Standard Deviation Calculator
One list, both answers: the population standard deviation ÷N and the sample standard deviation ÷(n−1) side by side, with the deviations laid out and the divisor question settled in the open.
Standard Deviation Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- 2 to 500 values; univariate only
- Both divisors printed — choosing between them is your question to answer
In short: For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the deviations are −3, −1, −1, −1, 0, 0, 2, 4 — squaring gives 32, so the population standard deviation is √(32/8) = 2 exactly, while the sample standard deviation divides by 7 instead: √(32/7) ≈ 2.138090. Neither is the “real” one — they answer different questions. If your eight numbers ARE the whole population, use ÷N; if they are a sample meant to speak for a larger one, ÷(n−1) corrects the underestimate that squaring-and-averaging an unknown mean bakes in.
Formula
σ = √(Σ(x−μ)² / N) · s = √(Σ(x−x̄)² / (n−1))
the two formulas differ only in the divisor — and the divisor is the entire statistical argument.
Worked Example
- Center. the mean x̄. Every deviation is measured from it, which is why the deviations must sum to exactly zero — the page checks that identity live.
- Square and sum. the squared deviations cannot cancel, which is the whole point of squaring: direction is noise here, size is the signal.
- Divide and root. ÷N if the list is the entire population, ÷(n−1) if it is a sample. Both are printed so the choice is always visible.
2, 4, 4, 4, 5, 5, 7, 9: SS = 9+1+1+1+0+0+4+16 = 32. Population: 32/8 = 4, σ = 2. Sample: 32/7 ≈ 4.571429, s ≈ 2.138090.
Strengths & Limits Of This Model
Where this engine is strong
- Population and sample results side by side, never merged
- The deviations table shows every squared term
Where it stops
- No grouped-data mode (frequency tables live on the grouped pages)
- No trimmed or winsorised variants
Practical Use Cases
Lab replicates
sample sd is what a replicate set is for
Manufacturing
population sd when you truly hold every unit’s measurement
Coursework
both printed, so the grader’s convention cannot surprise you
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Standard Deviation Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why two different divisors?
÷N measures the spread of the list you typed, full stop. ÷(n−1) estimates the spread of a larger population from a sample, and it divides by n−1 because the deviations were measured from the sample’s own mean — which already absorbed one degree of freedom. The correction is called Bessel’s, and the page prints both results so you never inherit a hidden convention.
Which one should I use?
Ask one question: is my list the whole story or a stand-in? Census, exam hall, full production run — ÷N. A poll, a replicate set, eight parts from a thousand — ÷(n−1). When in doubt in statistics coursework, the sample version is almost always the one being asked for.
Why do the deviations have to sum to zero?
Because the mean is the balance point: what sits below must exactly offset what sits above. The page computes the sum and shows it is 0 — if rounding ever left a residue it would be displayed, not hidden. It is the cheapest possible check that the mean was computed right.
Why square instead of just averaging the deviations?
Because the raw deviations sum to zero by construction — averaging them always gives 0 and tells you nothing. Squaring makes every contribution positive so the spread cannot cancel itself away. Absolute values would work too (that is the mean absolute deviation) but they are harder algebra, and the square is what the normal distribution is built on.
What happens to the units?
Squaring squares them: a list in centimetres gives a variance in cm² and a standard deviation back in cm. That round trip — square to kill the signs, root to recover the units — is why the standard deviation, not the variance, is the number you can hold up against the data.
How big a list can it take?
Five hundred values, comfortably. The arithmetic is one pass for the mean and one for the squares; precision is protected by the computational identity, and a two-value list is refused as a spread with no freedom (n−1 = 1 is fine, n = 1 is not a spread question).
Is this the same standard deviation Excel calls STDEV?
Excel’s STDEV is the sample one (÷n−1) and STDEVP is the population one (÷N). The page prints both labelled, which is exactly the guard against the classic spreadsheet mix-up.
Does an outlier change the answer much?
Yes — and that is informative, not a bug. Squared deviations weight a far point by the SQUARE of its distance, so one wild value can dominate the sum. The deviations table prints every term, so you can see which value is doing the damage.