About the standard deviation calculator
Standard deviation measures how spread out a set of numbers is around its mean. This calculator takes a list of values and returns the standard deviation and variance for a sample or a whole population, along with the mean, sum of squares, standard error and a working table of deviations.
A small standard deviation means the values cluster near the mean; a large one means they are widely scattered.
Who should use the standard deviation calculator
- Students in statistics and the sciences.
- Analysts summarising variability in survey, test or quality-control data.
- Anyone who needs to know whether a sample is consistent or scattered.
How to use the standard deviation calculator: what each input means
- Data values
- Numbers separated by commas, spaces or new lines. At least two values are needed.
- Sample or population
- Choose sample if the data are a subset of a larger group, population if they are every member of it.
How the standard deviation calculator works: method and formula
Each value's difference from the mean is squared, the squares are added, and the total is divided by n for a population or n − 1 for a sample. The standard deviation is the square root of that variance. Dividing by n − 1 (Bessel's correction) compensates for the fact that a sample tends to understate the spread of the population it came from.
- Population
- σ = √(Σ(x − μ)² ÷ N)
- Sample
- s = √(Σ(x − x̄)² ÷ (n − 1))
- Standard error of the mean
- SE = s ÷ √n
How to read your standard deviation calculator result
For roughly normal data, about 68% of values lie within one standard deviation of the mean and about 95% within two. The coefficient of variation, standard deviation divided by the mean, lets you compare spread between data sets with different units or scales.
Standard Deviation Calculator worked example
The data set 10, 12, 23, 23, 16, 23, 21, 16.
- Mean = 144 ÷ 8 = 18.
- Sum of squared deviations = 192.
- Sample variance = 192 ÷ 7 = 27.428571, so s = 5.2372294.
- Population variance = 192 ÷ 8 = 24, so σ = 4.8989795.
Sample standard deviation ≈ 5.2372; population standard deviation ≈ 4.8990.
Practical uses of the standard deviation calculator
- Reporting variability in lab measurements.
- Comparing consistency between two groups or products.
- Preparing inputs for confidence intervals and z-scores.
Assumptions and limitations of this calculator
- Standard deviation is sensitive to outliers.
- A single value has no sample standard deviation, so two or more numbers are required.
- Interpretation with the 68–95 rule assumes data that are close to normally distributed.