Enter numbers separated by commas (complete population data)
Result
Step-by-Step Derivation
Population Standard Deviation Formula
Population SD σ = √[Σ(x - μ)² / N]
μ = Population Mean = Σx / N
σ² = Σ(x - μ)² / N (Population Variance)
Population standard deviation measures the spread of an entire population in the same units as the data.
⚠Population standard deviation uses N in the denominator. For sample data, use sample standard deviation with n-1.
What is Population Standard Deviation?
Population standard deviation (σ) is the most common measure of data spread. It tells you how much individual data points deviate from the population mean, in the same units as the original data.
Population Mean (μ)
The average value of all data points in the population.
Population SD (σ)
Measure of spread in original units. Square root of variance.
Variance (σ²)
Average squared deviation. σ is easier to interpret.
68-95-99.7 Rule
~68% within μ±σ, ~95% within μ±2σ, ~99.7% within μ±3σ.
💡 Example: Population: 4, 6, 8, 10, 12, 14. Mean = 9, Variance = 11.67, Std Dev = 3.42.
Population standard deviation (σ) measures the spread of an entire population. It is the square root of population variance. Formula: σ = √[Σ(x - μ)² / N].
What does standard deviation tell us?▼
Standard deviation measures data dispersion. A small σ means data points cluster near the mean; a large σ means data is spread out. It is in the same units as the original data.
When should I use population standard deviation?▼
Use when you have complete population data (all members of a group). Examples: all student test scores, all employee salaries, all products from a batch.
How is standard deviation used in statistics?▼
Standard deviation is used in hypothesis testing, confidence intervals, quality control (6σ), risk analysis, and comparing data distributions.
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