Find the variance of a set of numbers.
Each value's distance from the mean is squared, and those squared distances are averaged.
Variance: Average of squared deviations from the mean
A variance calculator measures how spread out a list of numbers is by averaging the squared distances of the values from their mean. This calculator uses the population formula, dividing by the number of values n, and also reports the mean and the standard deviation.
Variance is the foundation of most of statistics. It appears in Class 11 statistics, in B.Com and BBA business statistics, and in finance, where the variance of returns measures risk. It is less intuitive than range because its units are squared, but it has mathematical properties that make it the natural quantity to work with, particularly when combining independent sources of variation. The question it answers is: on average, how far are the values from the mean, in squared terms?
1. Find the mean: x̄ = Σx ÷ n.
2. Subtract the mean from each value to get the deviations x − x̄.
3. Square each deviation: (x − x̄)².
4. Add the squared deviations: Σ(x − x̄)².
5. Divide by n for population variance: σ² = Σ(x − x̄)² ÷ n.
6. A shortcut that gives the same result is σ² = (Σx² ÷ n) − x̄².
The deviations from the mean always add up to zero, because the mean is the balance point of the data. So averaging raw deviations tells you nothing. Taking absolute values would work but is awkward algebraically. Squaring makes every deviation positive and weights large deviations more heavily, which matches the idea that a value far from the mean is more unusual. Squared deviations also add neatly: for independent quantities, the variance of a sum is the sum of the variances.
Dividing by n gives the population variance, correct when your list is the entire group you care about, such as all five innings of one player. When the list is a sample used to estimate a larger population's variance, statisticians divide by n − 1 instead. Deviations measured from the sample mean are slightly too small on average, and n − 1 corrects this bias. This calculator divides by n. For small samples used for inference, use the sample standard deviation calculator.
Expanding Σ(x − x̄)² gives Σx² − 2x̄Σx + n x̄². Since Σx = n x̄, this becomes Σx² − 2n x̄² + n x̄² = Σx² − n x̄². Dividing by n gives σ² = Σx² ÷ n − x̄², the mean of the squares minus the square of the mean. It is quicker by hand for whole-number data, but with large numbers and small spread it can lose precision on a calculator, so the definition is safer in software.
Dhruv, a school cricketer, scored 6, 8, 10, 12 and 14 runs in his last five innings in the inter-house league and wants to measure how consistent he has been.
Answer: Variance 8; Standard deviation 2.8284; Mean 10
Forgetting to square the deviations, which always gives a total of zero.
Dividing by n − 1 when the textbook question expects the population formula, or by n when it expects the sample formula.
Reporting variance in the original units; its units are squared, such as runs² or ₹².
Subtracting the median instead of the mean from each value.
Measuring the consistency of a player's scores or a student's marks.
Measuring investment risk as the variance of returns.
Quality control of weights, lengths or fill volumes in manufacturing.
Class 11 and business statistics exercises.
As the input to standard deviation, correlation and regression.
Why square the deviations instead of just averaging them?
If you simply averaged the differences from the mean, the ones above and below would cancel out to zero. Squaring makes every difference positive first, so the real spread is properly captured.