Variance Calculator
Calculate the variance, standard deviation, and sum of squared deviations for a data set. See also Standard Deviation Calculator and Mean Calculator.
How to Calculate Variance
Variance measures how far each value in a data set is from the mean. To calculate: (1) find the mean of the data, (2) subtract the mean from each value to get the deviation, (3) square each deviation, (4) find the average of the squared deviations. For a sample, divide by n−1 (Bessel's correction); for a population, divide by n.
Variance Formulas
Population Variance: σ² = Σ(xᵢ − μ)² / N
Sample Variance: s² = Σ(xᵢ − x̄)² / (n − 1)
Standard Deviation = √Variance
Example Calculation
Data: 6, 7, 3, 9, 5, 8
Mean = 38 / 6 ≈ 6.333333
Deviations²: 0.111, 0.444, 11.111, 7.111, 1.778, 2.778
Σ(xᵢ − x̄)² ≈ 23.333
Sample Variance = 23.333 / 5 ≈ 4.666667
Sample Std Dev ≈ 2.160247
Variance Reference Table
| Concept | Formula | Measures |
|---|---|---|
| Variance | Σ(xᵢ−x̄)²/(n−1) | Squared spread |
| Std Deviation | √Variance | Spread in original units |
| Range | Max − Min | Total spread |
| IQR | Q3 − Q1 | Middle 50% spread |
Solved Examples
Example 1: Battery Lifespan Testing
A manufacturer tests 6 batteries (hours until failure): 48, 52, 45, 50, 47, 54. Calculate the sample variance.
Mean = (48+52+45+50+47+54)/6 = 296/6 = 49.333
Deviations: -1.333, 2.667, -4.333, 0.667, -2.333, 4.667
Squared: 1.778, 7.111, 18.778, 0.444, 5.444, 21.778
Sum of squares = 55.333
Sample variance = 55.333/(6-1) = 11.067 hours squared
Answer: The sample variance is 11.07 hours squared. The sample standard deviation is sqrt(11.07) = 3.33 hours, meaning battery life typically varies by about 3.3 hours from the mean.
Example 2: Portfolio Risk Assessment
Monthly returns (%) for a stock over 5 months: 3.2, -1.5, 4.8, 2.1, -0.6. Calculate the variance of returns.
Mean = (3.2 + (-1.5) + 4.8 + 2.1 + (-0.6))/5 = 8.0/5 = 1.6%
Deviations: 1.6, -3.1, 3.2, 0.5, -2.2
Squared: 2.56, 9.61, 10.24, 0.25, 4.84
Sum = 27.50
Sample variance = 27.50/4 = 6.875 (% squared)
Answer: The variance of returns is 6.875 (percentage points squared). In finance, variance measures investment risk. Higher variance means more unpredictable returns.
Example 3: Comparing Variability Between Groups
Reaction times (ms) for caffeine group: 210, 195, 220, 205, 200. Placebo group: 250, 230, 280, 240, 260. Compare their variances.
Caffeine: mean = 206, deviations: 4,-11,14,-1,-6
Variance_caffeine = (16+121+196+1+36)/4 = 370/4 = 92.5
Placebo: mean = 252, deviations: -2,-22,28,-12,8
Variance_placebo = (4+484+784+144+64)/4 = 1480/4 = 370
Answer: The placebo group variance (370) is 4 times larger than the caffeine group (92.5), suggesting caffeine not only lowers reaction time but also produces more consistent performance.
Practice Questions
Question 1
Quiz scores for 5 students: 8, 6, 9, 7, 10. Find the population variance and sample variance.
Answer: Mean = 40/5 = 8. Squared deviations: 0, 4, 1, 1, 4. Sum = 10. Population variance = 10/5 = 2.0. Sample variance = 10/4 = 2.5.
Question 2
If you add 10 to every value in a dataset, how does the variance change?
Answer: The variance does not change. Adding a constant shifts all values equally, so deviations from the mean remain the same. If the original variance is 25, adding 10 to every value still gives variance = 25.
Question 3
A dataset has variance 16. If each value is multiplied by 3, what is the new variance?
Answer: When multiplying by constant k, variance is multiplied by k squared. New variance = 16 x 9 = 144. (Standard deviation would be multiplied by 3: from 4 to 12.)
Common Mistakes
Confusing variance with standard deviation
Variance is in squared units (e.g., cm squared), making it hard to interpret directly. Standard deviation (the square root of variance) is in the original units. Report SD for interpretability, but variance is needed for many statistical formulas.
Using the wrong denominator (n vs n-1)
For sample variance, divide by n-1 (Bessel correction). For population variance, divide by n. Using n for a sample systematically underestimates the true population variance, especially for small samples.
Thinking variance can be negative
Variance is always non-negative because it is a sum of squared deviations. A variance of zero means all values are identical. If you calculate a negative variance, you have made an arithmetic error.
Key Takeaways
- Variance measures the average squared distance from the mean and quantifies data spread.
- Sample variance (s squared) divides by n-1; population variance (sigma squared) divides by n.
- Variance is additive for independent variables: Var(X+Y) = Var(X) + Var(Y).
- Adding a constant does not change variance; multiplying by k multiplies variance by k squared.
- Variance is essential in ANOVA, regression, portfolio theory, and hypothesis testing.
Frequently Asked Questions
Why do we square the deviations?
Squaring ensures all deviations are positive (negative deviations don't cancel out positive ones) and gives more weight to larger deviations, making variance sensitive to outliers.
Why divide by n−1 for sample variance?
Dividing by n−1 (Bessel's correction) corrects the bias that occurs when estimating population variance from a sample. It produces an unbiased estimate of the population variance.
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units, while standard deviation is in the same units as the original data, making it easier to interpret.
Can variance be zero?
Yes. Variance is zero when all values in the data set are identical, meaning there is no spread at all.