Variance Calculator
Calculate variance, standard deviation, and coefficient of variation for your data set. Supports both sample and population variance with step-by-step solutions.
s^2 = Sum(x - x-bar)^2 / (n-1)Results
Sample Variance
27.4286
s^2
Standard Deviation
5.2372
s
Enter Data
Variance Type
Sample variance uses n-1 (Bessel's correction) to provide an unbiased estimate when your data is a sample from a larger population.
Results Summary
Sample Variance
27.4286
Standard Deviation
5.2372
Mean
18.0000
Coefficient of Variation
29.10%
Min
10
Max
23
Range
13
Step-by-Step Calculation
Step 1: Calculate the Mean
Mean = Sum of all values / Count
Mean = 144.00 / 8
Mean = 18.0000
Step 2: Calculate Deviations from Mean
| Value (x) | x - Mean | (x - Mean)^2 |
|---|---|---|
| 10 | -8.0000 | 64.0000 |
| 12 | -6.0000 | 36.0000 |
| 23 | +5.0000 | 25.0000 |
| 23 | +5.0000 | 25.0000 |
| 16 | -2.0000 | 4.0000 |
| 23 | +5.0000 | 25.0000 |
| 21 | +3.0000 | 9.0000 |
| 16 | -2.0000 | 4.0000 |
| Sum | - | 192.0000 |
Step 3: Calculate Variance
Sample Variance (s^2) = Sum of squared deviations / (n - 1)
s^2 = 192.0000 / (8 - 1)
s^2 = 192.0000 / 7
s^2 = 27.4286
Step 4: Calculate Standard Deviation
Standard Deviation = sqrt(Variance)
s = sqrt(27.4286)
s = 5.2372
Data Distribution & Deviations
Red dashed line shows the mean (18.0000). Bars further from the line indicate larger deviations.
Sample vs Population Comparison
Sample (n-1)
Use when estimating from a sample of a larger population
Population (N)
Use when you have all data points in the population
Variance Formulas
Population Variance
sigma^2 = Sum(x - mu)^2 / N
Divide by total count N
Sample Variance
s^2 = Sum(x - x-bar)^2 / (n-1)
Divide by n-1 (Bessel's correction)
Standard Deviation
SD = sqrt(Variance)
Same units as original data
Coefficient of Variation
CV = (SD / Mean) x 100%
Relative variability measure
Results
Sample Variance
27.4286
Standard Deviation
5.2372
?How Do You Calculate Variance?
Variance measures how spread out data is from the mean. Population variance: sigma-squared = Sum of (x - mu)^2 / N. Sample variance: s^2 = Sum of (x - x-bar)^2 / (n-1). Standard deviation is the square root of variance. Use population variance when you have all data points; use sample variance when working with a sample from a larger population (dividing by n-1 corrects for bias).
What is Variance?
Variance is a statistical measure of the spread or dispersion of a set of data points around their mean. It quantifies how much the values differ from the average. A low variance indicates that data points are clustered close to the mean, while a high variance indicates data is spread out. Variance is the average of squared deviations from the mean, and standard deviation is its square root, expressing spread in the original data units.
Key Facts About Variance
- Population variance (sigma-squared): divide by N (total count)
- Sample variance (s-squared): divide by n-1 (Bessel's correction for bias)
- Standard deviation = square root of variance (same units as data)
- Variance is always non-negative (>= 0)
- Variance of 0 means all values are identical
- Coefficient of Variation (CV) = (std dev / mean) x 100%
- Adding a constant to all data does not change variance
- Multiplying all data by k multiplies variance by k^2
Quick Answer
Variance measures how spread out data is from the mean. Population variance: sigma-squared = Sum of (x - mu)^2 / N. Sample variance: s^2 = Sum of (x - x-bar)^2 / (n-1). Standard deviation is the square root of variance. Use population variance when you have all data points; use sample variance when working with a sample from a larger population (dividing by n-1 corrects for bias).
Frequently Asked Questions
Variance measures how spread out data is from the mean. It is calculated as the average of squared deviations from the mean. A variance of 0 means all values are identical; higher variance means more spread.
Population variance divides by N (total count) when you have all data points. Sample variance divides by n-1 (called Bessel's correction) to provide an unbiased estimate when working with a sample from a larger population.
Standard deviation is the square root of variance. While variance is in squared units, standard deviation is in the same units as the original data, making it easier to interpret and compare with the mean.
Coefficient of Variation (CV) = (standard deviation / mean) x 100%. It expresses variability as a percentage of the mean, allowing comparison of variability between datasets with different scales or units.
Use population variance when you have data for the entire population (all possible values). Use sample variance when you have a sample and want to estimate the variance of the larger population. Most real-world applications use sample variance.
Last updated: 2025-01-15
Results
Sample Variance
27.4286
s^2
Standard Deviation
5.2372
s