What is standard deviation and what does it measure?

Standard deviation (σ or s) measures how spread out data values are around the mean. A low standard deviation means values cluster close to the mean; a high standard deviation means values are widely spread. Example: test scores with a mean of 70 and SD of 5 are tightly clustered (most scores 65–75). The same mean with SD of 20 means scores range widely (many between 50–90).

What is the difference between population and sample standard deviation?

Population SD (σ) is used when you have data for an entire population: divide by N (the count). Sample SD (s) is used when your data is a sample from a larger population: divide by N−1 (Bessel's correction) to produce an unbiased estimate. In practice, use sample SD (N−1) whenever your data represents a subset of all possible observations — which is almost always. Most calculators default to sample SD.

What does the 68-95-99.7 rule mean?

In a normal distribution, approximately 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is the empirical rule. Example: adult male heights with mean 70 inches and SD 3 inches → 68% are 67–73 inches; 95% are 64–76 inches; 99.7% are 61–79 inches.

What is the difference between standard deviation and variance?

Variance is the average of squared differences from the mean: σ² = Σ(x−μ)²/N. Standard deviation is the square root of variance: σ = √(Σ(x−μ)²/N). Variance is in squared units (e.g., inches²); standard deviation is in the same units as the original data (inches), making it more interpretable. Both measure spread, but SD is more commonly reported because it's in the original unit of measurement.

Standard Deviation Calculator

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Standard Deviation — Formula & Guide

Population SD (σ)

σ = √[ Σ(xᵢ − μ)² / N ]
Use when your data is the entire population.

Sample SD (s)

s = √[ Σ(xᵢ − x̄)² / (n−1) ]
Use when your data is a sample from a larger population (Bessel's correction).

Frequently Asked Questions

How do I calculate standard deviation by hand?

For data {2, 4, 4, 4, 5, 5, 7, 9}: (1) Find mean = (2+4+4+4+5+5+7+9)/8 = 5. (2) Subtract mean, square each: (2−5)²=9, (4−5)²=1×3=3, (5−5)²=0×2=0, (7−5)²=4, (9−5)²=16. (3) Sum = 9+3+3+3+0+0+4+16 = 38. Wait: (4−5)² = 1, three of them = 3. Sum = 9+1+1+1+0+0+4+16 = 32. (4) Divide by N−1=7 → 32/7 ≈ 4.57. (5) √4.57 ≈ 2.14.

Is a higher or lower standard deviation better?

Neither is inherently better — it depends on context. Low SD means consistent, predictable values (good for quality control or test scoring). High SD means high variability (sometimes desirable in investment returns meaning upside potential, but often indicates inconsistency).

Worked example: 2, 4, 4, 4, 5, 5, 7, 9

Mean = (2+4+4+4+5+5+7+9) / 8 = 5
Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16 → sum = 32
Population variance = 32 / 8 = 4
SD = √4 = 2

Related Calculators

References

Pearson, K. (1894). "Contributions to the Mathematical Theory of Evolution." Phil. Trans. Royal Society A, 185.
Freedman, D., Pisani, R. & Purves, R. (2007). Statistics (4th ed.). Norton.

What Standard Deviation Tells You

Standard deviation (σ or s) measures how spread out data values are around the mean. A low SD means values cluster tightly; a high SD means they spread widely. For a normal distribution, 68% of values fall within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD (the empirical rule). Use population SD (σ, divide by N) when you have all data; use sample SD (s, divide by N−1) when working with a sample — Bessel's correction (N−1) reduces bias in the sample estimate.

Practical uses: quality control (6-sigma means defects occur less than 3.4 per million), finance (SD of returns = volatility; higher SD = riskier investment), education (standardized test scores use SD to compare), and weather forecasting (temperature variability). A coefficient of variation (CV = SD/mean × 100%) compares variability between datasets with different scales — useful when comparing the consistency of apples to oranges.

Empirical Rule (Normal Distribution)

Range	% of Data	Example (mean=100, SD=15)
Mean ± 1 SD	68.3%	85 to 115
Mean ± 2 SD	95.4%	70 to 130
Mean ± 3 SD	99.7%	55 to 145
Outside ± 3 SD	0.3%	< 55 or > 145 (rare)