How to Calculate Variance with Solved Examples?
Variance measures how far observations spread from their mean. This page covers population and sample variance, direct and shortcut formulas, frequency distributions, transformation rules, and solved variance examples. Use the formulas to calculate variance for individual observations and data given with frequencies.
What Is Variance?
For observations x₁, x₂, ..., xₙ with mean x̄, variance is calculated by squaring each deviation (xᵢ − x̄), adding the squared deviations, and dividing by the number of observations for population variance. Squaring makes every contribution non-negative. For data 2, 4, 6, 8, the mean is 5 and the population variance is (9 + 1 + 1 + 9) ÷ 4 = 5.
Variance Video Lessons
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Range, Variance and Standard Deviation
Learn how to calculate and interpret range, variance, and standard deviation as measures of dispersion in statistics, including the role of each measure in describing data spread.
Variance Formula & Tricks
Important Formulas
Use this when the data represents the complete population. Here, μ is the population mean and N is the number of observations.
Use this when the observations form a sample. Here, x̄ is the sample mean and n is the sample size.
For population variance, calculate the mean of the squares and subtract the square of the mean.
Use this shortcut when x has frequency f. The total frequency is Σf.
A is an assumed mean and d is the deviation from it. This reduces calculation size when the observations are large.
Quick Tricks
Instead of finding every deviation, calculate Σx² and Σx, then apply variance = mean of squares − square of mean.
Adding or subtracting the same constant from every observation does not change variance. Multiplying every observation by k multiplies variance by k².
Variance Concepts
Population and Sample Variance
For a complete data set, use σ² = Σ(x − μ)²/N. For a sample drawn from a larger population, use s² = Σ(x − x̄)²/(n − 1). The denominators differ, so the two answers are generally not equal. For data 2, 4, 6, 8, the squared deviation sum about the mean 5 is 20; population variance is 20/4 = 5, while sample variance is 20/3.
Direct Method for Individual Observations
For data x₁, x₂, ..., xₙ, first find x̄ = Σx/n. Then calculate (xᵢ − x̄)² for every observation. The population variance is the total squared deviation divided by n. This method is suitable when the values and mean are small.
Variance from a Frequency Distribution
The mean is x̄ = Σfx/Σf. The population variance can be calculated as σ² = Σf(x − x̄)²/Σf or by the shortcut σ² = Σfx²/Σf − (Σfx/Σf)². Frequency counts each repeated occurrence of a value.
Properties of Variance
If y = x + c or y = x − c, then Var(y) = Var(x). If y = kx, then Var(y) = k²Var(x). More generally, Var(a + bx) = b²Var(x). Variance is always non-negative and equals zero only when all observations are identical.
Relation Between Variance and Standard Deviation
The relation is σ = √σ² for population data and s = √s² for sample data. Therefore, variance is expressed in squared units, while standard deviation is expressed in the original units.
Variance Data Table
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Population and Sample Variance
The denominator distinguishes population variance from sample variance.
| Measure | Formula | Denominator |
|---|---|---|
| Population variance | σ² = Σ(x − μ)² / N | N |
| Sample variance | s² = Σ(x − x̄)² / (n − 1) | n − 1 |
Note: Use population variance for a complete data set and sample variance when the data is a sample.
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Variance Revision Points
Key formulas, properties and methods for quick revision.
- Variance is the average squared deviation from the mean.
- Population variance: σ² = Σ(x − μ)²/N.
- Sample variance: s² = Σ(x − x̄)²/(n − 1).
- Shortcut: variance = mean of squares − square of mean.
- For frequency data: σ² = Σfx²/Σf − (Σfx/Σf)².
- Var(x + c) = Var(x), and Var(kx) = k²Var(x).
- Variance is never negative and is zero only when all values are equal.
- Standard deviation is the square root of variance.
Variance FAQs
What is the variance formula for individual observations?
For population data, σ² = Σ(x − μ)²/N. The shortcut form is σ² = Σx²/N − (Σx/N)².
What is the variance of 2, 4, 6, and 8?
The mean is 5, and the squared deviation sum is 20. Population variance = 20/4 = 5.
How is sample variance different from population variance?
Population variance divides by N, while sample variance divides by n − 1. For squared deviation sum 20 from four observations, the answers are 5 and 20/3 respectively.
What happens to variance when 5 is added to every observation?
It remains unchanged because Var(x + c) = Var(x). Adding a constant shifts the data but does not change its spread.
What happens to variance when every observation is multiplied by 3?
Variance becomes nine times the original variance because Var(3x) = 3²Var(x) = 9Var(x).
How do you calculate variance from a frequency table?
Use σ² = Σfx²/Σf − (Σfx/Σf)² for population data, where x is the value and f is its frequency.