Variance Calculator | Population & Sample
Calculate population variance and sample variance from a dataset with step-by-step workings.
Quick Examples
Tip: Shift+Enter to calculate. Separate values with commas, spaces, or semicolons.
What Is the Variance Calculator | Population & Sample?
Variance measures how far a set of numbers is spread out from their average (mean). A variance of 0 means all values are identical; larger variance means the data is more spread out. Standard deviation, the square root of variance, is expressed in the same units as the original data, making it easier to interpret.
The critical distinction between population and sample statistics affects the denominator: population formulas divide by n, while sample formulas divide by n − 1 (Bessel's correction). The correction removes bias when estimating the population variance from a subset of the data.
Why n−1 and not n for sample variance?
Formula
Statistical Formulas Reference
| Statistic | Formula | Denominator | Use when |
|---|---|---|---|
| Mean x̄ | Σxᵢ / n | n | Always |
| Population Variance σ² | Σ(xᵢ−x̄)² / n | n | You have all data (census) |
| Sample Variance s² | Σ(xᵢ−x̄)² / (n−1) | n−1 | You have a sample (survey) |
| Population SD σ | √[Σ(xᵢ−x̄)² / n] | n | Full population known |
| Sample SD s | √[Σ(xᵢ−x̄)² / (n−1)] | n−1 | Estimating from sample |
| z-score | (xᵢ − x̄) / σ | σ | Standardizing values |
| CV (coeff. var.) | s / |x̄| × 100% | x̄ | Comparing dispersions |
| Skewness (Pearson) | μ₃ / σ³ | σ³ | Measuring distribution shape |
| IQR | Q3 − Q1 | — | Robust spread measure |
How to Use
- 1Enter your dataset in the text area, separate values with commas, spaces, semicolons, or newlines.
- 2Select a preset (Classic example, Exam scores, Stock returns, All same) to see results instantly.
- 3Click Calculate or press Shift+Enter to compute all statistics.
- 4The primary cards show population variance σ², sample variance s², population SD σ, and sample SD s.
- 5The Descriptive Statistics panel shows mean, median, mode, min, max, range, Q1, Q3, IQR, CV, and skewness.
- 6The Interpretation panel translates the numbers into plain English (±1σ range, skewness description).
- 7Toggle Show Data Breakdown Table to see each value's deviation, squared deviation, and z-score.
Example Calculation
Example: Dataset 2, 4, 4, 4, 5, 5, 7, 9 (n = 8)
Understanding Variance | Population & Sample
Variance and standard deviation are the two most important measures of dispersion in statistics. Where the mean tells you the centre of a dataset, variance and SD tell you how spread out the values are around that centre. Two datasets can have identical means yet completely different distributions, only measures of spread distinguish them.
In the real world, standard deviation appears everywhere: financial analysts use it to measure investment risk (higher SD = more volatile returns); quality engineers use it in Six Sigma to measure process consistency (a ±3σ process has 99.73% of output within spec); scientists use it to report measurement uncertainty; and educators use it to compare score distributions across different tests.
The choice between population and sample statistics is frequently misunderstood. If you have test scores for every student in a specific class, those are the population, use σ² (divide by n). If you surveyed 30 students to estimate scores for all 1 000 students in a school, those 30 are a sample, use s² (divide by n−1). Using the wrong formula doesn't cause a large numerical error for big datasets, but it matters for small samples (n < 30) where Bessel's correction is statistically significant.
The z-score table in this calculator flags values with |z| > 2 in red, a common threshold for identifying potential outliers under the assumption of a normal distribution. Always investigate flagged values before removing them: they may be measurement errors, or they may be the most important data points in your set.
Frequently Asked Questions
When should I use population variance vs sample variance?
The choice depends on whether your data represents the whole group or a subset:
- ›Population (÷n): you measured every member, a complete census with no estimation needed.
- ›Sample (÷n−1): you measured a subset, Bessel's correction removes bias in estimation.
- ›Example: all test scores in one class → population; random sample from all classes → sample.
- ›When n is large (thousands), the difference between ÷n and ÷(n−1) becomes negligible.
What does the coefficient of variation (CV) tell me?
- ›CV = (sample SD / |mean|) × 100%, dimensionless measure of relative spread.
- ›Low CV (<15%): data is tightly clustered around the mean.
- ›High CV (>30%): data is widely dispersed relative to its average.
- ›CV is meaningless if the mean is near zero (division by near-zero inflates it artificially).
- ›Use CV to compare variability: e.g., compare test score variability across different subjects.
What does a z-score tell me about individual data points?
- ›z = (xᵢ − x̄) / σ, signed distance from the mean in standard deviation units.
- ›z = 0: exactly at the mean; z = +1: one SD above; z = −2: two SDs below.
- ›In a normal distribution: 68% within ±1 SD, 95% within ±2 SD, 99.7% within ±3 SD.
- ›The data breakdown table flags values with |z| > 2 in red (potential outliers).
What does skewness indicate about the shape of my data?
- ›Skewness ≈ 0 (−0.5 to +0.5): roughly symmetric, normal or near-normal distribution.
- ›Positive skewness (right-skewed): long right tail; mean > median (e.g., income distribution).
- ›Negative skewness (left-skewed): long left tail; mean < median (e.g., failure times).
- ›High skewness (|s| > 1) suggests mean and SD may not fully represent the data.
Why is variance in squared units and what is the point of standard deviation?
- ›Squaring deviations eliminates cancellation between positive and negative differences.
- ›Side effect: variance is in squared units (m², kg², $²), hard to interpret directly.
- ›Standard deviation = √variance, restores the original unit for interpretability.
- ›Example: dataset in kg → variance in kg², standard deviation in kg (same unit as data).
- ›Variance is preferred in mathematical derivations (additive for independent variables); SD for communication.