Standard Deviation Calculator
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Calculate mean, median, mode, range, variance, and standard deviation from any data set.
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Standard deviation is a measure of how spread out the values in a data set are from the mean (average). A low standard deviation means the values cluster closely around the mean. A high standard deviation means the values are spread widely. It is one of the most commonly used statistics in science, finance, quality control, and research.
In a normal distribution (bell curve), about 68% of values fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. This is known as the 68-95-99.7 rule or the empirical rule.
When you have data for an entire population, you divide the sum of squared deviations by N (the count). When you have a sample drawn from a larger population, you divide by N-1 instead. This adjustment, known as Bessel's correction, compensates for the fact that a sample tends to underestimate the true population variance. In most real-world research situations, you are working with a sample rather than a complete population, so N-1 is the appropriate choice.
Context determines what a "high" or "low" standard deviation means. A standard deviation of 5 inches in human height data represents enormous variation, but a standard deviation of 5 in test scores out of 100 might be quite tight. The coefficient of variation (standard deviation divided by the mean, expressed as a percentage) allows comparison of variability across data sets with different scales and units.