Variance Calculator

What is Variance?

Variance is a statistical measurement of the spread between numbers in a data set. It measures how far each number in the set is from the mean (average) and thus from every other number in the set. Variance is the square of standard deviation.

Variance Formulas

Population Variance (sigma squared)

Used when you have data for the entire population:

Variance = sum((xi - mean)^2) / N

Where N is the total number of values in the population.

Sample Variance (s squared)

Used when you have data from a sample of the population:

Variance = sum((xi - mean)^2) / (n - 1)

Where (n - 1) is called Bessel's correction for unbiased estimation.

How to Calculate Variance

1. Find the mean: Add all data values and divide by count
2. Calculate deviations: Subtract the mean from each value
3. Square the deviations: Square each deviation value
4. Sum squared deviations: Add all the squared deviations
5. Divide: For population, divide by N; for sample, divide by (n-1)

Example Calculation

For data set: 4, 8, 6, 5, 3

Step 1: Mean = (4 + 8 + 6 + 5 + 3) / 5 = 5.2

Step 2: Deviations from mean:
4 - 5.2 = -1.2
8 - 5.2 = 2.8
6 - 5.2 = 0.8
5 - 5.2 = -0.2
3 - 5.2 = -2.2

Step 3: Squared deviations:
(-1.2)^2 = 1.44
(2.8)^2 = 7.84
(0.8)^2 = 0.64
(-0.2)^2 = 0.04
(-2.2)^2 = 4.84

Step 4: Sum = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8

Step 5: 
Population Variance = 14.8 / 5 = 2.96
Sample Variance = 14.8 / 4 = 3.7

Variance vs Standard Deviation

• Standard deviation is the square root of variance
• Standard deviation is in the same units as the data
• Variance is in squared units (harder to interpret directly)
• Variance is useful for mathematical calculations
• Standard deviation is more commonly used for interpretation

Properties of Variance

• Variance is always non-negative (>= 0)
• Variance of a constant is zero
• Variance is not affected by adding a constant to all values
• Multiplying all values by constant c multiplies variance by c squared

Applications of Variance

Finance

Measuring investment risk, portfolio volatility, and price fluctuations.

Quality Control

Assessing manufacturing consistency and product uniformity.

Research

Statistical analysis, ANOVA tests, and hypothesis testing.