OneTaskBy Tim King · Updated
Standard Deviation Calculator
Sample and Population SD, Variance and Mean
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How the Standard Deviation Calculator works
This standard deviation calculator measures how spread out a list of numbers is. Paste values separated by commas, spaces or new lines and choose sample or population. You get both standard deviations and variances, the mean, the standard error, the coefficient of variation and the ranges one and two standard deviations either side of the mean.
The standard deviation is the typical distance of a value from the mean. A small one means the data clusters tightly, a large one means it is scattered. It is the basis of error bars, quality control limits, exam grade boundaries and investment risk. Use the sample version when your numbers are a selection from a bigger group, which is nearly always, and the population version only when you have every member.
How the standard deviation calculator works
The formula for the population standard deviation is σ = √(Σ(x − mean)² / n). The sample standard deviation divides by n − 1 instead, Bessel's correction, which makes up for the sample mean sitting closer to its own data than the true mean would. Variance is the square before the root. The standard error is s / √n, and the coefficient of variation is SD over mean.
First example. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5 and the squared deviations sum to 32, so the population SD is √(32 / 8) = 2.
Second example. The same data as a sample gives s = √(32 / 7) = 2.13809, slightly larger because it divides by 7.
How many values fall within 1 SD? In a normal distribution about 68%, 95% within 2 SD and 99.7% within 3 SD.
Share of a normal distribution within k standard deviations of the mean, with Chebyshev's minimum
| k (SDs) | Within ±k SD | Outside | Rarity outside | Any distribution, at least |
|---|---|---|---|---|
| 0.25 | 19.74125% | 80.3% | 1 in 1.25 | No guarantee |
| 0.5 | 38.29249% | 61.7% | 1 in 1.62 | No guarantee |
| 0.75 | 54.67454% | 45.3% | 1 in 2.21 | No guarantee |
| 1 | 68.26895% | 31.7% | 1 in 3.15 | No guarantee |
| 1.25 | 78.87003% | 21.1% | 1 in 4.73 | 36.0% |
| 1.5 | 86.63855% | 13.4% | 1 in 7.48 | 55.6% |
| 1.75 | 91.98818% | 8.01% | 1 in 12.5 | 67.3% |
| 2 | 95.44999% | 4.55% | 1 in 22 | 75.0% |
| 2.25 | 97.55511% | 2.44% | 1 in 40.9 | 80.2% |
| 2.5 | 98.75806% | 1.24% | 1 in 80.5 | 84.0% |
| 2.75 | 99.40404% | 0.596% | 1 in 168 | 86.8% |
| 3 | 99.73001% | 0.270% | 1 in 370 | 88.9% |
| 3.25 | 99.88458% | 0.115% | 1 in 866 | 90.5% |
| 3.5 | 99.95347% | 0.0465% | 1 in 2,150 | 91.8% |
| 3.75 | 99.98231% | 0.0177% | 1 in 5,650 | 92.9% |
| 4 | 99.99366% | 0.00634% | 1 in 15,800 | 93.8% |
| 4.25 | 99.99786% | 0.00214% | 1 in 46,700 | 94.5% |
| 4.5 | 99.99932% | 0.000680% | 1 in 147,000 | 95.1% |
| 4.75 | 99.9998% | 0.000204% | 1 in 491,000 | 95.6% |
| 5 | 99.99994% | 0.0000574% | 1 in 1,740,000 | 96.0% |
Tips and pitfalls
- Sample is the safer default. Survey results, test batches and experiments are samples.
- Same units as the data. Variance is in squared units; the SD is back in the original ones.
- Outliers inflate it. One extreme value can double the SD of a small data set.
- Compare spreads with the CV. The coefficient of variation lets you compare data on different scales.
FAQ
How do you calculate standard deviation?
Find the mean, subtract it from each value and square the result. Add the squares, divide by n for a population or n − 1 for a sample, and take the square root. For 1, 3, 5 the mean is 3, the squares sum to 8, and the sample SD is √(8 / 2) = 2.
Should I use sample or population standard deviation?
Use population only when your data includes every member of the group you care about, such as the marks of the whole class. Use sample when it is a subset you want to generalise from, such as a survey of 500 voters. When unsure, sample is the standard choice in research.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean, and the standard deviation is its square root. The SD is easier to read because it is in the same units as the data: heights in cm give an SD in cm but a variance in cm². Both measure spread.
What is a good standard deviation?
It depends on the mean and the context. An SD of 5 is tiny for house prices but huge for body temperature. The coefficient of variation, SD divided by mean, puts it in proportion: under about 10% is usually considered low spread, and above 30% is high.
What is the standard error of the mean?
It estimates how far the sample mean is likely to be from the true mean, found as s / √n. It shrinks as the sample grows, so quadrupling the sample halves it. Roughly, the true mean lies within two standard errors of the sample mean about 95% of the time.
See also: all Statistics tools, Arithmetic, Number Theory, Algebra and Geometry, Trigonometry and Calculus.
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