Standard deviation measures how spread out your data is around the mean. Knowing the hand method makes the number meaningful; a standard deviation calculator gives you both the population and sample value instantly. We'll do both.
On this page
The two formulas
Population: σ = √( Σ(x − x̄)² / n )
Sample: s = √( Σ(x − x̄)² / (n − 1) )They differ only in the divisor — n versus n − 1.
By hand, step by step
Use the data set 2, 4, 4, 4, 5, 5, 7, 9 (n = 8).
- Mean: x̄ = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5.
- Deviations from mean: −3, −1, −1, −1, 0, 0, 2, 4.
- Square them: 9, 1, 1, 1, 0, 0, 4, 16.
- Sum of squares: Σ(x − x̄)² = 32.
- Divide and root: σ = √(32/8) = √4 = 2; s = √(32/7) ≈ 2.138.
σ or s?
If your eight numbers are the whole group, use σ = 2. If they're a sample from a larger group and you're estimating the group's spread, use s ≈ 2.138. The n − 1 in the sample formula corrects the bias that arises from estimating the mean from the same data.
On the calculator
Switch to STAT mode, choose 1-VAR, and enter the values:
2, 4, 4, 4, 5, 5, 7, 9You'll see σ = 2 and s ≈ 2.138 alongside the mean — matching the hand calculation exactly. The complete STAT walkthrough is in the statistics calculator pillar guide.
Frequently asked questions
What is the formula for standard deviation?
σ = √(Σ(x − x̄)²/n) for a population; s = √(Σ(x − x̄)²/(n − 1)) for a sample.
Divide by n or n−1?
n for a whole population (σ); n−1 for a sample (s), which removes estimation bias.
How do I get SD on the calculator?
STAT mode → 1-VAR → enter values; σ and s are shown together.
Check your standard deviation
Enter your data in STAT mode and read σ and s at a glance.
Open the standard deviation calculator →