Descriptive statistics Updated November 2026

Standard deviation

Standard deviation measures how far values typically sit from the mean. Subtract the mean from each value, square the results, average them (dividing by n - 1 for a sample or n for a population) and take the square root.

Syntax

s = sqrt( sum((x - mean)^2) / (n - 1) )

Part	What it is
`x`	each value
`mean`	the average of the values
`n`	the number of values; use n - 1 for a sample, n for a whole population

Examples

Data: 2, 4, 4, 4, 5, 5, 7, 9  (n = 8, mean = 5)
squared deviations: 9, 1, 1, 1, 0, 0, 4, 16  (sum = 32)
population sd = sqrt(32 / 8) = sqrt(4)

Result: 2

Excel: =STDEV.P(A1:A8)

Same data as a sample
sample sd = sqrt(32 / 7) = sqrt(4.5714)

Result: 2.138 (rounded)

Excel: =STDEV.S(A1:A8)

Data: 1, 2, 3, 4, 5  (mean = 3)
squared deviations: 4, 1, 0, 1, 4  (sum = 10)
sample sd = sqrt(10 / 4)

Result: 1.581 (rounded)

Common mistakes

Using STDEV.P on a sample understates the spread; use STDEV.S (n - 1) unless you truly have every member of the population.
Forgetting to square the deviations lets positives and negatives cancel to zero; square first, then average.
Reporting the variance as the standard deviation; the variance is in squared units, so take the square root.
Comparing standard deviations of data measured on different scales; use the coefficient of variation instead.

Practice Standard deviation

These drills check your answer on live data, free, no account needed.

Sample standard deviationA random sample of 6 minutes to resolve a support ticket (minutes): 46, 35, 28, 23, 26, 17. What is the sample standard deviation? Round to 2 decimal places.
Population standard deviationThese are the figures for every store in a small retail chain, so they are the whole population, not a sample. Weekly revenue in thousands of dollars: 63, 50, 42, 37, 39. What is the population standard deviation? Round to 2 decimal places.

Questions

What is the difference between STDEV.S and STDEV.P in Excel?

STDEV.S divides by n - 1 and estimates the spread of a population from a sample. STDEV.P divides by n and describes the spread of the exact data you have. STDEV.S is always slightly larger.

What does a standard deviation of 0 mean?

Every value is identical. The larger the standard deviation, the more spread out the values are around the mean.

Why divide by n - 1 for a sample?

Deviations are measured from the sample mean, which sits closer to the sample values than the true population mean does, so the squared deviations are a little too small. Dividing by n - 1 (Bessel's correction) removes that bias.

VarianceHow to calculate variance step by step, why the sample version divides by n - 1, how it relates to standard deviation, and Excel VAR.S and VAR.P.
z-scoreThe z-score formula z = (x - mean) / sd, worked examples, how to read a z-score as a percentile with the normal table, and the Excel function STANDARDIZE.
Coefficient of variationThe coefficient of variation (CV) is the standard deviation divided by the mean, as a percent. Formula, worked examples, Excel and when it misleads.
Empirical rule (68-95-99.7)The empirical rule: 68% of normal data lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Worked examples for intervals and tails.