Standard Deviation & Variance Calculator
Calculate standard deviation, variance, mean, median, and range from a data set. Supports both sample and population standard deviation.
Standard Deviation
Results update instantly as you type
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Range
12 to 22 (span: 10)
Coefficient of Variation
20.33%
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Data Distribution with Standard Deviation
Data Set (sorted)
Step 1: Calculate the Mean (x̄)
x̄ = (12 + 15 + 18 + 22 + 14 + 19 + 16 + 21) ÷ 8= 17.125
Step 2: Deviations & Squared Deviations
Table of (xᵢ − x̄) and (xᵢ − x̄)²
| i | xᵢ | xᵢ − x̄ | (xᵢ − x̄)² |
|---|---|---|---|
| 1 | 12 | -5.125 | 26.265625 |
| 2 | 15 | -2.125 | 4.515625 |
| 3 | 18 | 0.875 | 0.765625 |
| 4 | 22 | 4.875 | 23.765625 |
| 5 | 14 | -3.125 | 9.765625 |
| 6 | 19 | 1.875 | 3.515625 |
| 7 | 16 | -1.125 | 1.265625 |
| 8 | 21 | 3.875 | 15.015625 |
| Sum | 0 | 84.875 | |
Step 3: Variance
σ² = 84.875 ÷ (n − 1) = 84.875 ÷ 7 = 12.125
Step 4: Standard Deviation
σ (or s) = √(12.125) = 3.482097
Mean
17.125
Median
17
Min / Max
12 / 22
CV
20.33%
The Formula
Standard deviation measures the spread of data around the mean. Population SD (σ) divides by N; sample SD (s) divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population parameter. The variance is the square of the standard deviation and represents the average squared distance from the mean.
Variable Definitions
Mean
The arithmetic average of the data set. μ is the population mean; x̄ is the sample mean.
Standard Deviation
The average distance of each data point from the mean. σ is population SD; s is sample SD.
Divisor
Population SD divides by N; sample SD divides by n-1 to correct for bias (Bessel's correction).
Coefficient of Variation
SD divided by the mean, expressed as a percentage. Allows comparison of variability across datasets with different units.
How to Use This Calculator
- 1
Enter your data set as comma-separated numbers (e.g., "12, 15, 18, 22, 14").
- 2
Select whether your data represents a sample (most common) or the full population.
- 3
View the standard deviation, variance, mean, median, range, and coefficient of variation.
- 4
Use the coefficient of variation to compare variability across datasets with different scales or units.
In a normal distribution, 68% of values fall within one standard deviation of the mean, 95% within two
Understanding the Concept
Standard deviation (SD) is the most widely used measure of statistical dispersion. A low SD means data is clustered close to the mean; a high SD means data is spread widely. In a normal distribution, approximately 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs — this is known as the empirical rule or the 68-95-99.7 rule. The choice between sample and population SD is critical: sample SD uses n-1 (Bessel's correction) because a sample tends to underestimate the true population variability. Without this correction, the sample SD would be biased. In practice, almost all real-world research uses sample SD since data is almost always a sample drawn from a larger population. The coefficient of variation (CV) normalizes the SD by the mean, allowing you to compare variability between datasets with different units — for example, comparing the consistency of test scores (points) with heights (centimeters). Finance uses SD to measure investment volatility: a stock with a higher SD has more price fluctuation and is considered riskier.
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