The standard deviation formula

The two formulas

Sample standard deviation — use this when your numbers are a subset drawn from a larger group:

s = √[ Σ(x − x̄)² / (n − 1) ]

Population standard deviation — use this when your numbers are the entire group:

σ = √[ Σ(x − μ)² / N ]

Read either from the inside out: find the mean, measure how far each value sits from it, square those distances, average them, then undo the squaring. The only difference is that the sample version divides by n − 1 where the population version divides by N. If you are working from a sample — which in coursework you almost always are — use the n − 1 version.

Two shortcuts worth knowing before the detail. Standard deviation is simply the square root of the variance, so if somebody has already handed you s² you are one operation from the answer: s = √(s²). And in a spreadsheet the whole thing collapses to =STDEV.S(A1:A100) for a sample or =STDEV.P(A1:A100) for a population.

Every term, explained

Why square the deviations at all?

Try averaging the raw deviations on 2, 4, 6. The mean is 4, so the deviations are −2, 0 and +2, and they sum to zero. That is not a coincidence of this example: the deviations from the mean sum to zero for every dataset, by the definition of the mean. An unsquared average of them would report zero spread for all data, which is useless.

Squaring solves that by making every term non-negative. It also has the side effect of weighting a deviation of 10 as a hundred times more significant than a deviation of 1, rather than ten times — which is why a single outlier can move a standard deviation so sharply. Taking the square root at the end undoes the inflation of scale, but not that reweighting.

Why n − 1 for a sample?

A sample's mean is calculated from that sample, so it sits as close to those particular numbers as any single point can. The true population mean is almost always somewhere slightly else, and further from your data on average. Measuring spread around the sample's own too-flattering centre therefore understates the population's real spread every time — the bias is systematic, not random.

Dividing by n − 1 instead of n inflates the result by exactly enough to cancel that bias out. The correction is named for Friedrich Bessel, and its size depends entirely on n:

For a large sample the choice barely matters. For a small one it matters a great deal, which is exactly when people are most likely to be working by hand and most likely to pick wrong.

A worked example, both ways

Data: 2, 4, 4, 4, 5, 5, 7, 9.

1. Sum = 40, count = 8, so the mean is 5.
2. Deviations: −3, −1, −1, −1, 0, 0, 2, 4.
3. Squared: 9, 1, 1, 1, 0, 0, 4, 16. Their sum, Σ(x − x̄)², is 32.
4. As a population: 32 ÷ 8 = 4, and √4 = σ = 2 exactly.
   As a sample: 32 ÷ 7 = 4.5714, and √4.5714 = s = 2.1381.

This is the standard textbook example precisely because the population answer comes out as a whole number. Paste it into the calculator above and toggle the mode to watch only the denominator change.

The shortcut formula, and why not to use it

Older textbooks give a one-pass rearrangement that avoids computing the mean first:

s² = [ Σx² − (Σx)² / n ] / (n − 1)

Algebraically it is identical. Numerically it is not. It subtracts two large, nearly equal quantities, and in floating-point arithmetic that step throws away most of the significant digits. On the values 1000000.1, 1000000.2, 1000000.3 — whose true variance is exactly 0.01 — the shortcut can return zero or even a negative number, which is impossible for a variance. The formulas at the top of this page, computed with Welford's algorithm, do not have this failure mode.

Every standard deviation formula, in one table

The same idea rearranged for whatever you happen to have — raw values, a frequency table, several groups, or a variance somebody already computed.

The formula for grouped data

When the data arrives as a frequency table rather than a list, you never see the individual values — only that some midpoint x occurred f times. Each squared deviation is therefore weighted by its frequency, and the count becomes Σf:

s = √[ Σf(x − x̄)² / (Σf − 1) ]  where  x̄ = Σfx / Σf

The answer is an approximation, because treating every value in a class as sitting exactly at the midpoint is a fiction — real values are spread across the interval. Expect it to run slightly low for heavily skewed classes. The grouped-data calculator shows the fx and fx² columns as it goes.

The formula to copy and paste (Word, Google Docs)

Plain Unicode, one line each — safe in Word, Google Docs, email or a comment:

• s = √(Σ(x − x̄)² / (n − 1))
• σ = √(Σ(x − μ)² / N)

In LaTeX:

• s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
• \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}

For a properly typeset version, both editors have an equation tool that accepts this syntax. In Google Docs choose Insert → Equation and type \sigma, \sqrt or \sum followed by a space; each turns into its symbol. In Word, press Alt+= to open an equation and paste the LaTeX line above — current versions of Word convert it when you choose LaTeX in the Equation ribbon.

The notation guide has the individual characters and how to type each one.

The same formula in Excel, Python and R

Every one of these applies the formulas at the top of this page; they differ only in which denominator they pick by default, and that default is the single most common source of a wrong answer.

NumPy and pandas disagreeing on the default is a genuine trap: the same column run through both returns two different numbers unless you set ddof explicitly. R has no population function at all, so you rescale sd() by hand. The Excel, Python and R guides work through each in full.