Standard deviation calculator

How do I calculate standard deviation?

Find the mean, subtract it from each value and square the differences, add them up, divide by n (population) or n − 1 (sample), and take the square root. For 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the squared differences add to 32, 32 ÷ 8 = 4, and √4 = 2.

Sample or population — which should I use?

Population if your numbers are the whole group you care about; sample if they are a sample from a bigger group. The sample version divides by n − 1, which gives 2.14 for the list above instead of 2.

What is variance?

The standard deviation squared — the average squared distance from the mean. It is 4 (population) or 4.57 (sample) for the list above.

What does standard deviation tell me?

How spread out the values are. For data shaped like a bell curve, about 68% of values fall within 1 SD of the mean and about 95% within 2 SD.

Why n − 1 for a sample?

A sample's own mean sits closer to its values than the true mean does, so dividing by n underestimates the spread. Dividing by n − 1 (Bessel's correction) makes the variance estimate unbiased.

What is a "good" or "high" standard deviation?

There is no universal threshold, because the standard deviation is in the same units as the data. A spread of 8 points is large for a 10-point quiz and small for a 1,000-point exam. To compare spread between groups with different means or units, divide by the mean to get the coefficient of variation: 10.11 ÷ 79.50 = 12.7% for the scores in the worked example.

How does one extreme value change the standard deviation?

Because the differences are squared, outliers have an outsized effect. Adding a single score of 20 to the six scores in the worked example moves the mean from 79.50 to 71.00 and the sample standard deviation from 10.11 to 24.31.

The formulas

The population standard deviation is σ = √(Σ(x − μ)² ÷ N), and the sample standard deviation is s = √(Σ(x − x̄)² ÷ (n − 1)). In both, you subtract the mean from each value, square the differences so that values below and above the mean do not cancel, add them up, divide, and take the square root to return to the original units. The quantity under the root is the variance. Both versions are shown by the calculator; the large number at the top is the sample version, which is the right one when your numbers are a sample from something larger.

Worked example: six test scores

The scores are 72, 85, 90, 64, 78, 88. The mean is 79.50. The deviations from the mean are -7.50, 5.50, 10.50, -15.50, -1.50, 8.50. Squaring and adding them gives 511.50. For the sample version divide by n − 1 = 5: variance 102.30, standard deviation 10.11. For the population version divide by n = 6: the standard deviation is 9.23. The population value is always smaller, and the gap shrinks as n grows.

How to read it: the empirical rule

For data that are roughly bell-shaped, about 68% of the values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three (OpenStax Introductory Statistics, section 2.7). For the scores above, one standard deviation on each side of 79.50 spans 69.4 to 89.6. The rule describes a normal distribution; it fails for strongly skewed data or for very small samples, where the percentages can differ a lot.

Common mistakes

Using the population formula for a sample understates the spread, especially for small n. Entering only some of the data, such as a single column of a table, answers a different question. Mixing units, such as hours and minutes, in one list makes the result meaningless. The standard deviation also says nothing about shape: two lists with the same value can be bell-shaped or have two separate clumps. Finally, do not confuse it with the standard error, which is the standard deviation divided by √n and describes the precision of the mean rather than the spread of the data.

Sources: OpenStax Introductory Statistics 2e, chapter 2 (section 2.7, measures of the spread of the data; the empirical rule) and chapter 7; NIST/SEMATECH e-Handbook of Statistical Methods on the standard deviation and the n − 1 divisor. Related: standard error calculator, z-score calculator and average calculator.