Core concept
What does standard deviation measure?
Standard deviation measures how far values typically lie from the mean. A smaller value indicates more consistent measurements, while a larger value indicates greater spread.
Formulas
Population and sample standard deviation
Population standard deviationσ = √[Σ(x − μ)² ÷ n]
Sample standard deviations = √[Σ(x − x̄)² ÷ (n − 1)]
Worked example
Analyze a repeated-measurement dataset
Consider the values 2, 4, 4, 4, 5, 5, 7, and 9.
- Calculate the mean: 40 ÷ 8 = 5.
- Find each value's difference from the mean.
- Square and add those differences.
- Divide by n for population variance.
- Take the square root to obtain σ = 2.
Choosing a formula
Should you use sample or population standard deviation?
Population
Use all relevant observations
Use population standard deviation when the dataset represents the complete group being analyzed.
Sample
Estimate a larger population
Use sample standard deviation when the observations are a sample from a broader population.
Laboratory analysis
Interpret variation between repeated trials
Standard deviation should be interpreted together with the mean, range, instrument resolution, sample size, and experimental method.
Learn how to place these statistics in context in the experimental results analysis guide.
Common mistakes
Standard-deviation calculation mistakes
- Using the population formula when the data is only a sample.
- Removing unusual results without a documented reason.
- Mixing measurements that use different units.
- Reporting standard deviation without the mean or sample size.
- Rounding intermediate calculations too early.
Interpretation limits
Assumptions and limitations
Standard deviation summarizes the spread of numerical observations around their mean, but it does not by itself show whether the data are accurate, unbiased, normally distributed, or free from outliers.
The population and sample formulas use different denominators. Choosing the wrong formula, combining unrelated datasets, or including extreme values can materially change the reported result.
Related tools
Related statistics calculators
Calculate the central values of a dataset with the Mean, Median, and Mode Calculator.
Report measurements with appropriate precision using the Significant Figures Calculator.
Questions and answers
Standard deviation FAQ
What is the difference between sample and population standard deviation?
Population standard deviation divides by n and is used when the dataset represents the entire population. Sample standard deviation divides by n minus 1 and is used when the data is a sample from a larger population.
What does a low standard deviation mean?
A low standard deviation means the values are closely grouped around the mean and show relatively little variation.
What does a high standard deviation mean?
A high standard deviation means the values are more widely spread around the mean and show greater variation.
Can standard deviation be calculated from one value?
Population standard deviation for one value is zero, but sample standard deviation is not defined because n minus 1 would be zero.