Standard Deviation Calculator

1. Paste a list of numbers (commas/spaces/new lines all work).
2. Choose Sample or Population.
3. Click Calculate (or turn on Auto-calculate).
4. Read the mean, variance, standard deviation, plus optional steps and the mini visual.

• Parse: extract numbers (and expand x:freq pairs).
• Compute mean & spread: uses a stable one-pass method (Welford) to reduce rounding error.
• Variance: population uses N; sample uses n−1.
• Standard deviation: √variance.

Example 1 — Classic dataset (Population)

Data: 2, 4, 4, 4, 5, 5, 7, 9

1. Mean: x̄ = (2+4+4+4+5+5+7+9)/8 = 40/8 = 5
2. Squared deviations sum: Σ(x−5)² = 9+1+1+1+0+0+4+16 = 32
3. Population variance: σ² = 32/8 = 4
4. Population SD: σ = √4 = 2

Example 2 — Same data (Sample)

Use the same dataset, but treat it as a sample.

1. We already have Σ(x−x̄)² = 32 and n = 8.
2. Sample variance: s² = 32/(8−1) = 32/7 ≈ 4.571
3. Sample SD: s = √4.571 ≈ 2.138
4. Interpretation: sample SD is a bit larger because it uses n−1.

Example 3 — Frequency-style input

Data: 10:3, 12:2, 15:1 (so the list is 10,10,10,12,12,15)

1. Expanded list has n = 6.
2. Mean: x̄ = (10+10+10+12+12+15)/6 = 69/6 = 11.5
3. Squared deviations: 3·(10−11.5)² + 2·(12−11.5)² + 1·(15−11.5)²
4. That is 3·2.25 + 2·0.25 + 12.25 = 6.75 + 0.5 + 12.25 = 19.5
5. Population variance: σ² = 19.5/6 = 3.25 → SD: σ = √3.25 ≈ 1.803