Standard Deviation & Variance Calculator
Compute sample and population standard deviation, variance, mean, and sum of squares for any list of numbers.
5.237
Divided by (n - 1)4.899
Divided by N27.429
Degrees of Freedom24.000
Total PopulationDataset Metrics Breakdown
What is Standard Deviation and Variance?
Standard Deviation is a core statistical metric measuring the degree of dispersion or spread of a set of values around their arithmetic mean. A low standard deviation indicates that data points cluster tightly near the average, while a high standard deviation reveals wide variability across the data range. Variance represents the expectation of squared deviations from the mean ($s^2$ or $sigma^2$).
Key Capabilities
- Dual Sample vs Population Calculations: Automatically computes both Sample ($n-1$ denominator) and Population ($N$ denominator) results.
- Universal Delimiter Parsing: Intelligently extracts numbers separated by commas, spaces, line breaks, or tabs.
- Detailed Step-by-Step Breakdown: Displays Total Sum ($Sigma x$), Arithmetic Mean ($ar{x}$), Count ($n$), and Sum of Squares ($SS$).
- High Precision Accuracy: Evaluates large numerical datasets with complete floating-point floating point rigor.
How to Calculate Standard Deviation
- Paste or type your numbers into the text box.
- Click Calculate Statistics.
- Review the Sample Standard Deviation ($s$) if your data is a sample of a wider group, or Population Standard Deviation ($sigma$) if you have the complete dataset.
Frequently Asked Questions
When should I use Sample vs Population Standard Deviation?
Use Sample Standard Deviation ($s$, dividing by $n-1$) whenever your numbers represent a subset or sample of a larger population (such as a survey of 100 people out of a city). Use Population Standard Deviation ($sigma$, dividing by $N$) only if your numbers include 100% of the entire target group.
Why is Bessel's correction (n - 1) used?
Sample means inherently underestimate variability relative to true population means. Dividing by $n-1$ mathematically cancels this bias, providing an unbiased estimator of variance.
Can standard deviation ever be negative?
No. Because standard deviation is the square root of squared deviations, it is always greater than or equal to zero ($s geq 0$). A standard deviation of zero means every single number in the dataset is identical.
Is this tool free?
Yes, it is 100% free with unlimited statistical analysis.