“The average test score was 75” tells you where the center is — but it says nothing about whether everyone scored near 75 or the class split between 40s and 100s. That’s the job of variance and standard deviation: they measure how spread out a dataset is. Learn to calculate them, and you’ll understand the single most-used summary of “typical distance from average” in all of statistics.
Variance vs. Standard Deviation
- Variance is the average of the squared distances from the mean. It captures spread precisely, but its units are squared (square-dollars, square-inches) — awkward to interpret.
- Standard deviation is simply the square root of the variance. Taking the root returns the spread to the original units, which is why it’s the number people actually report.
Always compute variance first, then take the square root. If a problem asks only for standard deviation, variance is still your mandatory middle step.
Population vs. Sample: Which Formula?
- Population (N): you measured every member of the group (all students in a class). Parameters are denoted μ and σ, and you divide by N.
- Sample (n): you measured a subset to estimate the whole (30 voters out of a city). Statistics are denoted x̄ and s, and you divide by n − 1.
Bessel’s correction and degrees of freedom
Dividing a sample’s squared deviations by plain n systematically underestimates the true population variance. Statisticians correct this bias by dividing by n − 1 instead — this is Bessel’s correction. The quantity n − 1 is called the degrees of freedom: the number of values still free to vary once the sample mean is fixed.
The Formulas
Population
σ^2 = (Σ(x – μ)^2)/(N) σ = √((Σ(x – μ)^2)/(N))
Sample
s^2 = (Σ(x – x̄)^2)/(n – 1) s = √((Σ(x – x̄)^2)/(n – 1))
Dividing by n or n − 1 is the single most tested distinction in this topic. Sample → n − 1. Population → N. Get that right and half the battle is won.
The 5-Step Calculation Framework
- Find the mean: add all values, divide by the count.
- Find each deviation: subtract the mean from every data point (x − x̄).
- Square each deviation: this kills the negative signs so below-average and above-average values can’t cancel out.
- Sum the squared deviations: add them all up — this total is Σ(x − x̄)².
- Divide, then square-root: divide by (n − 1) for a sample or N for a population to get variance, then take the square root for standard deviation.
Worked Example 1: Sample Standard Deviation
Find the sample variance and standard deviation of: 3, 7, 8, 10, 12
Step 1: Mean
Sum = 3 + 7 + 8 + 10 + 12 = 40; n = 5; x̄ = 40 ÷ 5 = 8
Steps 2–3: Deviations and their squares
- 3: (3 − 8) = −5 → (−5)² = 25
- 7: (7 − 8) = −1 → (−1)² = 1
- 8: (8 − 8) = 0 → (0)² = 0
- 10: (10 − 8) = 2 → (2)² = 4
- 12: (12 − 8) = 4 → (4)² = 16
Step 4: Sum of squared deviations
Σ(x − x̄)² = 25 + 1 + 0 + 4 + 16 = 46
Step 5: Divide by n − 1 and take the root
Sample variance: s² = 46 ÷ 4 = 11.5
Sample standard deviation: s = √11.5 ≈ 3.39
Worked Example 2: Population Standard Deviation
A teacher has the scores of her entire class of 5 students: 4, 8, 15, 16, 23. Find σ.
Mean: μ = (4 + 8 + 15 + 16 + 23) ÷ 5 = 66 ÷ 5 = 13.2
Squared deviations:
- (4 − 13.2)² = (−9.2)² = 84.64
- (8 − 13.2)² = (−5.2)² = 27.04
- (15 − 13.2)² = (1.8)² = 3.24
- (16 − 13.2)² = (2.8)² = 7.84
- (23 − 13.2)² = (9.8)² = 96.04
Sum = 84.64 + 27.04 + 3.24 + 7.84 + 96.04 = 218.8
Population variance: σ² = 218.8 ÷ 5 = 43.76
Population standard deviation: σ = √43.76 ≈ 6.62
Notice: the same numbers treated as a sample would give s² = 218.8 ÷ 4 = 54.7 and s ≈ 7.40 — a bigger spread estimate, exactly as Bessel’s correction intends.
3 Mistakes to Avoid
- Dividing by n for sample data: samples use n − 1. Dividing by n gives a biased (too small) variance.
- Thinking variance can be negative: squared deviations are always non-negative, so their average is too. A negative variance means an arithmetic error somewhere.
- Mixing up units: variance is in squared units; standard deviation is in the original units. Report the one the question asks for.
Quick sanity check: the standard deviation can never exceed the range of the data. If your range is 9 but your s comes out as 40, retrace your steps — something went wrong.