Variance Calculator

Data Input

Separate values with commas, spaces, tabs, or new lines
0 values entered

Enter data values to see results

What This Variance Calculator Does

Variance tells you how spread out a set of numbers is: how far, on average, the values sit from their mean once each distance is squared. This calculator takes a list of numbers and works that out for you. It finds the mean, subtracts it from every value, squares the differences, adds them up and divides, then shows each of those steps so you can follow or check the working.

It is written for students checking homework, teachers preparing examples, and anyone who needs a quick, reliable figure for a report or a spreadsheet. You get both versions of the answer: sample variance (s²), which divides by n − 1 and is the one to use when your numbers are a sample from a larger group, and population variance (σ²), which divides by n and is for when you have every value in the group. The standard deviation, mean, range and sum of squares come with it.

To use it, type or paste your numbers into the box above, separated by commas, spaces or new lines. You can paste a column straight from Excel or Google Sheets. Choose Sample or Population, and the results and chart update as you type. Everything is calculated in your browser, so your data is not uploaded anywhere.

Variance Formula

Understand the mathematics behind variance calculations.

Interactive: Hover over or tap any symbol in the formula to see what it represents.

= ( − ) / −
Click any symbol in the formula above to see its role in the variance equation.
Sample Sample Variance Formula
s² = Σ ( xᵢ − x̄ ) ² / ( n − 1 )

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.

Population Population Variance Formula
σ² = Σ ( xᵢ − μ ) ² / n

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

🔢 Data Raw numbers
→
⚖️ Mean Balance point
→
↔️ Deviations xᵢ − mean
→
⬛ Squared (xᵢ − mean)²
→
✅ Variance σ² or s²
The only difference: population variance divides by n, sample variance divides by n − 1. Everything else in the variance equation is identical. The n − 1 correction matters most with small sample sizes — for very large samples, the difference between population variance and sample variance becomes negligible. Software like Excel, R, Python, SPSS, and Google Sheets all use these same formulas.

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.

Whole Population (N) Sample (n)
20 of 20 data points used
Population
σ² = Σ(xᵢ − μ)² / n
Divides by n because you have every data point in the whole population. The mean (μ) is exact — no correction needed.
σ² Population Variance
  • 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
s² Sample Variance
  • 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.

1

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.

x̄ = Σxᵢ / n
0 3 4 5 6 8 x̄ = 5.2
2

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.

dᵢ = xᵢ − x̄
3

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.

(xᵢ − x̄)²
4

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.

Σ(xᵢ − x̄)²
5

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.

σ² = 14.8 / 5 = 2.96 s² = 14.8 / 4 = 3.70

Step-by-Step Variance Example

Let's calculate the variance of an 8-value data set: 4, 8, 6, 5, 3, 7, 2, 9.

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.

0 3 6 9 4 8 6 5 3 7 2 9 x̄ = 5.5
Data Set
4 8 6 5 3 7 2 9
Step 1 Step 1: Find the Mean
x̄ = (4 + 8 + 6 + 5 + 3 + 7 + 2 + 9) / 8
Mean (x̄) = (4 + 8 + 6 + 5 + 3 + 7 + 2 + 9) / 8 = 44 / 8 = 5.5

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.

Step 2 Step 2: Subtract the Mean and Square
xᵢ Deviation (x − x̄) Squared Deviation (x − x̄)²
4−1.52.25
82.56.25
60.50.25
5−0.50.25
3−2.56.25
71.52.25
2−3.512.25
93.512.25

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.

Step 3 Step 3: Sum the Squared Deviations
Σ(xᵢ − x̄)² = 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 + 12.25 + 12.25
Sum of Squares = 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 + 12.25 + 12.25 = 42

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.

Step 4 Step 4: Divide by n (population) or n − 1 (sample)
Population Variance (σ²) = 42 / 8 = 5.25
Population σ²
Sample Variance (s²) = 42 / 7 = 6
Sample s² (Sample)
Standard deviation is the square root of variance. For this data set: population σ = √5.25 ≈ 2.291, sample s = √6 ≈ 2.449. Standard deviation uses the same units as the original data, which makes it easier to interpret than variance. You can calculate variance and standard deviation together in Excel using VAR.S (sample) and VAR.P (population) functions, or in Google Sheets, R, Python, and SPSS.

How to Calculate Variance By Hand

Follow these checklist steps to calculate variance manually without a calculator.

Scratch Pad
Data: {3, 7, 5, 9, 1}
n = 5
Sum = 3+7+5+9+1 = 25
Mean = 25 ÷ 5 = 5
Deviations: −2, 2, 0, 4, −4
Squared: 4, 4, 0, 16, 16
Sum of squares = 40
σ² = 40/5 = 8   s² = 40/4 = 10
Step 1 Write down your dataset

List all numbers clearly. Count how many values you have (n).

Example: Data set = {3, 7, 5, 9, 1} → n = 5
Step 2 Calculate the arithmetic mean

Add all the numbers and divide by n. Note this value.

Sum = 3 + 7 + 5 + 9 + 1 = 25. Mean = 25 ÷ 5 = 5.
Step 3 Subtract the mean from each number

Subtract the mean from each raw value. Some will be negative.

3−5 = −2, 7−5 = 2, 5−5 = 0, 9−5 = 4, 1−5 = −4
Step 4 Square each of the differences

Multiply each difference by itself. All values must be positive.

