Standard Deviation Calculator
Calculate standard deviation
Results
| Count (N) | — |
|---|---|
| Sum (∑x) | — |
| Mean (Mean, μ or x̄) | — |
| Sum of Squares (SS) | — |
| Sample Std. Deviation (s) | — |
| Sample Variance (s²) | — |
| Population Std. Deviation (σ) | — |
| Population Variance (σ²) | — |
Standard deviation is one of those stats everyone quotes but few people want to compute by hand—especially when the dataset has twenty values and your instructor wants every squared deviation written out. This calculator returns both sample and population results plus a step-by-step trace so you can verify the work without rebuilding the table in a spreadsheet.
How to Use This Standard Deviation Calculator
- Enter your dataset. Paste numbers separated by commas, spaces, tabs, or line breaks. Large lists are supported—the parser reads every token it finds.
- Press Calculate. The results table shows count, sum, mean, sum of squares, sample variance and SD, and population variance and SD.
- Review the step-by-step block. Below the table, the tool lists the mean calculation, each deviation squared (first 15 rows for long lists), SS, variance for both types, and the final square roots.
- Need more than spread? For quartiles, outliers, and SEM in one pass, use the Statistics Calculator. To standardize a single score against a known mean and SD, try the Z-score Calculator.
Standard Deviation Formulas and Practical Applications
Standard deviation answers a simple question: on average, how far are values from the center? Variance squares those distances first, which makes the units harder to interpret—SD puts you back in the original measurement scale.
Sum of squares and mean
x̄ = Σx / N
SS = Σ(x − x̄)²
Example: for 10, 12, 23, 23, 16, 23, 21, 16, the mean is 18. Each value minus 18 is squared and summed to produce SS—the engine room for everything that follows.
Population variance and SD
σ² = SS / N
σ = √σ²
Use population formulas when your list is the entire group—every shipment in a lot, every sensor on one machine. Dividing by N treats each point as a full member of the population.
Sample variance and SD
s² = SS / (N − 1)
s = √s²
Sample SD applies Bessel's correction: dividing by N − 1 instead of N compensates for using the sample mean in place of the unknown population mean. That is why a classroom sample almost always reports s, not σ.
Where teams use SD day to day
- Checking whether batch weights stay within a tight band around the target mean.
- Comparing variability between two production lines before changing a process.
- Feeding sample SD into a confidence interval for a survey mean.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population SD divides SS by N; sample SD divides by N − 1. The sample version is slightly larger on average, which helps when you are estimating spread for a bigger group you have not measured fully.
How is standard deviation calculated step-by-step?
Mean first, then squared deviations, then SS, then variance (pick the right denominator), then square root. The step-by-step panel on this page walks through exactly that sequence with your numbers filled in.
What is the Sum of Squares (SS)?
SS is the running total of (x − x̄)² across all points. Both variance formulas start from the same SS—the only difference is whether you divide by N or N − 1.
Why do sample stats show undefined when N is 1?
With a single value there is no spread to estimate, and N − 1 = 0 makes division impossible. Population SD still computes (it will be zero), but sample SD and variance need at least two points.
Does a higher standard deviation always mean bad data?
High SD means wider spread—not automatically wrong. Context matters: low SD on temperature logs may mean a stable process; high SD on sales by region may be expected. See the full math calculators hub for related tools.