD, 0.50.

Sample or population: which one?

Almost always you hold a sample, so choose Sample. The reason for the n − 1 is that deviations are measured from the sample mean, which sits closer to the sample's own values than the unknown true mean would. Dividing by n would therefore understate the variance on average, and dividing by n − 1 (Bessel's correction) removes that bias.

Choose Population only when you really have every value, such as the final marks of all students in one class that you are describing and not generalising from. For small data sets the choice matters a lot; for hundreds of values the two answers are nearly identical.

How to interpret the standard deviation

For data that follow a normal (bell-shaped) distribution, the proportion of values within a given number of standard deviations of the mean is fixed:

For any distribution at all, Chebyshev's inequality still guarantees that at least 75% of values lie within 2 SDs and at least 88.9% within 3 SDs. Whether an SD is "large" depends on the mean: a standard deviation of 5 is huge for heights in metres and tiny for annual incomes. The coefficient of variation divides SD by the mean to make that comparison fair.

To place one value on this scale, compute its z-score: the number of standard deviations it lies from the mean.

Common mistakes

• Mixing up SD and SEM. The SD describes how spread out individual values are; the SEM describes how precisely the mean is estimated. It shrinks as the sample grows, while the SD does not.
• Using the population formula on a sample, which gives a result that is too small, especially for fewer than 30 values.
• Ignoring outliers. One extreme value inflates the SD because deviations are squared.
• Assuming normality. The 68-95-99.7 pattern only holds for roughly bell-shaped data.
• Typing percentages as proportions. In proportion mode a rate of 5% is entered as 5 with the rate option, or as 0.05 with the proportion option.

To work with the squared version of the same measure, see the variance calculator. Quoting a result with its uncertainty is covered by the confidence interval calculator.