(−2)² = 4, 2² = 4, 0² = 0, 4² = 16, (−4)² = 16
Step 5 Add all the squared values

Sum the squared differences. This is your Sum of Squares.

Sum of Squares = 4 + 4 + 0 + 16 + 16 = 40
Step 6 Divide by n − 1 (sample) or n (population)

This final step gives you the variance.

σ² = 40 ÷ 5 = 8 (Population)  |  s² = 40 ÷ 4 = 10 (Sample)
Tip: Standard deviation is just the square root of the variance you calculated in the final step.

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.

Population size (n): 5
3 100
Population σ²
Sum of Squares / 5
—
Sample s²
Sum of Squares / 4
—
Difference
s² − σ²
—

Convergence: s² approaches σ² as n grows

% n 0% 50% 100% 3 25 50 75 100
At n = 5, sample variance is 25% larger than population variance

More Variance Examples

Four more data sets worked through to both the population and the sample variance.

1Quiz Scores

Data: 7, 8, 10, 6, 9

Working: Mean = 8. Deviations: −1, 0, 2, −2, 1. Squared: 1, 0, 4, 4, 1. Sum of squares = 10.

Population variance = 10 / 5 = 2 | Sample variance = 10 / 4 = 2.5

2Delivery Times (days)

Data: 3, 5, 2, 4, 6, 3

Working: Mean = 23 / 6 ≈ 3.833. Deviations: −0.833, 1.167, −1.833, 0.167, 2.167, −0.833. Sum of squared deviations ≈ 10.833.

Population variance ≈ 10.833 / 6 ≈ 1.806 | Sample variance ≈ 10.833 / 5 ≈ 2.167

3Daily Calorie Intake

Data: 2100, 1950, 2200, 2050, 1900

Working: Mean = 2040. Squared deviations: 3600, 8100, 25600, 100, 19600. Sum of squares = 57000.

Population variance = 11,400 | Sample variance = 14,250

4Battery Life (hours)

Data: 8.2, 7.9, 8.5, 8, 8.3, 8.1

Working: Mean ≈ 8.167. Sum of squared deviations ≈ 0.2333.

Population variance ≈ 0.0389 | Sample variance ≈ 0.0467

What Makes Variance Larger or Smaller?

If a result looks surprising, one of these is usually the reason:

  • Sample size. Variance calculated from a handful of values can change a lot from one sample to the next. With more data, sample variance settles closer to the true population value.
  • Outliers. Deviations are squared, so a single extreme value contributes far more than its share. Check unusual values before interpreting the result.
  • Measurement precision. Rounding measurements coarsely (for example to the nearest whole number) can hide real differences and shrink the variance.
  • Mixed groups. Combining distinct groups, such as adults and children, inflates the variance compared with the variance inside each group.
  • Scale of the data. Multiplying every value by 10 multiplies the variance by 100. Compare variances only between data sets on the same scale, or use the coefficient of variation.

Explore Our Full Suite of Calculators

9 calculators in total: the variance calculator above plus 8 tools for related measures, each with worked examples, formulas and FAQs.

Variance Calculator σ SD x̄ Mean s² Sample σ² Pop. CV Coeff. % % Var sp² Pooled ♠ Poker
Click to open →

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.

What is the variance of 2, 4, 6, 8, 10?

Mean = 30 / 5 = 6. Deviations: −4, −2, 0, 2, 4. Squared: 16, 4, 0, 4, 16. Sum of squares = 40. Population variance σ² = 40 / 5 = 8 and sample variance s² = 40 / 4 = 10. The standard deviations are σ = √8 ≈ 2.828 and s = √10 ≈ 3.162.

What is the variance of 5, 10, 15, 20, 25?

Mean = 75 / 5 = 15. Deviations: −10, −5, 0, 5, 10. Squared: 100, 25, 0, 25, 100. Sum of squares = 250. Population variance σ² = 250 / 5 = 50 and sample variance s² = 250 / 4 = 62.5. The standard deviations are σ ≈ 7.071 and s ≈ 7.906.

Can I find the variance from the standard deviation?

Yes. Variance is the standard deviation squared. If the standard deviation is 4, the variance is 4² = 16. In the other direction, the standard deviation is the square root of the variance.

How do I find the variance of a frequency table?

Multiply each value by its frequency and divide the total by the sum of the frequencies to get the mean. Then, for each value, multiply its squared deviation from the mean by its frequency, add these up, and divide by N (population) or N − 1 (sample), where N is the total frequency. You can also type the repeated values into the calculator directly.

What does a variance of zero mean?

Every value is the same. For {5, 5, 5, 5} the mean is 5 and every deviation is 0, so the variance is 0. Any difference between values gives a positive variance.

What does a high variance mean?

The values are spread far from the mean. Whether that is "high" depends on the scale of the data, so compare it with the mean (using the coefficient of variation) or with other data sets measured the same way.

Can I calculate variance with only two numbers?

Yes. For 3 and 7 the mean is 5, each deviation is ±2 and the sum of squares is 8. The population variance is 8 / 2 = 4 and the sample variance is 8 / 1 = 8. Sample variance needs at least two values; with one value it would divide by zero.

Trending Articles & Guides

Explore our top tutorials, mathematical breakdowns, and practical variance examples.

View All Articles