Variance Calculator
The mean squared deviation, two formulas deep: the definitional sum and the computational shortcut shown to agree, with the population and sample variances kept honestly apart — and the squared units said out loud.
Variance 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
- No decomposition (between/within) — that is ANOVA ground
In short: Variance is the standard deviation’s square and its source. For 2, 4, 4, 4, 5, 5, 7, 9 the squared deviations sum to 32, so the population variance is 32/8 = 4 exactly and the sample variance is 32/7 ≈ 4.571429. The page computes that sum twice on purpose: the definitional route (subtract the mean, square, add) and the computational shortcut (Σx² − (Σx)²/n, no mean needed) — when they agree, the arithmetic is proven, and when they drift you are looking at floating-point cancellation, which the page names rather than hides.
Formula
σ² = Σ(x−μ)² / N · s² = Σ(x−x̄)² / (n−1)
computational: Σx² − (Σx)²/n
the shortcut needs no mean at all — which is why calculators have always loved it, and why its subtraction deserves a second opinion.
Worked Example
- Definitional. deviations from the mean, squared, summed — the formula the definition actually says.
- Computational. the same sum via Σx² − (Σx)²/n. The page runs both and compares them.
- Divide twice. ÷N for the population, ÷(n−1) for the sample — the same split as the standard deviation page, one square earlier.
2, 4, 4, 4, 5, 5, 7, 9: Σx = 40, Σx² = 212. Computational: 212 − 1600/8 = 32. Definitional: the squared deviations 9+1+1+1+0+0+4+16 = 32. Population variance 4; sample variance 32/7 ≈ 4.571429.
Strengths & Limits Of This Model
Where this engine is strong
- Both formulas computed and compared on your own data
- The units-are-squared card is on the page, not in a footnote
Where it stops
- No weighted or grouped variance
- No probability-distribution variances
Practical Use Cases
Theory work
variance is what adds across independent variables — standard deviations do not
Portfolio and process risk
the square that drives every spread formula downstream
Checking a shortcut
both formulas printed, agreement shown
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.
Variance Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does variance get its own page when it is just a squared standard deviation?
Because statistics runs on variance, not on sigma: independent variances ADD (the foundation of every error analysis), variances decompose into between-group and within-group parts (the foundation of ANOVA), and the correlation coefficient is a variance ratio in disguise. The square root is the display layer; the algebra all happens here.
Why two formulas for the same number?
The definitional formula says what variance MEANS — average squared distance from the mean. The computational one says how to compute it without ever forming the mean, which made it the hand-calculation and early-computer favourite. The page runs both and compares: agreement is a proof, drift is floating-point cancellation and gets named.
Is the computational formula ever wrong?
It is algebraically identical. Numerically it can drift when the mean is huge compared with the spread — subtracting two nearly equal large numbers eats digits. The page prints both values so a drift is visible instead of silent.
Why is the sample variance bigger than the population variance?
On the same list, dividing by the smaller number (n−1) gives the bigger value — 32/7 beats 32/8. That is not a penalty, it is honesty: a sample’s spread around its own fitted mean systematically understates the population’s spread, and Bessel’s correction pays the shortfall back on average.
What are the units of a variance?
Squared — a list in centimetres has its variance in cm². The page says it in the units card because it is the number-one source of nonsense when variances meet raw data. Root it to get back to the data’s units; that rooted value is the standard deviation.
Can a variance be negative?
No — and a negative result from the computational formula is arithmetic noise, not a discovery. It happens when Σx² and (Σx)²/n are nearly equal and cancellation bites. The page floors the comparison at zero and says when the two routes disagreed by more than display rounding.
Do the values have to be a sample or a population?
The page computes both, always. The data does not know what it is — only you know whether the list is everything or a stand-in, and the two cards keep the question in front of you.
How does this connect to the standard deviation page?
Same list, same squared sum, one square root apart. That page is the one to quote next to data (units match); this one is the one to add and decompose (the algebra works). Both print population and sample — the pair travels together.