Understanding Variance in Statistics
Variance is a fundamental statistical measure that quantifies the spread or dispersion of a dataset around its mean. It helps us understand how far a set of numbers are spread out from their average value.
Population vs. Sample Variance
There are two types of variance calculations:
- Population Variance (σ²): Used when you have data for an entire population. Divides the sum of squared differences by N (total count).
- Sample Variance (s²): Used when working with a sample of a larger population. Divides by (n-1) instead of n to provide an unbiased estimate.
How to Calculate Variance
The steps to calculate variance are:
- 1. Calculate the mean (average) of your numbers
- 2. Subtract the mean from each number to get differences
- 3. Square each difference
- 4. Sum all squared differences
- 5. Divide by n-1 for sample variance or n for population variance
Applications of Variance
Variance is used in many real-world applications:
- • Financial risk assessment and portfolio management
- • Quality control in manufacturing
- • Research and experimental design
- • Weather forecasting and climate studies
- • Educational assessment and testing
Example Calculation
Let's calculate the sample variance for the numbers: 2, 4, 4, 4, 5, 5, 7, 9
- 1. Mean = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) ÷ 8 = 5
- 2. Differences from mean: -3, -1, -1, -1, 0, 0, 2, 4
- 3. Square differences: 9, 1, 1, 1, 0, 0, 4, 16
- 4. Sum squared differences: 32
- 5. Divide by (n-1) = 7 for sample variance: 32 ÷ 7 ≈ 4.57