Variance Calculator
Enter a data set to find the sample variance and population variance, with the sum of squares and every step of the working.
- s² and σ²
- Sum of squares
- Step-by-step
- Paste from Excel
Your data
7 values.
Results
Sample variance (s²)
5.8095
Population variance (σ²)
4.9796
Sum of squares (SS)
34.8571
Standard deviation (s)
2.4103
Center
- Count (n)
- 7
- Sum (Σx)
- 99
- Mean
- 14.1429
- Median
- 14
- Mode
- No mode
Spread
- Sample standard deviation (s)
- 2.4103
- Population standard deviation (σ)
- 2.2315
- Sample variance (s²)
- 5.8095
- Population variance (σ²)
- 4.9796
- Sum of squares (SS)
- 34.8571
- Mean absolute deviation
- 1.8776
- Standard error of the mean
- 0.911
- Coefficient of variation (s ÷ mean)
- 17.04%
- Minimum
- 11
- Maximum
- 18
- Range
- 7
Quartiles and outliers
- Q1
- 12
- Q3
- 16
- Interquartile range (IQR)
- 4
- Five number summary
- 11, 12, 14, 16, 18
- Outlier fences
- 6 and 22
- Outliers (1.5 × IQR)
- None
Empirical rule (using s)
- Within 1 SD (68% if normal)
- 5 of 7 (71.4%)
- Within 2 SD (95% if normal)
- 7 of 7 (100%)
- Within 3 SD (99.7% if normal)
- 7 of 7 (100%)
Step-by-step working
Count the values
n = 7
Find the mean
x̄ = Σx ÷ n
Σx = 12 + 15 + 11 + 18 + 14 + 16 + 13 = 99
x̄ = 99 ÷ 7 = 14.1429
Subtract the mean from each value and square the result
(x − x̄)²
(12 − 14.1429)² = (−2.1429)² = 4.5918
(15 − 14.1429)² = (0.8571)² = 0.7347
(11 − 14.1429)² = (−3.1429)² = 9.8776
…and the same for every value. The table lists them all.
Add up the squared deviations (sum of squares)
SS = Σ(x − x̄)²
SS = 4.5918 + 0.7347 + 9.8776 + 14.8776 + 0.0204 + 3.449 + 1.3061 = 34.8571
Divide by n − 1 to get the sample variance
s² = SS ÷ (n − 1)
s² = 34.8571 ÷ 6 = 5.8095
Optional: take the square root for the standard deviation
s = √s²
s = √5.8095 = 2.4103
| # | x | x − x̄ | (x − x̄)² | z |
|---|---|---|---|---|
| 1 | 12 | −2.1429 | 4.5918 | −0.889 |
| 2 | 15 | 0.8571 | 0.7347 | 0.3556 |
| 3 | 11 | −3.1429 | 9.8776 | −1.3039 |
| 4 | 18 | 3.8571 | 14.8776 | 1.6003 |
| 5 | 14 | −0.1429 | 0.0204 | −0.0593 |
| 6 | 16 | 1.8571 | 3.449 | 0.7705 |
| 7 | 13 | −1.1429 | 1.3061 | −0.4742 |
| Σ | 99 | 0 | 34.8571 |
Rows with |z| above 3 are highlighted. The Σ row of x shows Σx.
What is variance?
Variance measures how far a set of numbers is spread out from its mean. It is the average of the squared deviations: subtract the mean from every value, square each difference so negatives do not cancel positives, add them up and divide. Divide by N for the population variance σ², or by n − 1 for the sample variance s², which corrects for the fact that a sample underestimates spread. Because the deviations are squared, variance is in squared units (cm², points²), which is why the standard deviation, its square root, is easier to interpret. Variance is still the quantity statistics is built on: it adds up neatly across independent variables and it is what ANOVA, regression and t-tests work with.
How to use the variance calculator
- Type or paste your numbers, separated by commas, spaces, tabs or new lines, or switch to Frequency table to enter values with counts.
- Choose Sample (divide by n − 1) or Population (divide by N). Both variances are shown either way.
- Read the variance, the sum of squares and the standard deviation at the top of the results.
- Follow the working: the mean, each squared deviation, the sum of squares and the division, all with your numbers.
Variance formulas
- Sample variance
s² = Σ(x − x̄)² ÷ (n − 1)
Estimates the variance of a population from a sample.
- Population variance
σ² = Σ(x − μ)² ÷ N
Exact variance when the data is the whole population.
- Sum of squares
SS = Σ(x − x̄)² = Σx² − (Σx)² ÷ n
The numerator of both formulas. Excel calls it DEVSQ.
- Variance and standard deviation
variance = SD², SD = √variance
Standard deviation is in the original units; variance is in squared units.
Sample or population: which one should you use?
Why use this variance calculator
- Sample and population side by sideYou get s and σ (and s² and σ²) at the same time, so there is no guessing which button a teacher or textbook expects. Switch the working between n − 1 and N with one click.
- Working shown with your numbersEvery answer comes with the formula and each line of arithmetic filled in with your data, plus a table of deviations, so you can check homework or copy the method into an exam answer.
- Frequency tables and grouped dataType each value once with how often it appears, or enter class intervals such as 10-19 and the calculator uses their midpoints. No need to repeat a value 30 times.
- Paste straight from a spreadsheetCopy a column from Excel, Google Sheets or Numbers and paste it. Commas, spaces, tabs and new lines all work, headers like “Score” are skipped, and multi-column pastes let you pick one column.
