What is Standard Deviation Calculator?
Standard deviation quantifies the amount of dispersion or variation of a set of values from their mathematical mean.
How to Use This Calculator
- Choose Sample (s) or Population (σ).
- Enter raw numbers separated by commas.
- Review standard deviation, variance, and sample mean.
The Mathematical Formula & Variables
Sample s = Sqrt( Sum((x - mean)^2) / (n - 1) ); Population sigma = Sqrt( Sum((x - mean)^2) / N )
Variables Definition
| Symbol / Variable | Name | Description |
|---|---|---|
n - 1 |
Bessel Correction | Compensates for bias in small sample sizes. |
Step-by-Step Worked Example
Scenario: Sample dataset: 12, 15, 18, 20, 21, 25 (n = 6).
- Mean = (12+15+18+20+21+25) / 6 = 111 / 6 = 18.5.
- Squared deviations: (12-18.5)²=42.25, (15-18.5)²=12.25, (18-18.5)²=0.25, (20-18.5)²=2.25, (21-18.5)²=6.25, (25-18.5)²=42.25.
- Sum of squared deviations = 105.5.
- Variance s² = 105.5 / (6 - 1) = 21.1.
- Standard deviation s = √21.1 = 4.59.
Result: Standard Deviation s = 4.60
Frequently Asked Questions
What is the empirical 68-95-99.7 rule?
In a normal bell curve distribution, approximately 68.2% of all observations fall within 1 standard deviation of the mean, 95.4% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations.