Standard deviation formula reference

Standard deviation formulas with the data role stated

Choose sample or population before selecting a denominator, preserve exact squared deviations through variance, and return to original units with the square root.

Population variance averages squared deviations from the population mean using N. The usual sample variance estimates population spread using squared deviations from the sample mean and denominator n minus 1. Standard deviation is the nonnegative square root of the matching variance. For Y = a + bX, the standard deviation changes by the absolute value of b while the shift a has no effect.

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Standard deviation formulas PDF

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Population spread

Describe the full population with the population mean and denominator N.

$\sigma^2=\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2$: Population varianceN is a positive integer. Mu is the arithmetic mean of the N population values.
Use population variance when the values represent the complete population being described.
$\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(x_i-\mu)^2}$: Population standard deviationN is a positive integer. Mu is the population mean. Use the nonnegative square root.
Use population standard deviation to express population spread in the original data units.
$\sigma^2=\frac{1}{N}\sum_{i=1}^{N}x_i^2-\mu^2$: Population variance computational formThe N values form the full population being described. Mu is the population mean of the same values.
Use this identity as an independent arithmetic check for population variance while retaining sufficient precision.

Sample spread

Estimate population spread from a sample with the n minus 1 denominator.

$s^2=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2$: Sample variancen is at least two. X bar is the arithmetic mean of the n sample observations.
Use this sample-variance definition when estimating spread from a sample with Bessel's correction.
$s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^2}$: Sample standard deviationn is at least two. X bar is the sample mean. Use the nonnegative square root.
Use sample standard deviation to express sample spread in the original data units.
$s^2=\frac{\sum_{i=1}^{n}x_i^2-n\bar{x}^{\,2}}{n-1}$: Sample variance computational formn is at least two. X bar is the sample mean of the same n observations.
Use this identity to check a sample-variance calculation while keeping exact or high-precision intermediate values.

Changes of scale and location

A shift leaves standard deviation unchanged, while scaling changes it by the absolute scale factor.

$\operatorname{SD}(a+bX)=|b|\operatorname{SD}(X)$: Standard deviation under a linear transformationa and b are constants. The standard deviation of X is finite.
Use this rule when every observation is shifted by a and scaled by b.

Notation

Symbols used on this sheet

Interpret each symbol with the conditions attached to its formula.

Symbol	Meaning
$N$	The number of values in the complete population.
$n$	The number of observations in the sample.
$\mu$	The population arithmetic mean.
$\bar{x}$	The sample arithmetic mean.
$\sigma^2$	Population variance, expressed in squared data units.
$s$	Sample standard deviation, expressed in the original data units.

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Short applications

These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.

Population standard deviation

The values are declared to be the complete population, so use N = 4.

2, 2, 6, 6

\mu=4,\quad\sigma=\sqrt{16/4}=2

Condition check: The computational identity E(X²) - μ² gives 20 - 16 = 4 for the variance.

Sample standard deviation

The sample label selects n minus 1 after the squared deviations total 8.

1, 3, 5

s=\sqrt{8/(3-1)}=2

Condition check: Multiplying s squared by n minus 1 returns 4 times 2 = 8.

Scale and shift

The shift changes location only; the negative scale changes distance by magnitude three.

\sigma_X=4,\quad Y=10-3X

\sigma_Y=|-3|\sigma_X=12

Condition check: The transformed variance is 9 times 16 = 144, whose square root is 12.

Use conditions, not memory alone

Common formula confusions

Using n for a sample estimate

Dividing sample squared deviations by n without explaining a different estimand.

Fix: For the usual sample variance estimator, divide by n minus 1 and require at least two observations.

Confusing variance and standard deviation

Reporting the squared-unit variance as though it were in the data's original units.

Fix: Take the nonnegative square root for standard deviation and label both quantities.

Rounding the mean early

Using a rounded center to calculate every later deviation.

Fix: Keep the center and squared-deviation total exact until the final displayed result.

Adding a shift to SD

Claiming Y = X + 10 increases SD by 10.

Fix: A common shift cancels when each transformed value is compared with the transformed mean.

Common questions

Standard deviation formulas with the data role stated FAQ

What is the sample standard deviation formula?

Subtract the sample mean from each of n observations, square and add the deviations, divide by n minus 1, and take the nonnegative square root.

When should I use population standard deviation?

Use it when the values are the complete population you intend to describe, not merely because the list is large.

Why is variance in squared units?

Its numerator squares each value-minus-mean deviation. The square root used for standard deviation returns the measure to the original units.

How does multiplying every value affect SD?

Multiplying by b changes standard deviation by absolute b. A negative multiplier reverses order but cannot make spread negative.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

Measures of Scale, NIST/SEMATECH e-Handbook of Statistical Methods
Measures of Location, NIST/SEMATECH e-Handbook of Statistical Methods

Written by Mathos AIPublished September 11, 2026