Variance Calculator
Compute population or sample variance from the sum of squared deviations and number of values. Useful in statistics coursework, quality control, and any analysis that requires measuring data spread.
About this calculator
Variance measures how far data points are spread from their mean. It is the average of squared deviations from the mean. For a population, variance σ² = Σ(xᵢ − μ)² / N, where N is the total count. For a sample, variance s² = Σ(xᵢ − x̄)² / (n − 1), using n − 1 (Bessel's correction) to produce an unbiased estimate of the population variance. In this calculator, you supply the sum of squared deviations Σ(xᵢ − x̄)² directly, so the formula becomes: sample variance = sum_squares / (n − 1) and population variance = sum_squares / n. The key difference is that sample variance divides by n − 1 because using n would systematically underestimate variability in the broader population. Variance is expressed in squared units; taking its square root gives the standard deviation, which is in the original units.
How to use
Suppose you have 6 data points and their sum of squared deviations from the mean is 150. For sample variance: s² = 150 / (6 − 1) = 150 / 5 = 30. For population variance: σ² = 150 / 6 = 25. Enter n = 6, sum of squared deviations = 150, and select your variance type. The calculator returns 30 for sample variance or 25 for population variance. The standard deviation would then be √30 ≈ 5.48 (sample) or √25 = 5.0 (population).
Frequently asked questions
When should I use sample variance instead of population variance?
Use sample variance when your data represents a subset of a larger population and you want to estimate the population's true variance. The division by n − 1 instead of n corrects for the fact that a sample tends to underrepresent the full spread of a population. Use population variance only when you have data for every member of the group you are analyzing, such as the exact scores of all students in one specific class rather than a random selection. In most real-world research and data analysis, sample variance is the appropriate choice.
What does a high variance mean in a dataset?
A high variance indicates that data points are widely spread out from the mean, meaning the dataset is highly variable or inconsistent. For example, if two factories produce bolts with the same average diameter but one has much higher variance, that factory's output is less predictable and more likely to include defective parts. Low variance means values cluster tightly around the mean, indicating consistency and precision. In finance, high variance in asset returns signals greater risk. Variance is best interpreted alongside context — what counts as 'high' depends entirely on the scale and nature of your data.
How is variance related to standard deviation?
Standard deviation is simply the square root of variance: SD = √variance. While variance is useful mathematically because squaring deviations makes them all positive and amplifies larger deviations, it is expressed in squared units which are hard to interpret intuitively. Standard deviation brings the measure back to the original units of measurement, making it easier to understand. For instance, if your data is in kilograms, the variance is in kg² but the standard deviation is in kg. Both measure spread, but standard deviation is more commonly reported in summaries and research papers.