Statistics
Variance and standard deviation calculator
One list of numbers, four answers: the sample variance and sample standard deviation, and the population variance and population standard deviation. The headline follows the mode you pick; the other mode's figure is printed underneath, and the SD is in the first row of results.
Separate numbers with commas, spaces or new lines, or paste a spreadsheet column.
Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.
Your results will appear here: the standard deviation first, then the rest of the summary and a chart.
Variance (sample)
16.9444
The variance is in squared units, so it is hard to read on its own. Its square root is the standard deviation, 4.116: your values typically sit about that far above or below their mean of 28.78.
Population variance (σ²): 15.0617, if these values are the whole group.
- Count (n)
- 9
- Mean (x̄)
- 28.7778
- Standard deviation (s)
- 4.11636
- Standard error
- 1.37212
- Minimum
- 22
- Q1 (25%)
- 27
- Median
- 29
- Q3 (75%)
- 31
- Maximum
- 35
- Range
- 13
More statistics (5)
- Relative SD (%RSD)
- 14.304%
- Coefficient of variation
- 0.14304
- Sum (Σx)
- 259
- Sum of squares, Σ(x − x̄)²
- 135.556
- IQR (Q3 − Q1)
- 4
Data distribution
Shaded bands mark ±1, ±2 and ±3 SD from the mean. 5 of 9 values (56%) fall within ±1 SD.
Chart as text
Mean 28.7778, sample standard deviation s = 4.11636, from 9 values between 22 and 35.
- Within ±1 SD (24.66 to 32.89): 5 of 9 values (56%). About 68% for normal data.
- Within ±2 SD: 9 (100%). About 95% for normal data.
- Within ±3 SD: 9 (100%). About 99.7% for normal data.
Show the working, step by step
All four figures for one data set
Nine commute times in minutes, 24, 31, 28, 35, 22, 30, 27, 33, 29, have a mean
of 28.7778 and a sum of squared deviations Σ(x − x̄)² = 135.556. Everything else follows from
those two numbers and the choice of denominator:
| Variance (min²) | Standard deviation (min) | |
|---|---|---|
| Sample (÷ n − 1 = 8) | s² = 16.9444 | s = 4.11636 |
| Population (÷ n = 9) | σ² = 15.0617 | σ = 3.88094 |
Read across a row and you move between variance and SD by squaring or rooting. Read down a column and you move between sample and population by rescaling by (n − 1)/n. Those are the only two relationships among the four.
The formulas side by side
s² = Σ(x − x̄)² / (n − 1) s = √s² σ² = Σ(x − μ)² / N σ = √σ²
Converting between sample and population without the raw data:
σ² = s² × (n − 1) / n s² = σ² × n / (n − 1)
For the commute data, 16.9444 × 8/9 = 15.0617, matching the table.
Units: why the variance looks so large
A variance of 16.9 next to a mean of 28.8 minutes can look alarming until you notice its units are minutes squared. It does not say commutes vary by 17 minutes; the SD says they typically vary by about 4. The same trap catches people with money: a variance in dollars² is not a dollar amount, and comparing it with a price is meaningless.
Units also explain why the variance is still used. Squared units add cleanly: if the drive to the station and the train ride vary independently, the variance of the whole journey is the sum of the two variances. That is what makes variance the working currency of ANOVA, regression and error propagation, while the SD is the number for people.
Choosing sample or population
If these nine commutes are a sample of the trips you will make this year, use s and s². If they are the only nine commutes you are describing — a week-and-a-bit you are reporting on and nothing more — use σ and σ². When in doubt, the sample version is the safer default, and the sample vs population guide explains the reasoning. Dedicated pages open in each mode: sample variance, population variance, sample SD and population SD.
For variance on its own, the variance calculator is the general tool; standard deviation vs variance covers when each is the right report.
Related calculators
-
Variance calculator
The same calculation with variance as the only headline.
-
Standard deviation calculator
SD first, with a dot plot and full working.
-
SD vs variance
When each is the right thing to report.
-
Mean and variance
The two parameters that pin down a distribution.
Common questions
How do I convert variance to standard deviation?
Take the square root. A variance of 16.9444 min² is a standard deviation of √16.9444 = 4.11636 min. Going the other way, square the standard deviation.
Why is variance in squared units?
Because it averages squared deviations. If the data is in minutes, each deviation is in minutes and each squared deviation is in minutes squared. The square root in the standard deviation undoes that, which is why the SD is the one people quote.
Which of the four numbers should I report?
Usually the sample standard deviation, since most data is a sample and the SD is in the data's own units. Report a variance when it feeds into something else — an ANOVA table, a pooled estimate, a variance of a sum — and the population versions only when the data is the entire group.
Does the square root of the sample variance equal the sample SD?
Yes — the sample SD is defined as √s². The same holds for the population pair, σ = √σ². What you cannot do is mix them: the square root of the population variance is not the sample SD.
Can I add standard deviations?
Not directly. For independent quantities it is the variances that add: Var(X + Y) = Var(X) + Var(Y). Add the variances, then take the square root. Two independent steps with SDs of 3 and 4 minutes give a total SD of 5 minutes, not 7.