Variance Calculator
Calculate population and sample variance, standard deviation, and mean of any comma-separated data set. Get instant statistical analysis with bar chart visualization.
About This Calculator
The Variance Calculator computes both population and sample variance, standard deviation, and mean for any data set. Enter comma-separated numbers to get a complete dispersion analysis with bar chart visualization.
Variance measures how far data points are spread from their mean. It is calculated by averaging the squared differences from the mean. Population variance divides by N (all data points), while sample variance divides by N-1 (Bessel's correction) to give an unbiased estimate. Standard deviation, the square root of variance, expresses dispersion in the original units and is more commonly used for interpretation.
For example, consider the data set 2, 4, 6, 8, 10. The mean is 6. The squared deviations are 16, 4, 0, 4, 16. The population variance is 40 ÷ 5 = 8, and the sample variance is 40 ÷ 4 = 10. The population standard deviation is √8 ≈ 2.828, and the sample standard deviation is √10 ≈ 3.162. This shows that with just 5 data points, the sample estimate is larger to account for not knowing the true population mean.
Variance analysis is used across many fields. In finance, it measures portfolio risk and volatility. In quality control, it tracks process consistency and defect rates. In education, it helps understand score distributions. In scientific research, it forms the foundation for ANOVA (Analysis of Variance) and hypothesis testing. A low variance indicates data points cluster tightly around the mean, while a high variance suggests greater spread and variability.
Regional Notes: Variance and standard deviation calculations follow the same mathematical definitions across India (CBSE), the United States (CCSS), and the United Kingdom (GCSE/A-Level). Bessel's correction (N-1) for sample variance is a universal standard in all three education systems.
Frequently Asked Questions
What is variance?
Variance measures how spread out data points are from the mean. It is the average of squared differences from the mean.
Population vs sample variance?
Population variance uses N in the denominator (sum of squared differences divided by N). Sample variance uses N-1 (Bessel's correction) to provide an unbiased estimate of the population variance.
What is standard deviation?
Standard deviation is the square root of variance. It measures dispersion in the same units as the original data, making it easier to interpret than variance.
How do you calculate variance?
Find the mean, subtract it from each value (deviations), square each deviation, sum them, and divide by N (population) or N-1 (sample).
What does a high variance indicate?
A high variance indicates data points are far from the mean and from each other, suggesting greater variability or spread in the data set.
Why use sample variance instead of population?
Sample variance uses N-1 because using N would underestimate the true population variance. Bessel's correction makes it unbiased when estimating from a sample.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. While variance is in squared units, standard deviation is in the same units as the original data, making it easier to interpret. For example, if data is in meters, variance is in meters squared and standard deviation is in meters.
Can variance be negative?
No, variance can never be negative. Since variance is the average of squared deviations (differences from the mean), all terms are non-negative. A variance of zero means all data points are identical, while positive values indicate spread.