Standard Deviation Calculator
Enter comma-separated numbers to calculate population and sample standard deviation, variance, mean, and standard error — with complete step-by-step working showing every deviation and squared deviation.
Enter your values above to see the results.
Tips & Notes
- ✓Standard deviation is always in the same units as your data. If your data is in kilograms, SD is in kilograms — not kilograms squared. Variance is in squared units; SD brings it back to the original scale.
- ✓Use sample SD (n−1) for almost all practical work. Population SD (N) is only appropriate when you have literally every member of the population — a complete census or all products in a batch.
- ✓Standard error (SE = s/√n) measures precision of the mean estimate, not spread of individual values. Larger samples produce smaller SE — the mean estimate becomes more reliable.
- ✓Doubling all values doubles the SD. Adding a constant to all values leaves SD unchanged. These properties are useful for checking calculations: shift data by subtracting the mean to simplify computation.
- ✓SD is sensitive to outliers — one extreme value inflates it significantly. For skewed data, report median and IQR alongside mean and SD to give a complete picture of the distribution.
Common Mistakes
- ✗Using population SD when sample SD is required. Nearly all real datasets are samples. Using N instead of n−1 produces a smaller, biased SD that underestimates true population spread. Excel's STDEV() uses n−1; STDEVP() uses N.
- ✗Confusing standard deviation with standard error. SD measures spread of individual data points. SE measures precision of the sample mean — how much the mean would vary across repeated samples. SE = SD/√n and is always smaller than SD.
- ✗Taking the square root of the variance before summing the squared deviations. The correct order: subtract mean, square each deviation, sum all squared deviations, divide by n or n−1, then take the square root.
- ✗Reporting SD without specifying whether it is population or sample. In published work, always state which formula was used. The difference grows larger with small sample sizes and matters significantly for n < 30.
- ✗Treating SD as a measure of skewness or shape. SD only measures spread — a dataset with SD = 5 could be perfectly symmetric or heavily skewed. Use skewness and kurtosis statistics to describe shape separately.
Standard Deviation Calculator Overview
Standard deviation is the most widely used measure of data spread. It tells you, on average, how far individual values stray from the mean. A small standard deviation means values cluster tightly around the mean. A large one means they are widely scattered. Standard deviation appears in confidence intervals, hypothesis tests, control charts, investment risk analysis, and anywhere data variability matters. This calculator handles both population and sample standard deviation with full step-by-step breakdown.
Population standard deviation — when you have all data:
σ = √[Σ(xᵢ − μ)² / N]
EX: Scores [70, 80, 85, 90, 75] → N=5, μ=80 → deviations: −10,0,5,10,−5 → squared: 100,0,25,100,25 → Σ=250 → σ=√(250/5)=√50≈7.07Sample standard deviation — when data is a subset:
s = √[Σ(xᵢ − x̄)² / (n − 1)]
EX: Same data as sample → s = √(250/4) = √62.5 ≈ 7.91 | Uses n−1=4 instead of N=5 for unbiased estimationStandard error — how much the sample mean varies:
SE = s / √n
EX: s = 7.91, n = 5 → SE = 7.91 / √5 = 7.91 / 2.236 ≈ 3.54 | SE tells you the expected variation in the mean if you repeated the sampleInterpreting standard deviation — the empirical rule for normal data:
| Range | Contains Approximately | Example (μ=80, σ=7) |
|---|---|---|
| μ ± 1σ | 68% of values | 73 to 87 |
| μ ± 2σ | 95% of values | 66 to 94 |
| μ ± 3σ | 99.7% of values | 59 to 101 |
Frequently Asked Questions
Variance (s²) is the average squared deviation from the mean. Standard deviation (s) is the square root of variance. Both measure the same spread, but SD is in the original units of your data — making it directly interpretable. Data with mean = 50 and SD = 10 means most values fall between 40 and 60. Variance of 100 conveys the same information but in squared units, which is harder to interpret intuitively.
Use population SD (σ, divides by N) when you have every member of the group — complete test scores for one class, all units in a production run. Use sample SD (s, divides by n−1) when you measured a subset and want to estimate the full population's spread — surveying 200 of 10,000 customers, testing 50 of 5,000 parts. In most practical work, you have sample data, so s is the right choice.
For roughly bell-shaped data, about 68% of values fall within 1 SD of the mean, and 95% within 2 SDs. Example: exam scores with mean=75 and SD=10. About 68% of students scored between 65 and 85; about 95% scored between 55 and 95. A score of 95 is 2 SDs above the mean — in the top 2.5%, which is unusual but not extremely rare.
All values in the dataset are identical. SD = 0 means there is zero variability — every observation equals the mean exactly. Example: data [7, 7, 7, 7, 7] has mean = 7 and SD = 0. In practice, SD = 0 in a real dataset usually indicates a data entry error or a measurement instrument with no precision. Any non-trivial dataset measured in the real world will have SD > 0.
Larger samples do not automatically reduce SD — more data gives a better estimate of the true spread, not less spread. However, larger samples do reduce the standard error (SE = s/√n). Doubling sample size from 25 to 100 cuts SE in half, making the mean estimate twice as precise. SD itself stabilizes as n grows, converging toward the true population SD.
Standard error SE = s/√n measures how precisely the sample mean estimates the population mean. SD measures spread of individual values around the mean. Example: a dataset with SD=20 and n=100 has SE=20/√100=2. This means the sample mean is expected to be within about ±2 units of the true population mean (68% confidence). SE shrinks with larger n; SD does not.