Standard Deviation Formula

The standard deviation formula is a recipe for converting a list of numbers into a single value that describes how spread out they are around the mean. Whichever version you use, the formula does the same thing in steps: find the mean, measure how far each value sits from the mean, square those distances, average them, and take the square root.

The result lands in the same units as your data, which is why standard deviation is easy to interpret. The intermediate value (the average of squared distances) is called the variance, and it is the bridge between the raw data and the final standard deviation.

For a longer plain-language explanation of what standard deviation means and how to interpret it, see What Is Standard Deviation?.

When your data set is the entire group you care about, use the population formula:

It looks intimidating, but it just spells out the recipe: take each value, subtract the population mean, square the difference, sum those squares, divide by the count, and take the square root.

When your data is a sample drawn from a larger population, use the sample formula:

It is identical to the population formula except for two things. The mean is written as x̄ (x-bar) instead of μ, and you divide by n − 1 instead of n. That n − 1 trick is called Bessel's correction.

The two formulas exist because they are doing slightly different jobs:

• Population standard deviation describes the spread of a complete data set. If you measured every member of the group you care about, this is the right formula.
• Sample standard deviation estimates the spread of a population from a sample of it. Dividing by n − 1 instead of n is Bessel's correction, and it is precise about what it fixes: for a random sample of independent observations it makes the sample variance an unbiased estimator of the population variance. Taking the square root afterwards does not preserve that property, so the sample standard deviation still carries a small downward bias. n − 1 is the right default, but it does not make s a perfectly unbiased estimate of σ.

Which formula applies depends on what your numbers are, not on which is safer. Use the population formula when your data is the entire group you are describing; use the sample formula when it is a subset you are generalising from. Spreadsheet and statistics packages commonly expose both — Excel as STDEV.S and STDEV.P — and several default to the sample form, so check which one a tool used before comparing results. The standard deviation calculator reports both side by side so you can pick the one that matches your context. For a fuller comparison with examples and decision rules for picking between them, see Sample vs Population Standard Deviation.

A short reference for every symbol in both formulas:

• σ (sigma). Population standard deviation. The capital Greek letter for “s.”
• s. Sample standard deviation.
• x. An individual value in your data set. The formulas apply the same step to each x in turn.
• μ (mu). Population mean.
• x̄ (x-bar). Sample mean.
• N. Total count in the population.
• n. Total count in the sample.
• n − 1. Sample size minus one. This is Bessel's correction, the only real difference between the population and sample formulas.
• Σ (capital sigma). Sum, applied across all the values.

Once the symbols are familiar, the formulas read like a recipe rather than a wall of math. For a fuller reference covering every symbol with short explanations of what each one represents, see Standard Deviation Symbol.

We will compute standard deviation for this small data set:

The next five sections walk through the calculation one step at a time. We will compute the population standard deviation first because the math is the cleanest, then show how the sample version differs at the end.

Sum every value and divide by the count.

(2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5

The mean is 5.

For every value in the data set, subtract the mean. These differences are called the deviations.

• 2 − 5 = −3
• 4 − 5 = −1
• 4 − 5 = −1
• 4 − 5 = −1
• 5 − 5 = 0
• 5 − 5 = 0
• 7 − 5 = 2
• 9 − 5 = 4

Some deviations are negative (values below the mean) and some are positive (values above the mean). That is expected.

Square each deviation. Squaring does two things at once: it removes the sign of each number (squares are always non-negative), and it gives more weight to values that are far from the mean.

• (−3)² = 9
• (−1)² = 1
• (−1)² = 1
• (−1)² = 1
• (0)² = 0
• (0)² = 0
• (2)² = 4
• (4)² = 16

Sum the squared deviations:

9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32

Variance is the average of the squared deviations. This is where the population and sample formulas diverge:

• Population variance: 32 / 8 = 4
• Sample variance: 32 / (8 − 1) = 32 / 7 ≈ 4.5714

Variance is in squared units, which is why it is rarely interpreted on its own. The square root in the next step brings the answer back into the same units as your data.

For a calculator that reports both variance and standard deviation in one step, see the variance calculator. For a side-by-side look at how the two measures relate and when to use each, see Variance vs Standard Deviation.

Take the square root of the variance.

• Population standard deviation: √4 = 2
• Sample standard deviation: √(32 / 7) ≈ 2.1381

For this data set, values land an average of 2 units (population) or about 2.1381 units (sample) from the mean of 5.

Textbook data usually arrives grouped: a list of values with how many times each one occurs, rather than the raw list. The formula does not change, but every term is weighted by its frequency, so the sum of squared differences becomes Σf(x − x̄)² and the count becomes N = Σf.

Take twenty results scored 2 to 6:

The mean is 74 / 20 = 3.7. The weighted sum of squared differences is 22.20. From there the two variants split on the divisor, exactly as they do for a raw list:

The two answers differ by 0.0274 here, which is small because twenty is a reasonably large N. The gap widens quickly as the sample shrinks: dividing by nineteen instead of twenty is a 5% change in the variance, and dividing by four instead of five would be 25%.

One warning specific to grouped data. If your table gives class intervals rather than single values, such as 0 to 9 and 10 to 19, you have to use the midpoint of each interval as x. The result is then an estimate rather than the exact standard deviation of the underlying data, because the midpoint stands in for values you no longer have.

Doing the math by hand once is a great way to understand the formula. After that, a calculator is faster and less error-prone, especially for longer data sets. Five steps and one square root for every list adds up.

The standard deviation calculator handles parsing (commas, spaces, or one number per line) and reports the mean, plus the standard deviation and variance for whichever variant you select. For more than a handful of numbers, it saves a real amount of arithmetic.

If your data already lives in a spreadsheet, see How to Calculate Standard Deviation in Excel for the STDEV.S and STDEV.P approach.

A few traps that catch people running the formula by hand:

• Skipping the square step. Without squaring, the deviations from the mean cancel out to zero. Squaring is what keeps positive and negative gaps from erasing each other.
• Dividing by N when you should divide by n − 1. For sample data (almost always), use n − 1. Dividing by N gives a slightly biased underestimate of the population spread.
• Forgetting the square root at the end. Without the square root, you have variance, not standard deviation. Variance is in squared units, which is why the square root brings the answer back to the original scale.
• Reporting too many decimal places. The formula is precise; the data usually is not. Two or three significant figures is enough for most contexts.
• Confusing s with σ in the same answer. Pick one, label it clearly. The standard deviation calculator shows both side by side so you can avoid the mismatch.

Standard deviation is also the unit a z score uses. A z score expresses how many standard deviations a single value sits from the mean of its data set, so once you have a standard deviation you can standardize any value against it. The z score calculator takes a raw score, mean, and standard deviation and returns the z score, and the Z Score Formula guide walks through the calculation and what positive, negative, and zero z scores mean.

• Population: σ = √(Σ(x − μ)² / N)
• Sample: s = √(Σ(x − x̄)² / (n − 1))
• The only practical difference is the denominator: divide by N for population, by n − 1 for sample.
• Five-step recipe: mean, distances, squares, average (variance), square root.
• Variance and standard deviation describe the same spread in different units; standard deviation is the square root of variance.
• For a fast number, use the standard deviation calculator.
• For the meaning side, see What Is Standard Deviation?.