Standard Deviation Calculator
Calculate sample and population standard deviation with step-by-step explanations
Data Input
Enter at least 2 values to calculate standard deviation. Maximum 30 values allowed.
Sample vs Population
Sample (s)
- • Uses n-1 (Bessel's correction)
- • Estimates population parameter
- • More common in practice
- • Unbiased estimator
Population (σ)
- • Uses N (total count)
- • True population parameter
- • Rarely available in practice
- • Exact calculation
Interpretation Guide
Low Standard Deviation
Data points close to the mean
High Standard Deviation
Data points spread out from mean
Zero Standard Deviation
All values are identical
Quick Tips
Standard deviation is always non-negative
Same units as the original data
Square root of variance
Key measure of data variability
Understanding Standard Deviation
What is Standard Deviation?
Standard deviation is a measure of variability that indicates how spread out data points are from the mean. It's the square root of variance and provides a measure of dispersion in the same units as the original data.
Key Properties
- •Always non-negative (σ ≥ 0)
- •Zero when all values are identical
- •Expressed in original data units
- •Square root of variance
Formulas
Population Standard Deviation
σ = √[Σ(xi - μ)² / N]
Used when you have data for entire population
Sample Standard Deviation
s = √[Σ(xi - x̄)² / (n-1)]
Uses Bessel's correction (n-1) for unbiased estimation
Empirical Rule (68-95-99.7 Rule)
For normally distributed data, the empirical rule states:
within 1σ of mean
within 2σ of mean
within 3σ of mean
Applications
Quality Control
Monitor process consistency and identify outliers
Risk Assessment
Measure uncertainty in financial and scientific data
Research Analysis
Describe data variability in experiments
Standardization
Convert data to z-scores for comparison