Variance Calculator – Population & Sample Variance
Calculate variance and standard deviation for a data set. Supports population and sample variance. Free online statistics calculator for instant results.
What Is Variance?
Variance measures the spread of a dataset — how far the values are from the mean. A low variance means data points cluster near the mean; a high variance means they are spread out widely.
Variance is calculated as the average of squared differences from the mean:
- Population variance (σ²): σ² = Σ(xᵢ − μ)² / N
- Sample variance (s²): s² = Σ(xᵢ − x̄)² / (N−1)
Where xᵢ is each data point, μ (or x̄) is the mean, and N is the number of values. The standard deviation is simply the square root of variance — it is in the same units as the original data, making it more interpretable.
Population vs. Sample Variance
The key difference is the denominator — N vs. (N−1) — known as Bessel's correction:
| Type | Denominator | Use When | Symbol |
|---|---|---|---|
| Population Variance | N | You have data on the entire population | σ² |
| Sample Variance | N−1 | You have a sample from a larger population | s² |
In practice, most real-world data is a sample. Using N−1 (sample variance) produces an unbiased estimate of the true population variance. Using N (population variance) on a sample systematically underestimates the true variance.
Example: Testing a new drug on 50 patients means using sample variance (s²). Analyzing all students in a classroom means using population variance (σ²).
Step-by-Step Variance Calculation
Given the data set: 4, 7, 13, 2, 8
- Calculate the mean: (4+7+13+2+8) ÷ 5 = 34/5 = 6.8
- Find deviations from mean: (4−6.8)=−2.8; (7−6.8)=0.2; (13−6.8)=6.2; (2−6.8)=−4.8; (8−6.8)=1.2
- Square the deviations: 7.84; 0.04; 38.44; 23.04; 1.44
- Sum of squares: 7.84+0.04+38.44+23.04+1.44 = 70.8
- Population variance: 70.8 ÷ 5 = 14.16
- Sample variance: 70.8 ÷ 4 = 17.7
- Standard deviation: √14.16 = 3.76 (population) or √17.7 = 4.21 (sample)
Practical Applications of Variance
| Field | Application | Example |
|---|---|---|
| Finance | Investment risk | High variance = more volatile stock returns |
| Manufacturing | Quality control | Low variance = consistent product dimensions |
| Medicine | Clinical trials | Measuring variability in patient responses |
| Sports science | Performance analysis | Variability in athlete performance over season |
| Education | Test score analysis | Understanding spread of student performance |
💡 Did you know?
- Variance was introduced by Ronald Fisher in 1918 — the same paper where he coined the term "variance."
- In finance, variance is the basis of Modern Portfolio Theory. A portfolio's variance depends not just on individual asset variance but on correlations between assets.
- The coefficient of variation (CV = standard deviation / mean × 100%) allows comparing variability across datasets with different units or scales.
Frequently Asked Questions
What is the difference between variance and standard deviation?
Variance is the average of squared deviations from the mean; standard deviation is its square root. Standard deviation is in the same units as the original data (e.g., dollars, kg, seconds), making it more interpretable. Variance is useful in mathematical operations (variances of independent variables add directly), while standard deviation is better for describing spread to a non-technical audience.
When should I use sample vs. population variance?
Use population variance when your data contains every member of the group you're analyzing (e.g., all employees in one company). Use sample variance when your data is a subset of a larger group (e.g., a survey of 500 voters to estimate all voters' opinions). In most real-world research and statistics, sample variance is appropriate.
Can variance be negative?
No. Variance is always zero or positive because it is calculated from squared values. Variance = 0 only when all data points are identical (no spread). A negative variance is mathematically impossible and indicates a calculation error.
What is a "high" or "low" variance?
High and low are relative to the scale and context of the data. A variance of 10 is "low" for human heights in cm but "high" for heights in meters. The coefficient of variation (SD / mean × 100%) is scale-independent and allows comparison across different datasets. In quality control, specifications define acceptable variance ranges for each measurement.