What standard deviation is
Standard deviation measures how much the values in a data set typically deviate from the mean. A low standard deviation means values are clustered close to the mean, while a high standard deviation means values are more spread out. It is one of the most widely used measures of spread in statistics, data science, and quality control.
Formulas used
Population variance: σ² = Σ(x − μ)² ÷ N
Sample variance: s² = Σ(x − x̄)² ÷ (n − 1)
Standard deviation: standard deviation = √variance
Coefficient of variation: CV = (standard deviation ÷ mean) × 100
Sample or population: which to choose
Use the population calculation when the data set represents EVERY element you want to analyze, like the grades of every student in one specific class. Use the sample calculation when the data is only a sample drawn from a larger group, like the responses of 200 customers out of thousands, used to estimate the behavior of the whole audience. Dividing by n-1 instead of n in the sample calculation is known as Bessel's correction, and it exists because a sample tends to slightly underestimate the true variability of the entire population.
Worked example
For the data set 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. Calculating the population deviation, the variance is 4 and the standard deviation is exactly 2, a classic example used in introductory statistics textbooks.
What the coefficient of variation is for
Comparing standard deviations directly only makes sense when the data sets have similar means. If one set has a mean of 1,000 and a standard deviation of 50, and another has a mean of 10 and a standard deviation of 5, the second set is proportionally far more spread out, even with a smaller absolute standard deviation. The coefficient of variation solves this by expressing spread as a percentage of the mean, letting you compare relative variability across data sets on different scales.
The empirical rule in a normal distribution
When a data set roughly follows a normal distribution, the classic "bell curve", the empirical rule, also called the 68-95-99.7 rule, describes how values are distributed around the mean: about 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This rule is widely used to spot outliers and to quickly understand the expected spread of a data set.