Standard Deviation Calculator
Paste values to calculate population standard deviation, sample standard deviation, variance, mean, and the sum of squared deviations with unit-aware labels.
Outputs
Sample + Population
Formula
Variance -> SD
Units
Original + Squared
Live calculator
Standard deviation inputs
Calculate population and sample standard deviation from the same dataset, including variance and squared-deviation checks.
Supports up to 500 numeric values. Separate values with commas, spaces, semicolons, pipes, or line breaks.
Use any measurement label. The calculator keeps mean, median, percentile, and standard deviation in that unit; variance is squared.
Sample standard deviation
2.645751 points
Population SD
2.366432 points
Sample variance
7 points^2
Population variance
5.6 points^2
Count
5
Deviation and variance table
| Measure | Value | Check |
|---|---|---|
| Mean | 5 points | Each value is compared with this center point. |
| Sum of squared deviations | 28 points^2 | The total of every (value - mean)^2 term. |
| Population variance | 5.6 points^2 | Divide squared deviations by N when the dataset is the whole population. |
| Sample variance | 7 points^2 | Divide by n - 1 when the dataset estimates a larger population. |
| Sample standard deviation | 2.645751 points | Square root of sample variance, returned in the original unit. |
Input values read by the calculator
2, 4, 4, 6, 9
Sorted values
2, 4, 4, 6, 9
Calculate spread with the right denominator
Population standard deviation
Divide squared deviations by N when the values are the whole population.
Sample standard deviation
Divide by n - 1 when the values are a sample estimating a broader population.
Variance checks
Keep squared deviations and variance visible before taking the square root.
Standard deviation formulas used on this page
Standard deviation is the square root of variance. The population and sample versions differ in the denominator.
Working formulas
Population variance
sigma^2 = sum((x - mu)^2) / N
Use when the dataset contains the complete population.
Sample variance
s^2 = sum((x - xbar)^2) / (n - 1)
Use when the dataset is a sample from a larger population.
Standard deviation
SD = sqrt(variance)
The square root brings the result back to the original unit.
Symbols
- x - value
- One observed value in the dataset.
- mu or xbar - mean
- The population mean or sample mean used as the center point.
- N or n - count
- Population count N or sample count n, depending on the method.
Sample and population spread shown separately
Dataset checks before answers
- Values can be pasted with commas, spaces, semicolons, pipes, or line breaks.
- Sorted-order previews make median, percentile, minimum, maximum, and range checks easier to audit.
- Odd-count, even-count, and decimal presets expose the edge cases that often cause manual mistakes.
- A custom unit label keeps scores, dollars, kilograms, centimeters, seconds, or dimensionless values clear.
Method labels on every result
- Sample and population standard deviation are separated because they use different denominators.
- Percentiles show both nearest-rank and interpolated methods instead of hiding the convention.
- Z-scores are marked unitless while the raw value, mean, and standard deviation retain the dataset unit.
- Copy and print controls help move checked results into worksheets, reports, and study notes.
Standard deviation support for study and reporting
Students
Check homework datasets while seeing sorted values, formulas, and the method behind each output.
Teachers
Build classroom examples for center, spread, percentile, z-score, and sample-versus-population lessons.
Analysts
Quickly validate small datasets before moving values into spreadsheets, notebooks, or reports.
How it works in three quick steps.
Enter numeric values
Paste the dataset or choose an odd, even, or decimal preset.
Review the mean
The calculator subtracts the mean from each value before squaring deviations.
Choose the right deviation
Use population standard deviation for complete populations and sample standard deviation for estimates.
Save or print spread calculations
Copy result summary
Copy the dataset count, main statistic, and method notes into assignments or analysis notes.
Print a checked worksheet
Print inputs, result cards, formula tables, FAQs, and related tools for offline review.
Verify sorted order
Use the sorted preview to confirm the exact values used for median and percentile positions.
About this standard deviation calculator
Toolarithm's Standard Deviation Calculator is built for users who need to understand spread, not just copy a final value. It calculates the mean, sum of squared deviations, population variance, population standard deviation, sample variance, and sample standard deviation from the same pasted dataset. The page keeps the two denominator choices visible because many statistics mistakes come from mixing sample and population formulas.
The calculator supports score datasets, measurements, small experiments, classroom examples, and quick spreadsheet verification. A custom unit label is included because standard deviation should be interpreted in the original unit, while variance belongs in squared units. Showing both values side by side makes the formula path easier to audit before the result is used in a z-score, percentile discussion, report, or lesson.
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