Standard Deviation Calculator

Standard deviation formulas

Sample standard deviation formula

Where s is the sample standard deviation, xᵢ is each observation, x̄ (x-bar) is the sample mean, n is the number of observations, and Σ means “sum over all observations”. The denominator n − 1 is Bessel’s correction: deviations are measured from the sample’s own mean, which uses up one degree of freedom and would otherwise bias the spread estimate low.

Population standard deviation formula

Where σ (sigma) is the population standard deviation, μ (mu) is the population mean, and N is the size of the complete population. Because the data set is the entire population, no correction is needed and the denominator is N itself.

Computational (shortcut) formula

Textbooks often teach this algebraically equivalent “shortcut” because it needs only two running totals (Σx and Σx²) on paper. In floating-point arithmetic, however, subtracting two large, nearly equal totals can cancel catastrophically and destroy precision — a dataset like 1 000 000 000 000 ± 2 breaks it badly. CalcDomain does not use the shortcut formula for computation. This page computes with Welford’s numerically stable one-pass algorithm and verifies the result with an independent compensated two-pass method; the shortcut form is shown here only because you will meet it in textbooks.

How to calculate standard deviation by hand

Use the same dataset the calculator loads as its example — 2, 4, 4, 4, 5, 5, 7, 9 — and work through the definition once by hand. (This doubles as this page’s worked example: load it above and the audit table reproduces every line.)

1. Mean: Σx = 2 + 4 + 4 + 4 + 5 + 5 + 7 + 9 = 40, and n = 8, so the mean is 40 ÷ 8 = 5.
2. Deviations: subtract the mean from each value: −3, −1, −1, −1, 0, 0, 2, 4. (They sum to 0 — a built-in check.)
3. Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16.
4. Sum of squares: SS = 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
5. Denominator: treating the eight values as a whole population, divide by N = 8. (As a sample you would divide by n − 1 = 7.)
6. Variance: σ² = 32 ÷ 8 = 4 (sample: s² = 32 ÷ 7 ≈ 4.5714).
7. Square root: σ = √4 = 2 (sample: s ≈ 2.1381).

Sample vs population standard deviation

How to calculate standard deviation depends on one question: is this data set everything, or a slice of something bigger? If you measured every unit of interest — every student in the class you are describing, every machine on the line — you have a population and divide the sum of squared deviations by n. If your numbers stand in for a larger group — a survey, a lab sample, a week of a process that runs all year — you have a sample and divide by n − 1.

The correction exists because a sample’s deviations are measured from its own mean rather than the true population mean, which makes them systematically a little too small. Dividing by n − 1 instead of n compensates, so the sample standard deviation is always slightly larger than the population figure on the same data — you can see both side by side in the comparison table above. As n grows, the two converge.

The symbols track the choice: s and s² (with mean x̄) for samples, σ and σ² (with mean μ) for populations. A sample of one value has no defined standard deviation at all: n − 1 = 0, so this calculator rejects it rather than inventing a number.

What does standard deviation tell you?

Standard deviation answers “how far do values typically sit from the mean?” — in the original units of your data. If the data are centimetres, so is the standard deviation; the variance, by contrast, is in squared units (cm²), which is why the root is the reported figure.

• SD = 0 means every value is identical. It is the minimum possible value; standard deviation can never be negative.
• A small SD (relative to the numbers involved) means values cluster tightly around the mean; a large SD means they spread widely. “Small” and “large” only make sense against the scale of the data — an SD of 5 is enormous for human body temperature in °C and negligible for national salaries.
• Scale dependence: multiplying every value by a constant multiplies the standard deviation by the same constant (adding a constant leaves it unchanged). Comparing SDs across datasets is only meaningful on comparable scales.
• No universal threshold: there is no globally “good” or “bad” standard deviation. Whether spread is acceptable depends entirely on the context — manufacturing tolerances, test-score dispersion and portfolio volatility all judge the same number differently.

Standard deviation and the 68–95–99.7 rule

For data that are approximately normally distributed, about 68% of values fall within one standard deviation of the mean, about 95% within two, and about 99.7% within three — the “empirical rule”.

The 68–95–99.7 rule applies to data that are approximately normally distributed. Standard deviation itself does not require a normal distribution — it is defined for any numeric dataset. Skewed, bimodal or heavy-tailed data have a perfectly valid standard deviation, but the 68/95/99.7 percentages will not hold for them. This page computes your standard deviation without assuming any distribution, and its spread map shows positions, not a bell curve.

Standard deviation vs variance

The two are one square root apart: SD = √variance. Report the standard deviation when you want a human-readable spread; keep the variance when you need to add spreads of independent quantities, because variances add and standard deviations do not.

Standard deviation vs standard error

The standard deviation describes the data: how much individual observations vary. The standard error of the mean describes an estimate: how uncertain the sample mean is as a stand-in for the population mean. Collect more data and the SD settles toward a stable value while the SEM keeps shrinking with √n — which is why error bars of the two kinds look so different on the same measurements.

This calculator reports SEM only in sample mode. When you declare the dataset a complete population, its mean is not an estimate of anything, so a standard error is not applicable — the result panel says so explicitly rather than printing a misleading number.

Reproduce this standard deviation

A result you can’t reproduce is a number you have to trust. Check this page’s output in the tool you already use — put the data in a column/list called A1:A8, data or x and run:

=STDEV.S(A1:A8)   — sample (divides by n − 1)
=STDEV.P(A1:A8)   — population (divides by n)

from statistics import stdev, pstdev

stdev(data)    # sample (n − 1)
pstdev(data)   # population (n)

np.std(data, ddof=1)   # sample (n − 1)
np.std(data, ddof=0)   # population (n, NumPy default)

sd(x)                            # sample (n − 1)
sqrt(mean((x - mean(x))^2))      # population (n)

Different software defaults to different denominators. Always verify whether the function uses n or n − 1. Excel’s and Google Sheets’ STDEV/STDEV.S and R’s sd() are sample functions (n − 1); NumPy’s np.std() defaults to the population convention (ddof=0); Python’s statistics module makes you choose explicitly.

Sources

Methodological references for the definitions, the algorithm and the software reproduction section (these document the method — they are not input datasets):

• B. P. Welford, “Note on a Method for Calculating Corrected Sums of Squares and Products”, Technometrics, Vol. 4, No. 3, 1962 — the primary algorithm used by this calculator
• NIST/SEMATECH e-Handbook of Statistical Methods — measures of scale (standard deviation and variance definitions)
• Python statistics module documentation — stdev() (sample) and pstdev() (population)
• NumPy documentation — numpy.std and the ddof (delta degrees of freedom) parameter
• R documentation — sd() in the stats package (sample convention, denominator n − 1)
• Microsoft Excel documentation — STDEV.S function (sample standard deviation)
• Google Sheets documentation — STDEV function (sample standard deviation)