Population
- Standard deviation
- 2
- Variance
- 4
÷ N
Standard deviation calculator
Paste one dataset and compare sample and population results with the working shown.
N → x̄ → Σ → σ / s
Enter your dataset
Calculated in your browser. The number list is not uploaded or saved.
Separate values with commas, spaces, semicolons, or new lines. Use a period for decimals. Scientific notation such as 1.2e3 is accepted.
σ and s
Results
- Count
- 8
- Mean
- 5
Sample
- Standard deviation
- 2.1380899353
- Variance
- 4.57142857143
÷ (n − 1)
Calculation steps
The calculator uses a numerically stable running update, then shows the equivalent deviation formulas below.
- 1. Mean = sum of values ÷ n = 5.
- 2. Sum of squared deviations, Σ(xᵢ − mean)² = 32.
- 3. Population variance = 32 ÷ 8 = 4; population SD = √4 = 2.
- 4. Sample variance = 32 ÷ (8 − 1) = 4.57142857143; sample SD = √4.57142857143 = 2.1380899353.
Deviation check
| # | Value | xᵢ − mean | (xᵢ − mean)² |
|---|---|---|---|
| 1 | 2 | -3 | 9 |
| 2 | 4 | -1 | 1 |
| 3 | 4 | -1 | 1 |
| 4 | 4 | -1 | 1 |
| 5 | 5 | 0 | 0 |
| 6 | 5 | 0 | 0 |
| 7 | 7 | 2 | 4 |
| 8 | 9 | 4 | 16 |
How to use this standard deviation calculator
- Paste or type observations into the Numbers box. Commas, spaces, semicolons, and line breaks separate values, so a spreadsheet column can be pasted directly. Write decimals with a period, for example 12.5. Minus signs and notation such as 3e-4 are accepted. Labels, currency symbols, percent signs, and decimal commas are not.
- Choose an example or replace the list, then select Calculate. The page checks every token and identifies the first invalid item instead of quietly discarding it. Your input stays in place for correction.
- Read both columns before recording a result. Population values use N in the denominator. Sample values use n − 1 and therefore require at least two observations. The mean and count are shared because they describe the same list regardless of which standard-deviation definition you choose.
- Open the steps and deviation table to audit the arithmetic. Copy Results creates a plain-text summary. It does not copy every row, change precision, save data, or send numbers to a server.
What standard deviation and variance mean
Standard deviation describes spread around the arithmetic mean. Zero means every observation is identical. A smaller value means observations lie closer to the mean; a larger value means more dispersion. Size only makes sense with the data’s unit and context: 2 seconds may matter in a sprint but not in a long manufacturing cycle.
Variance is the average squared distance from the mean, using N or n − 1. Squaring stops positive and negative deviations from cancelling, but changes the unit: centimeters become square centimeters. Standard deviation takes the square root and returns to the original unit.
The results answer different setup questions. Use population SD for every member of the group being described. Use sample SD for a subset estimating variability in a wider population. The n − 1 denominator is Bessel’s correction. This descriptive tool does not decide whether sampling was representative or perform statistical inference.
Worked standard deviation examples
Textbook population with a known SD
For 2, 4, 4, 4, 5, 5, 7, 9, n is 8, the mean is 5, and squared deviations total 32. As a population, variance is 32 ÷ 8 = 4 and SD is 2. As a sample, variance is 32 ÷ 7 ≈ 4.57142857143 and SD ≈ 2.1380899353. Only the denominator changes.
A small sample
Three production times of 12.1, 12.4, and 12.7 minutes have mean 12.4 and squared-deviation sum 0.18. Population variance is 0.06 and SD is about 0.244949. If the runs estimate future variability, sample variance is 0.09 and SD is 0.3. The dataset’s purpose determines the reported result.
One observation
For the single value 7, population mean is 7, variance is 0, and SD is 0 because the complete one-member population has no spread. Sample results are unavailable: n − 1 is zero and one observation cannot estimate variability.
Where the result is useful
- Compare consistency of delivery times, machine cycles, or laboratory readings when units and conditions match.
- Summarize how quiz scores or response times cluster around their mean. Pair SD with the mean, count, and context instead of treating it as a quality score.
- Check hand calculations or spreadsheet output, making sure its population or sample function uses the intended denominator.
- Inspect source data when spread changes sharply. Real diversity, process changes, outliers, mixed units, or entry mistakes are possible; SD alone cannot choose the cause.
Input limits, precision, and common errors
- The limit is 10,000 finite numbers and 100,000 characters. The first invalid token is reported; words or symbols are never removed to create a hidden smaller dataset.
- Use one unit throughout. Mixing seconds and minutes, or percentages and decimal proportions, gives a valid but misleading calculation.
- A comma always separates values. Use 1.5 for one and a half and 1200 without a thousands separator.
- Results show up to 12 significant digits. JavaScript double precision and a stable running-variance algorithm are used, but extreme or nearly identical values beyond that precision can lose detail.
- Only 50 deviation rows are displayed for long lists, but all accepted values contribute. Copy Results includes only the summary.
- Outliers have strong influence because deviations are squared. Review the source and study design before deleting an observation.
- The tool does not produce confidence intervals, tests, z-scores, normality checks, outlier verdicts, forecasts, or AI interpretation.
Frequently asked questions
Should I use sample or population standard deviation?
Use population for the entire group being described and sample for a subset estimating a larger population. Follow and label any formula specified by an assignment or reporting standard.
Why is sample standard deviation larger?
Sample variance divides by n − 1 instead of n to compensate for a sample’s tendency to underestimate population variance. The difference shrinks as n grows.
Why does a one-number sample have no result?
Its denominator is n − 1 = 0, and one value contains no between-observation variation. A complete one-member population can legitimately have SD zero.
Can I paste from Excel or Google Sheets?
Yes. Paste a numbers-only column, row, or block. Lines, tabs, spaces, commas, and semicolons separate values. Remove headers and use decimal periods.
Are my numbers stored or analyzed elsewhere?
No. Parsing, calculation, formatting, and copying happen in this browser. The tool creates no account or history and does not send the list as analytics data.