- Every summary statistic at onceMean, median, mode, range, quartiles, IQR, outliers, standard error, coefficient of variation, mean absolute deviation and z-scores for each value appear together.
- Private, free and instantCalculations run in your browser as you type. Nothing is sent to a server, there is no sign-up, no limit on how many numbers you enter and no “show steps” paywall.
Common uses for a variance calculator
- Statistics homework. Check answers for AP Statistics, intro stats and lab reports, and copy the step-by-step working.
- Surveys and research. Describe survey responses or experimental groups before running a t-test or building a confidence interval.
- Finance and investing. Measure the volatility of monthly returns or prices; a higher standard deviation means a riskier, bumpier series.
- Quality control. Track how much part sizes, fill weights or process times vary, and spot values outside the usual range.
- Science labs. Summarise repeated measurements, report mean ± standard deviation, and find the standard error for error bars.
- Checking software output. Compare what Excel, SPSS, R or a TI-84 reports with a transparent calculation.
The same statistics in Excel, Google Sheets and a TI-84
To check an answer or redo it in a spreadsheet, use these functions (they work the same in Google Sheets). On a TI-83 or TI-84, enter the data in L1, press STAT, CALC, 1-Var Stats; for a frequency table put the counts in L2 and set FreqList to L2.
| Statistic | Excel and Google Sheets | TI-84 (1-Var Stats) |
|---|---|---|
| Sample standard deviation (s) | =STDEV.S(A2:A20) | Sx |
| Population standard deviation (σ) | =STDEV.P(A2:A20) | σx |
| Sample variance (s²) | =VAR.S(A2:A20) | Sx² |
| Population variance (σ²) | =VAR.P(A2:A20) | σx² |
| Mean | =AVERAGE(A2:A20) | x̄ |
| Median | =MEDIAN(A2:A20) | Med |
| Mode | =MODE.MULT(A2:A20) | Not shown |
| Q1 and Q3 | =QUARTILE.INC(A2:A20, 1) and 3, or QUARTILE.EXC | Q1, Q3 (textbook method) |
| Sum of squares (SS) | =DEVSQ(A2:A20) | Not shown (Σx² is) |
| Mean absolute deviation | =AVEDEV(A2:A20) | Not shown |
| Standard error of the mean | =STDEV.S(A2:A20)/SQRT(COUNT(A2:A20)) | Sx ÷ √n |
| Z-score of a value | =STANDARDIZE(x, mean, sd) | (x − x̄) ÷ Sx |
| Percentile from a z-score | =NORM.S.DIST(z, TRUE) | normalcdf(−1E99, z) |
What this calculator does that most others leave out
We compared the top standard deviation, variance, z-score and quartile calculators in October 2026. Most return one number for one mode, show the formula without your numbers, or cap the number of values. This one was built to cover the gaps.
- Both versions at once. Most calculators make you pick sample or population first. Here s, σ, s² and σ² are always on screen.
- Working with your own numbers. Not just the formula: the mean, every squared deviation, the sum of squares and the final square root are written out with your data.
- Frequency and grouped data. Enter value and count pairs or class intervals instead of typing repeated numbers.
- Quartiles that match your class. Choose the textbook / TI-84 method or Excel’s QUARTILE.INC or QUARTILE.EXC.
- Z-scores and outliers. Every value gets a z-score, and values beyond 1.5 × IQR or 3 standard deviations are flagged.
- No limits, fully private. Paste thousands of values or open a CSV file. Everything is processed on your device and never sent anywhere.
More math and statistics tools
Only need the mean, median and mode, or a weighted average for grades? Use the average calculator. For percentages and percent change, try the percentage calculator, and to type formulas like σ = √(Σ(x − μ)² ÷ N) for a report, use the online math editor. Browse all student tools and calculators.
Worked example: variance of 7 measurements
Data: 12, 15, 11, 18, 14, 16, 13. 1. Mean: x̄ = 99 ÷ 7 = 14.1429. 2. Squared deviations: (12 − 14.1429)² = 4.5918, (15 − 14.1429)² = 0.7347, and so on. 3. Sum of squares: SS = 34.8571. 4. Sample variance: s² = 34.8571 ÷ 6 = 5.8095. 5. Population variance: σ² = 34.8571 ÷ 7 = 4.9796. The sample standard deviation is √5.8095 = 2.4103.
What to expect from the result
Variance is shown in squared units of your data. Values are rounded to the decimal places you choose, but every result is calculated at full precision first. The sample variance needs at least two values. If your data comes as a probability distribution (values with probabilities rather than counts), multiply the probabilities into counts or use the population result with frequencies that match the proportions.
Variance Calculator questions
What is the difference between sample and population variance?
Population variance σ² divides the sum of squared deviations by N, the number of values, and describes a complete group. Sample variance s² divides by n − 1 and estimates the variance of a bigger population from a sample. For the same data, the sample variance is always a little larger.
How do I find variance from standard deviation?
Square it. A standard deviation of 3 means a variance of 9. Going the other way, the standard deviation is the square root of the variance.
Why is variance squared?
Deviations from the mean add up to zero, so they are squared before averaging to stop positives and negatives cancelling. Squaring also weights big deviations more, which gives variance the mathematical properties that most statistical tests rely on.
How do I calculate variance in Excel?
Use =VAR.S(A2:A20) for a sample and =VAR.P(A2:A20) for a population. Google Sheets uses the same functions. =DEVSQ() returns the sum of squares.
Is my data sent to a server?
No. The calculator runs entirely in your browser. Numbers you type, paste or open from a file never leave your device, which makes it safe for unpublished research, grades and business data.