Standard Deviation Calculator

Calculate population and sample standard deviation, variance, mean, and more from any dataset. See step-by-step calculation with formulas explained.

📈 Standard Deviation Calculator

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Results (8 values)

Mean (x̄)5.50

Population Std Dev (σ)2.29

Sample Std Dev (s)2.45

Population Variance (σ²)5.25

Sample Variance (s²)6.00

Steps

Mean (x̄) = 4 + 8 + 6 + 5 + 3 + 7 + 9 + 2 / 8 = 5.50

Σ(xᵢ − x̄)² = (4 − 5.50)² + (8 − 5.50)² + (6 − 5.50)² + (5 − 5.50)² + (3 − 5.50)² + (7 − 5.50)² + (9 − 5.50)² + (2 − 5.50)² = 42.00

Population Variance (σ²) = 42.00 / 8 = 5.25

Sample Variance (s²) = 42.00 / 7 = 6.00

💡 What is Standard Deviation and Why Does It Matter?

Standard deviation measures how spread out the values in a dataset are from the mean. A low standard deviation means the data points are clustered close to the mean; a high standard deviation means they are spread out over a wider range. It is the most widely used measure of variability in statistics.

There are two versions: Population standard deviation (σ) divides by N (the total population size), and Sample standard deviation (s) divides by N−1 (using Bessel's correction to provide an unbiased estimate). Use population SD when you have data for the entire group you're studying. Use sample SD when your data is a sample drawn from a larger population — which is the more common scenario in research.

Standard deviation is critical in science, finance, quality control, and data analysis. In finance, it measures investment risk (volatility). In manufacturing, it underpins Six Sigma quality standards. In science, it determines whether experimental results are statistically significant. The 68-95-99.7 rule states that in a normal distribution, 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs.

Data: 4, 8, 6, 5, 3, 7, 9, 2 → Mean = 5.50, Population SD (σ) ≈ 2.29, Sample SD (s) ≈ 2.45. Most values fall between 3.05 and 7.95 (within 1 SD).

σ = √[Σ(xᵢ − x̄)² / N]

Where:

σ = Population standard deviation
s = Sample standard deviation (divide by N−1)
xᵢ = Each data value
x̄ = Mean (average) of the data
N = Number of data points

📝 Worked Example

Data: {2, 4, 4, 4, 5, 5, 7, 9}

Mean = 5, Σ(xᵢ−5)² = 32

= σ = √(32/8) = √4 = 2

Sample SD

s = √(32/7)

= s ≈ 2.14