Population Variance Calculator - σ², σ, and Sample Variance From Raw Data

A population variance calculator is a descriptive-statistics tool that takes every value in a complete dataset and returns σ², the average squared deviation from the population mean μ. It also reports σ, the matching sample variance and sample standard deviation, the mean, count, and range so you can read the spread alongside the central value. The same calculator accepts a frequency column, so you can compute population variance from a grouped frequency table without expanding the data yourself.

• • Introductory statistics homework: Compute σ², σ, mean, and the sum of squared deviations from a list of test scores, measurements, or survey responses.
• • Quality-control reporting: When you have measurements for every unit in a finite production batch, treat the batch as the population and report σ² and σ as the actual process spread.
• • Comparing population and sample variance: Show σ² (divide by N) and s² (divide by N − 1) side by side so Bessel's correction is visible on small datasets.
• • Frequency-table descriptive statistics: Paste a frequency column for grouped data (for example, ages × counts) and the calculator applies each frequency as a weight.

Population variance is the right choice when the dataset is the entire population you care about — every part in a manufactured batch, every student in a small class, every measurement in a controlled experiment. When the data is a sample drawn from a larger population, the sample variance s² (which divides by N − 1 instead of N) is the unbiased estimator, and our standard deviation calculator returns both numbers side by side so you can decide which one to report.

The population variance calculator computes the mean of the dataset, sums the squared deviations from that mean, and divides by N. The result is σ², and its square root is the population standard deviation σ. The same script also computes the sample variance s² by dividing by N − 1, so both forms are visible in one result panel.

• xi: Each individual value in the dataset (entered in the Data Values box).
• μ: Population mean — sum of every value divided by N. Reported as the 'Mean (μ)' result.
• N: Total count of values in the dataset. With a frequency column, N is the sum of the frequencies (the weighted count).
• σ²: Population variance — the primary result. Sum of squared deviations from μ, divided by N. Units are the square of the input units.
• s²: Sample variance — same numerator as σ², divided by N − 1. Used when the dataset is a sample drawn from a larger population.

The same arithmetic runs on a frequency column. If you enter values 10, 20, 30 with frequencies 2, 3, 5, the calculator treats the dataset as 10, 10, 20, 20, 20, 30, 30, 30, 30, 30 — without forcing you to expand it by hand.

According to the NIST/SEMATECH e-Handbook of Statistical Methods, Section 1.3.5.6 (Measures of Scale), variance is the arithmetic average of the squared distance from the mean — the same definition this calculator applies as σ² = Σ(xi − μ)² / N for a complete dataset.

According to Wikipedia, Variance, sample variance divides the squared deviations by n − 1 (Bessel's correction) while population variance divides by n, which is the convention this calculator applies so the two results can be compared side by side.

Four small definitions explain every number the population variance calculator produces.

If you also need the median of the same dataset, our median calculator returns it alongside the variance, so you do not have to copy the data into a second tool.

Five short steps take you from a list of numbers to a complete variance report.

1. 1 Paste your data values: Type or paste every number into the Data Values box, separated by commas, spaces, or new lines. Non-numeric tokens are ignored.
2. 2 Add an optional frequency column: If your data is grouped, paste matching frequencies in the Frequency box. Leave it blank when each value should be weighted equally.
3. 3 Pick a decimal precision: Choose 2, 4, or 6 decimal places. Four decimals is a good default for homework; 6 decimals helps when matching a textbook answer.
4. 4 Read the variance and standard deviation results: The panel reports σ², σ, s², s, mean μ, count N, minimum, maximum, range, and Σ(xi − μ)².
5. 5 Sanity check with the squared-deviation sum: Compare Σ(xi − μ)² with the value you compute by hand. σ² is that sum divided by N, and s² is that sum divided by N − 1.

Paste '4, 8, 6, 5, 3' into the Data Values box and leave the Frequency box blank. The calculator returns μ = 5.2, σ² = 2.96, σ ≈ 1.7205, s² = 3.7, s ≈ 1.9235, count = 5, and Σ(xi − μ)² = 14.8. This follows the σ² = Σ(xi − μ)² / N and s² = Σ(xi − x̄)² / (N − 1) formulas given in the LibreTexts Introductory Statistics (Shafer and Zhang), Section 2.3 — Measures of Variability, which defines both forms side by side for the same dataset.

A purpose-built population variance calculator saves time and removes the arithmetic errors that come from doing the squared-deviation sum by hand.

• • Both variances in one panel: See σ² (divide by N) and s² (divide by N − 1) side by side, so you can decide which convention matches your dataset without switching tools.
• • Built-in frequency table support: Paste a frequency column and the calculator expands the dataset for you, avoiding the manual error of listing 200 identical values by hand.
• • Transparent calculation breakdown: The Results panel shows μ, N, Σ(xi − μ)², σ², and σ in one place, so you can explain how the variance is built up from the mean instead of trusting a black-box result.
• • Works on real-world inputs: Accepts negative numbers, decimals, scientific notation, and messy separators. Non-numeric tokens are dropped silently with the accepted count reported next to the result.

When the next step in your analysis is a confidence interval for the population mean or a hypothesis test, our confidence interval calculator uses σ² (or s²) as the variance input, so the result here can flow directly into the next calculation.

Three variables determine the variance, and two limitations tell you when to double-check the result or reach for a different tool.

• • The calculator treats the dataset as a flat list (or weighted list). It does not compute the variance of a probability distribution or the conditional variance of a regression model — those need a separate tool.
• • The dataset is treated as numeric. Categorical codes such as 'yes' or 'no' must be encoded as numbers (for example, 1 and 0) before they will be accepted.

If the goal is to standardize an individual value using the resulting standard deviation, our Z-score calculator takes μ and σ as inputs and returns the Z-score plus an estimated percentile. When you need the spread between two variables instead of within a single variable, our covariance calculator returns the covariance and the correlation coefficient from paired datasets.