Mathematics.

descriptive statistics

Variance and Standard Deviation

Statistics25 minDifficulty3 out of 10

You should know: descriptive statistics

Overview

Variance measures how spread out a set of numbers is by averaging the squared deviations from the mean. The standard deviation is the square root of the variance, which brings the measure back to the original units of the data, making it easier to interpret. A population's variance is denoted σ², and its standard deviation σ; sample statistics use s² and s, with the sample variance dividing by n − 1 (Bessel's correction) rather than n to give an unbiased estimate of the population variance. Both measures are foundational to statistics: larger values indicate data spread widely around the mean, while values near zero indicate data clustered tightly around it.

Intuition

Squaring each deviation from the mean before averaging serves two purposes: it makes negative and positive deviations both contribute positively (so they don't cancel out), and it penalizes large deviations more heavily than small ones. Taking the square root at the end (to get the standard deviation) undoes the squaring's distortion of units — variance of a data set measured in meters is in square meters, but the standard deviation is back in meters, so it can be compared directly to the data and the mean.

Formal Definition

Definition

For a population of size N with mean μ, and for a sample of size n with mean x̄:

σ2=1N∑i=1N(xi−μ)2\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(x_i - \mu)^2
Population variance
s2=1n−1∑i=1n(xi−xˉ)2s^2 = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2
Sample variance (Bessel's correction)
σ=σ2,s=s2\sigma = \sqrt{\sigma^2}, \qquad s = \sqrt{s^2}
Standard deviation
Var⁡(X)=E[X2]−(E[X])2\operatorname{Var}(X) = E[X^2] - (E[X])^2
Computational shortcut formula

Worked Examples

  1. Compute the mean.

    μ=2+4+6+84=204=5\mu = \frac{2+4+6+8}{4} = \frac{20}{4} = 5

  2. Compute squared deviations from the mean: (2-5)²=9, (4-5)²=1, (6-5)²=1, (8-5)²=9.

    ∑(xi−μ)2=9+1+1+9=20\sum (x_i-\mu)^2 = 9+1+1+9 = 20

  3. Divide by N=4 for population variance, then take the square root.

    σ2=204=5,σ=5≈2.236\sigma^2 = \frac{20}{4} = 5, \quad \sigma = \sqrt{5} \approx 2.236

Answer: σ² = 5, σ ≈ 2.236.

Practice Problems

Difficulty 3/10

Find the population variance and standard deviation of {1, 3, 5, 7, 9}.

Difficulty 4/10

A sample of 4 test scores is {70, 80, 90, 100}. Find the sample variance.

Difficulty 5/10

Two machines fill bottles with mean volume 500 mL each. Machine A has σ = 2 mL, machine B has σ = 8 mL. Which machine is more consistent, and why?

Quiz

Why does the sample variance formula divide by n − 1 instead of n?
If a data set has variance 16, its standard deviation is:
Standard deviation is preferred over variance for interpretation mainly because:

Summary

  • Variance averages squared deviations from the mean; standard deviation is its square root, restoring the original units.
  • Population variance divides by N; sample variance divides by n − 1 (Bessel's correction) to give an unbiased estimator.
  • Larger standard deviation means data is more spread out around the mean; smaller means data clusters tightly around it.

References

  1. WebsiteWikipedia — Variance