Statistics Tool
Variance Calculator
Calculate Variance and Standard Deviation for population or sample.
Variance (s²)
27.4286
Standard Deviation
5.2372
Σ The Formula
Var = Σ(x - μ)² / N
Real World Examples
Data
10, 12, 23, 23, 16
# About This Calculator
Variance is a statistical measurement of the spread between numbers in a data set. More specifically, variance measures how far each number in the set is from the mean (average), and thus from every other number in the set. Variance is often denoted by the symbol $\sigma^2$ (sigma squared).
It is the square of the Standard Deviation. While standard deviation is closer to the original units of data (making it more intuitive), variance is mathematically cleaner and is used extensively in advanced statistics (ANOVA, regression analysis) and probability theory.
Formulas
- Population Variance ($\sigma^2$): $\frac{\Sigma(x - \mu)^2}{N}$
- Sample Variance ($s^2$): $\frac{\Sigma(x - \bar{x})^2}{N - 1}$
How To Use
- Type or paste your data set into the text area.
- Choose **Sample** if the data is a subset, or **Population** if it's the whole group.
- The Variance and Standard Deviation update automatically.
Frequently Asked Questions
Why is variance squared?+
We square the differences to make them all positive. If we didn't square them, the negative differences (values below mean) would cancel out the positive ones (values above mean), and the average difference would always be zero!
When should I use variance vs standard deviation?+
Use Standard Deviation when reporting to people—it's in the same units as the data (e.g., 'test scores +/- 10 points'). Use Variance for calculations and statistical proofs, as it has better mathematical properties (variances can be added linearly in some cases).
Can variance be negative?+
No. Since variance is the average of SQUARED differences, and squares are always non-negative, variance must be zero or positive. A variance of zero means all numbers are identical.
What is high variance?+
High variance means the data is very spread out around the mean. Low variance means data is clustered tightly. In finance, high variance assets (crypto, startups) are riskier than low variance assets (bonds).
Why divide by N-1 for Sample Variance?+
This is 'Bessel's Correction'. When using a sample to estimate population variance, dividing by N underestimates the true spread. Dividing by N-1 compensates for this bias, giving a more accurate estimate of the population's true variance.
Is Variance Calculator free to use?+
Yes, Variance Calculator on Matheric is completely free to use. We believe in accessible education and utility for everyone.