Variance Calculator
Data Input
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A variance calculator is a tool (online or built into software) that automatically computes the variance of a data set for you — so you don't have to manually calculate the mean, find deviations, square them, and average them yourself.
What you typically input:
- • A list of numbers (e.g., 2, 4, 4, 4, 5, 5, 7, 9)
What it outputs:
- • The mean
- • The variance (often both population and sample variance, since they differ slightly)
- • Usually the standard deviation too, since it's directly derived from variance
Variance Formula
Understand the mathematics behind variance calculations.
Interactive: Hover over or tap any symbol in the formula to see what it represents.
Uses n − 1 (Bessel's correction) for an unbiased variance estimate from sampled data. This adjustment accounts for degrees of freedom when the true population mean is unknown.
Divides by n directly. No correction needed because the population mean (μ) is the true mean of the whole population.
How Data Flows Through the Formula
Population Variance and Sample Variance Calculator
Population variance and sample variance differ in one place: the divisor. Population variance divides the sum of squared differences by n (total count). Sample variance divides by n − 1 — Bessel's correction — to produce an unbiased variance estimate from sampled data. This correction matters most with small samples. For large datasets, the difference shrinks.
- Uses every data point in the whole population
- Divides sum of squares by n
- Mean symbol: μ (mu) — the true population mean
- Used in census data, full datasets, quality control
- Uses a subset of sampled data from the population
- Divides sum of squares by n − 1 (degrees of freedom)
- Mean symbol: x̄ (x-bar) — estimated mean
- Used in surveys, experimental design, clinical trials
How to Calculate Variance
Variance measures how far each number in a data set sits from the mean. These are the variance calculation steps.
To calculate variance, find the average of the squared differences from the mean. The process is the same for population variance and sample variance — the only difference is whether you divide by n or n − 1 at the end. This statistical variance calculation works for any data set, from survey analysis to machine learning feature selection.
Try it: Enter numbers and watch each step
Type 3–8 numbers separated by commas. The steps below will calculate variance with your data in real time.
Find the Mean (Average)
Add all values in the data set and divide by the count. For [4, 8, 6, 5, 3], the mean is (4+8+6+5+3) ÷ 5 = 5.2. This mean is the center point for every variance calculation.
Subtract Mean to Get Deviations
Subtract the mean from each number to find its deviation. Negative deviations mean the value is below the mean; positive values are above. The sum of deviations always equals 0.
Square Each Deviation
Square each deviation to make it positive. This ensures that deviations below and above the mean don't cancel each other out. Squaring also gives more weight to outliers.
Sum the Squared Deviations
Add all the squared deviations together. This sum is called the Sum of Squares (SS). For our data, it is 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8.
Divide to Get Variance
Divide the Sum of Squares by N (for population) or n − 1 (for sample). For sample variance, we divide 14.8 by 4 to get 3.7. For population variance, we divide by 5 to get 2.96.
Step-by-Step Variance Example
Let's calculate the variance of a sample dataset: 4, 8, 6, 5, 3.
Consider the data set: 4, 8, 6, 5, 3, 7, 2, 9. We'll calculate both sample variance (s²) and population variance (σ²) for these 8 values. Watch the bar chart below — it shows each data point, the mean line, and the squared deviations as you scroll through each step.
The mean is the center of the data set — every deviation is measured from this point. In the bar chart above, the dashed line marks x̄ = 5.5.
| xᵢ | Deviation (x − x̄) | Squared Deviation (x − x̄)² |
|---|---|---|
| 4 | −1.5 | 2.25 |
| 8 | 2.5 | 6.25 |
| 6 | 0.5 | 0.25 |
| 5 | −0.5 | 0.25 |
| 3 | −2.5 | 6.25 |
| 7 | 1.5 | 2.25 |
| 2 | −3.5 | 12.25 |
| 9 | 3.5 | 12.25 |
Each row shows one data point, its deviation from the mean, and the squared deviation. Notice how values far from x̄ (like 2 and 9) produce much larger squared differences.
This sum of squares (42) represents the total squared distance of all data points from the mean. It's the numerator in both variance formulas.
How to Calculate Variance By Hand
Follow these checklist steps to calculate variance manually without a calculator.
Follow these checklist steps to calculate variance manually without a calculator.
Step 1 Write down your dataset
List all numbers clearly. Count how many values you have (n).
Step 2 Calculate the arithmetic mean
Add all the numbers and divide by n. Note this value.
Step 3 Subtract the mean from each number
Subtract the mean from each raw value. Some will be negative.
Step 4 Square each of the differences
Multiply each difference by itself. All values must be positive.
Step 5 Add all the squared values
Sum the squared differences. This is your Sum of Squares.
Step 6 Divide by n − 1 (sample) or n (population)
This final step gives you the variance.
Whole Population Variance Calculation
When the data set includes every member of the group, use population variance — divide by n, not n − 1.
Population variance applies when your data set includes every observation from the whole population — not a sample. Examples: all employees in a company, every test score in a class, all products in a batch during quality control. The formula uses μ (the true population mean) and divides by n (the full count). No Bessel's correction. No degrees of freedom adjustment.
Interactive: How Sample Size Affects Variance
Drag the slider to see how the difference between σ² and s² changes with population size. At small n, the gap is large. As n grows, sample variance converges toward population variance.
Convergence: s² approaches σ² as n grows
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Frequently Asked Questions
Everything you need to know about variance.
What is the variance of 6 7 5 9 12 15?
The population variance of 6, 7, 5, 9, 12, 15 is 12.33 (σ² = 74/6) and the sample variance is 14.8 (s² = 74/5).
How to calculate variance of 1, 2, 3, 4, 5?
For the numbers 1, 2, 3, 4, 5, the population variance is 2 (10/5) and the sample variance is 2.5 (10/4).
How to calculate variance in Excel?
In Excel, use =VAR.S(range) to calculate sample variance or =VAR.P(range) to calculate population variance.
How to calculate sample variance?
Subtract the sample mean from each number, square the differences, sum them up, and divide by n − 1 (s² = Σ(xᵢ − x̄)² / (n − 1)).
How to calculate percentage of variance?
Calculate the coefficient of variation by dividing the standard deviation by the mean and multiplying by 100% (CV = (s / x̄) × 100%).
What is variance in statistics?
Variance is a statistical metric that measures how far data points are spread out from their average (mean).
What is the difference between sample variance and population variance?
Sample variance divides by n − 1 to correct for sampling bias, while population variance divides by n.
What is the variance formula?
The sample variance formula is s² = Σ(xᵢ − x̄)² / (n − 1) and the population variance formula is σ² = Σ(xᵢ − μ)² / n.
What is the relationship between variance and standard deviation?
Standard deviation is simply the square root of variance (s = √s² and σ = √σ²).
Can variance be negative?
No, variance can never be negative because it is calculated by summing squared differences.
Why do we use n-1 instead of n for sample variance?
Dividing by n − 1 (Bessel's correction) removes bias when estimating population variance from a sample.
How many data points do I need to calculate variance?
You need at least 1 data point for population variance and at least 2 data points for sample variance.
What is the coefficient of variation?
The coefficient of variation (CV = (s / x̄) × 100%) measures relative variability as a percentage of the mean.