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Standard Deviation Calculator

Enter your data set and instantly get mean, variance, standard deviation, range, median, mode, and a visual distribution chart with step-by-step breakdown.

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Population Standard Deviation
📈 Data Distribution
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Standard Deviation: Measuring Data Spread

Standard deviation is one of the most important measures in statistics. It quantifies the amount of variation or dispersion in a data set. A low standard deviation means data points are clustered close to the mean, while a high standard deviation indicates data is spread out over a wider range. Understanding standard deviation is essential for data analysis, quality control, scientific research, finance, and virtually any field that deals with numerical data.

Population vs. Sample Standard Deviation

Population Standard Deviation (σ):
  σ = √[ ∑(xᵢ - μ)² / N ]
  Use when data = entire population

Sample Standard Deviation (s):
  s = √[ ∑(xᵢ - x̄)² / (n-1) ]
  Use when data = sample from population
  (n-1) is Bessel's correction for bias

Variance = (Standard Deviation)²

Coefficient of Variation (CV):
  CV = (SD / Mean) × 100%
  Allows comparison between different scales

The Empirical Rule (68-95-99.7):
  68% of data within ±1 SD of mean
  95% of data within ±2 SD of mean
  99.7% of data within ±3 SD of mean

Applications of Standard Deviation

In finance, standard deviation measures investment risk and volatility. A stock with a high SD has more price fluctuation and thus more risk. Portfolio diversification aims to reduce overall SD. In quality control, Six Sigma methodology uses standard deviation to measure process capability: a Six Sigma process has only 3.4 defects per million opportunities. In science, SD quantifies experimental uncertainty and reproducibility. In education, standardized test scores are often reported as standard deviations from the mean (z-scores).

When to Use Population vs. Sample
Use population SD (divide by N) when your data represents every member of the group you are analyzing (e.g., test scores of all students in a class). Use sample SD (divide by n-1) when your data is a subset drawn from a larger group (e.g., surveying 100 people out of a city). When in doubt, use sample SD as it provides a more conservative (slightly larger) estimate and corrects for the bias inherent in estimating population parameters from samples.

Frequently Asked Questions

What is standard deviation? ▾
Standard deviation measures how spread out numbers are from the average. Low SD = data clustered near mean. High SD = data spread widely. It is the square root of variance.
What is the difference between population and sample SD? ▾
Population SD divides by N (total count). Sample SD divides by n-1 (Bessel's correction) to account for estimating from a subset. Sample SD is slightly larger.
What is variance? ▾
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance. Variance is in squared units while SD is in the same units as the data.
What is the empirical rule? ▾
For normal distributions: 68% of data falls within 1 SD of mean, 95% within 2 SD, 99.7% within 3 SD. Also called the 68-95-99.7 rule. Useful for identifying outliers.
What is a good standard deviation? ▾
There is no universal "good" SD. It depends on context. Compare using coefficient of variation (CV = SD/mean x 100%). Generally, lower CV means more consistent data. In finance, lower SD means lower risk.
Can standard deviation be negative? ▾
No. SD is always zero or positive because it is the square root of variance (sum of squared differences). SD = 0 means all values are identical. It can never be negative.