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Standard Deviation Calculator

Calculate standard deviation, variance, mean, median, and coefficient of variation for a data set. Supports population and sample.

Sources: Khan Academy -- Variance and Standard Deviation · Math is Fun -- Standard Deviation

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How to Calculate Standard Deviation

Standard deviation is one of the most important statistical measures: it indicates how much the values in a data set deviate from their mean. This calculator computes standard deviation, variance, mean, median, range, and coefficient of variation.

Standard Deviation Formula

For a population of N values: sigma = sqrt(sum((xi - mu)^2) / N)

For a sample of N values (Bessel's correction): s = sqrt(sum((xi - x_bar)^2) / (N-1))

Population vs. Sample: Which to Choose?

If your data represents the entire population of interest (e.g., grades of all students in a class), use population standard deviation (divide by N). If your data is a subset from a larger population (e.g., a survey of 1,000 people to estimate opinions of an entire country), use sample standard deviation (divide by N-1).

The Empirical Rule (68-95-99.7)

For normally distributed data, standard deviation has a very practical interpretation: 68% of data falls within +/-1 sigma, 95% within +/-2 sigma, and 99.7% within +/-3 sigma. A value more than 3 standard deviations from the mean is considered an outlier.

Practical Applications

Standard deviation is used everywhere: in industrial quality control, financial analysis (market volatility), scientific research, sports (performance analysis), meteorology, and education (grade distribution).

Frequently Asked Questions

What is the difference between population and sample standard deviation?▼
Population standard deviation (sigma) divides by N and is used when you have all data from the population. Sample standard deviation (s) divides by N-1 (Bessel's correction) and is used when data is a subset. In practice, sample standard deviation is used most often.
What does standard deviation indicate?▼
Standard deviation measures how much values deviate from the mean. A low standard deviation indicates data clustered near the mean; a high one indicates wide dispersion. For a normal distribution, about 68% of data falls within +/-1 SD of the mean.
What is the coefficient of variation?▼
The coefficient of variation (CV) is the ratio of standard deviation to mean, expressed as a percentage. It allows comparing the variability of datasets with different scales. For example, a CV of 10% indicates relatively low variability regardless of the unit of measurement.