Standard Deviation Calculator
Calculate population and sample standard deviation, variance, mean, sum, and margin of error with step-by-step solutions
Check Standard Deviation Calculator
Paste a dataset to measure spread around the mean, then choose population or sample logic based on whether your numbers are the whole group or a sample from a larger group.
How the result is built
Formula, Variables, And Worked Example
Use N when the dataset is the full population and n - 1 when the dataset is a sample used to estimate population spread.
Each x value and the mean use the original data unit. Standard deviation uses that same unit, while variance uses squared units. N and n are counts.
For 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5. The squared deviations sum to 32. Population variance is 32 / 8 = 4 and sigma = 2; sample variance is 32 / 7 = 4.5714 and sample s is about 2.1381.
- Choose population only when every member of the group is included.
- Do not compare standard deviations from datasets with different units without context.
- Report variance in squared units and standard deviation in the original data unit.
- Outliers can dominate squared deviations, so inspect the data plot before summarizing spread.
- Standard deviation describes spread, not whether the data is normally distributed.
Standard Deviation Formula, Assumptions, And Mistakes
When to use standard deviation
Use standard deviation when you need to describe how far values tend to sit from the mean. It is useful for repeated measurements, classroom scores, quality checks, finance returns, and any dataset where spread is as important as the average.
Population vs sample
Choose population when the dataset contains every value in the group. Choose sample when the values represent a larger group; the n - 1 denominator corrects the variance estimate. If you are unsure, sample standard deviation is usually the safer report for collected data.
What the sample confidence interval means
In sample mode, the page uses a two-sided 95% Student t critical value with n - 1 degrees of freedom. That makes the interval wider for small samples than the fixed 1.96 normal approximation. The interval assumes independent observations and is not a guarantee for strongly biased, dependent, or otherwise unsuitable samples.
Related tools
Use the Mean Median Mode Calculator for center, the Z-Score Calculator for individual standardized values, and the Confidence Interval Calculator when spread feeds an uncertainty estimate.