Variance Calculator — Population & Sample
Calculate population variance and sample variance from a dataset with step-by-step workings.
What Is the Variance Calculator — Population & Sample?
The Variance Calculator computes the variance and standard deviation of a dataset, for both population (all data available) and sample (data is a subset) cases. Enter your data values to get variance, standard deviation, mean, and a step-by-step breakdown of the sum of squared deviations.
Formula
How to Use
Enter your data values separated by commas. Select whether this is a population (you have all the data) or a sample (your data is a subset of a larger population). The calculator computes mean, each squared deviation, their sum, and the variance and standard deviation.
Example Calculation
Data: 2, 4, 4, 4, 5, 5, 7, 9. n=8, Mean=5. Deviations: −3,−1,−1,−1,0,0,2,4. Squared: 9,1,1,1,0,0,4,16. Sum=32. Population variance σ²=32/8=4. σ=2. Sample variance s²=32/7≈4.57. s≈2.14.
Understanding Variance — Population & Sample
Variance measures how spread out data values are around their mean. A high variance indicates that data points are scattered widely; a low variance indicates they cluster close to the mean. As the foundational measure of statistical dispersion, variance appears in probability theory, inferential statistics, quality control, and machine learning.
The distinction between population variance (σ²) and sample variance (s²) is crucial in statistics. When analyzing a complete population, divide by N. When working with a sample drawn from a larger population, divide by (n−1) — Bessel's correction — to obtain an unbiased estimate of the population variance. Most real-world analyses use the sample formula.
Standard deviation — the square root of variance — is the most widely reported measure of dispersion because it has the same units as the original data. It defines the spread of the bell curve in a normal distribution: approximately 68% of values fall within one standard deviation of the mean, and 95% within two standard deviations.
Frequently Asked Questions
Why does the sample variance divide by (n−1) instead of n?
Dividing by (n−1) makes the sample variance an unbiased estimator of the population variance. Using n instead underestimates the true variance — the denominator (n−1) is called Bessel's correction.
What is the difference between variance and standard deviation?
Variance is the average squared deviation from the mean. Standard deviation is its square root, returning to the same units as the original data. Standard deviation is more interpretable; variance is more mathematically convenient.
What does a large variance tell you?
A large variance means data points are spread far from the mean — high variability. A small variance means data clusters close to the mean. Zero variance means all values are identical.
What is a coefficient of variation?
The coefficient of variation (CV = standard deviation / mean × 100%) normalizes spread by the mean, allowing comparison of variability across datasets with different units or scales.
Is this calculator free?
Yes, completely free with no registration needed.
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