Standard Deviation Calculator
Calculate population and sample standard deviation with variance, mean, and step-by-step calculations. Perfect for statistics, data analysis, and research applications.
Standard Deviation & Variance Calculator
Common Standard Deviation Examples
Click on these links to see instant calculations with common data sets:
Standard Deviation & Variance
Standard deviation measures how spread out data points are from the mean. Variance is the square of standard deviation. Both are fundamental measures of data dispersion in statistics.
Standard Deviation Formulas
Sample Standard Deviation: s = √(Σ(x - x̄)² / (N-1))
Population Variance: σ² = Σ(x - μ)² / N
Sample Variance: s² = Σ(x - x̄)² / (N-1)
Where: μ = population mean, x̄ = sample mean, N = count
Standard Deviation Examples
| Data Set | Mean | Population σ | Sample s |
|---|---|---|---|
| 1,2,3,4,5 | 3 | 1.414 | 1.581 |
| 10,12,14,16,18 | 14 | 2.828 | 3.162 |
| 100,100,100,100 | 100 | 0 | 0 |
| 85,90,78,92,88 | 86.6 | 4.97 | 5.55 |
| 1,10,5,8,3 | 5.4 | 3.29 | 3.68 |
- Statistics: Analyze data spread and variability in research and surveys
- Quality Control: Monitor manufacturing processes and product consistency
- Finance: Measure investment risk and portfolio volatility
- Education: Analyze test scores and student performance distributions
- Science: Evaluate experimental data and measurement precision
Frequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out data points are from the mean (average). A low standard deviation means data points are close to the mean, while a high standard deviation means data points are spread out over a wider range.
What is the difference between population and sample standard deviation?
Population standard deviation uses the entire dataset and divides by N. Sample standard deviation uses a subset of data and divides by N-1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
How do you calculate standard deviation?
To calculate standard deviation: 1) Find the mean, 2) Calculate squared differences from the mean, 3) Find the average of squared differences (variance), 4) Take the square root of variance. For sample data, divide by N-1 instead of N.