Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, and empirical distribution ranges with complete step-by-step statistical formulas.

Data Configuration

•

Separate numbers with commas, spaces, or line breaks (minimum 2 numbers).

Standard Deviation (s)
7.8638(Variance: 61.8393)

Calculated across 8 data points with mean x̄ = 23.125.

Relative VariabilityCV: 34.01%

Mean (x̄)

23.125

Count (n)

8

Sum (∑x)

185

Data Range

23

(12 to 35)

Empirical Normal Distribution (68–95–99.7 Rule)

68.2% of Data (±1σ)

15.26 → 30.99

95.4% of Data (±2σ)

7.4 → 38.85

99.7% of Data (±3σ)

-0.47 → 46.72

1. Sample Standard Deviation (s) — Bessel's Correction
s=√
∑ (xᵢ − x̄)²n − 1
2. Population Standard Deviation (σ)
σ=√
∑ (xᵢ − μ)²N
Step-by-Step Calculation Breakdown
Step 1: Calculate the Mean (x̄) for Dataset [ 10, 20, 30, 40 ]
x̄=
10 + 20 + 30 + 404
=25.00
Step 2: Sum of Squared Differences ∑ (xᵢ − x̄)²
SS=(10 − 25)² + (20 − 25)² + (30 − 25)² + (40 − 25)²=225 + 25 + 25 + 225=500.00
Step 3: Sample Variance & Standard Deviation (n − 1 = 3)
s²=500 / 3 = 166.67→s=√166.67=12.91

Key Statistical Concepts

  • Variance: The average of squared deviations from the mean. Expressed in squared original units.
  • Standard Deviation: The square root of variance, reverting the scale back into the original units of measurement.
  • Coefficient of Variation (CV): The ratio of standard deviation to the mean, ideal for comparing variability across disparate unit scales.

Frequently Asked Questions

What does standard deviation tell you?
Standard deviation measures how spread out your data values are from the arithmetic mean. A low standard deviation indicates values cluster tightly near the average, while a high standard deviation indicates widespread dispersion across a broader range.
When should I use population vs sample standard deviation?
Use Population (σ) when your dataset includes every single member of the target group. Use Sample (s) when your dataset is a random subset of a larger population. Sample uses Bessel's correction (dividing by n − 1 instead of n) to avoid underestimating true population variance.
What is a good coefficient of variation (CV)?
The coefficient of variation (CV = σ / μ × 100%) expresses standard deviation as a percentage of the mean. A CV below 15% indicates low dispersion, 15%–30% indicates moderate variability, and above 30% indicates high variability.
What is the empirical rule (68–95–99.7 rule)?
For normally distributed bell-curve data: approx. 68.2% of data points fall within ±1 standard deviation of the mean, 95.4% fall within ±2 standard deviations, and 99.7% fall within ±3 standard deviations.

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