Standard deviation measures how spread out numbers are from their mean. Two datasets can share the same average but differ wildly in spread — standard deviation is how you see the difference.
The formulas
Population: σ = √( Σ(x − μ)² ÷ N )
Sample: s = √( Σ(x − x̄)² ÷ (n − 1) )
Use the sample formula when your data is a sample of a larger group — dividing by n − 1 corrects for underestimating spread.
How to calculate it step by step
For the sample data 4, 8, 6, 5, 3 (x̄ = 5.2):
- Find the mean: (4 + 8 + 6 + 5 + 3) ÷ 5 = 5.2
- Subtract the mean from each value: −1.2, 2.8, 0.8, −0.2, −2.2
- Square each deviation: 1.44, 7.84, 0.64, 0.04, 4.84
- Sum the squares: 14.8
- Divide by n − 1: 14.8 ÷ 4 = 3.7 (this is the variance)
- Take the square root: √3.7 ≈ 1.92
Why the square and root?
Squaring makes every deviation positive and punishes large outliers more than small ones; the root brings the answer back into the units of the original data.
Interpreting the result
- Low standard deviation — values cluster near the mean (consistent test scores)
- High standard deviation — values are widely spread (volatile prices)
- In a normal distribution, roughly 68% of values fall within 1 standard deviation of the mean and 95% within 2
Mean vs standard deviation
Dataset A: 5, 5, 5, 5 — mean 5, σ = 0 (no spread). Dataset B: 1, 3, 5, 7, 9 — mean 5, s ≈ 3.16 (wide spread).
Same average, completely different story.
Put it into practice
Paste any numbers into the free Standard Deviation Calculator — it returns sample and population SD, variance and mean instantly. For the average itself, use the Mean, Median & Mode Calculator.
