Standard Deviation Calculator: Sample and Population, Step by Step
Paste or type your numbers. You get the standard deviation, variance and mean, with each step of the calculation and a deviations table.
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Sample SD s
2.13809
- Sample variance s²
- 4.571429
- Mean
- 5
- Count n
- 8
- Population SD σ (if whole population)
- 2
Sample formula divides by n − 1 (Bessel’s correction): use it when the data is a sample from a bigger group.
Show working
- n = 8, Σx = 40
- x̄ = Σx / n = 40 / 8 = 5
- Σ(x − x̄)² = (2 − 5)² + (4 − 5)² + (4 − 5)² + (4 − 5)² + (5 − 5)² + (5 − 5)² + (7 − 5)² + (9 − 5)² = 32
- s² = Σ(x − x̄)² / (n − 1) = 32 / 7 = 4.571429
- s = √4.571429 = 2.13809
Deviations table
| x | x − mean | (x − mean)² |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
How to use
- Type or paste numbers separated by commas, spaces or new lines.
- Choose sample (n − 1) or population (n).
- Read the standard deviation and open “Show working”.
Good to know
Steps: find the mean, subtract it from each value, square those deviations, add them, divide by n (population) or n − 1 (sample), and take the square root.
Sample or population? If your numbers are the whole group (all 40 students in a class), use population σ. If they are a sample used to estimate a bigger group, use sample s, which divides by n − 1 to avoid underestimating the spread.
Standard deviation is in the same units as the data (marks, ₹, cm). Roughly, in bell-shaped data about 68% of values lie within one SD of the mean and about 95% within two.
Questions
What is the SD of 2, 4, 4, 4, 5, 5, 7, 9?
Mean 5, sum of squared deviations 32. Population SD = √(32/8) = 2; sample SD = √(32/7) ≈ 2.138.
Why divide by n − 1 for a sample?
A sample’s values sit closer to their own mean than to the true population mean, so dividing by n − 1 corrects the bias (Bessel’s correction).
What does a high standard deviation mean?
The values are widely spread out from the average. A low SD means they are bunched close to it.
Which one does NCERT use?
Class 11 statistics uses the population formula (divide by n) for given data. Use “population” to match textbook answers.
Is my data stored or sent anywhere?
No. The calculation runs entirely in your browser; nothing you type leaves your device.