About the Standard Deviation Calculator
Two classes can have the same average test score while one has marks clustered tightly around it and the other has marks scattered from very low to very high. The average alone cannot tell them apart. The standard deviation measures that spread: how far, typically, values lie from the mean. It is the most widely used measure of variability in statistics, science, finance and quality control.
This standard deviation calculator finds the sample or population standard deviation of any list of numbers. It shows the mean, the variance, the sum of squared deviations and every step of the working, gives the result for the other formula too, counts how many values lie within one standard deviation, and reports the coefficient of variation.
How to Use the Standard Deviation Calculator
Enter your numbers, separated by commas, spaces or new lines.
Choose whether the data is a sample from a larger group or the whole population you are interested in.
The standard deviation appears with the full working.
The Formulas
mean x̄ = Σx ÷ n
sum of squares SS = Σ(x − x̄)²
sample SD s = √( SS ÷ (n − 1) )
population SD σ = √( SS ÷ n )The only difference is the divisor: n − 1 for a sample, n for a population.
Step-by-Step Example
The values 2, 4, 4, 4, 5, 5, 7, 9.
Mean: 40 ÷ 8 = 5
Deviations: −3, −1, −1, −1, 0, 0, 2, 4
Squares: 9, 1, 1, 1, 0, 0, 4, 16 → sum 32
Population: 32 ÷ 8 = 4 → σ = √4 = 2
Sample: 32 ÷ 7 = 4.57143 → s = √4.57143 = 2.13809As a population the standard deviation is 2; as a sample it is 2.13809. Six of the eight values lie within one standard deviation of the mean.
Sample or Population?
This is the choice people most often get wrong.
Use the population formula when your data includes every member of the group you want to describe — every employee of a small company, every game in a finished season.
Use the sample formula when the data is a sample drawn to estimate something about a larger group — 50 customers surveyed out of thousands, 20 parts tested from a production run. A sample's values sit slightly closer to their own mean than to the true population mean, so dividing by n would underestimate the spread. Dividing by n − 1, known as Bessel's correction, fixes that bias in the variance. With large samples the two formulas give almost the same answer; with small samples the difference matters.
Interpreting the Standard Deviation
The standard deviation is in the same units as the data, which makes it easy to read: a standard deviation of 2 marks, 3 centimetres or 5 minutes. For data that is roughly bell-shaped, about 68% of values lie within one standard deviation of the mean, about 95% within two and about 99.7% within three — the empirical rule. Values more than two or three standard deviations away are unusual and worth checking.
Coefficient of Variation
The coefficient of variation divides the standard deviation by the mean and expresses it as a percentage. It lets you compare the variability of data measured on different scales or in different units: a standard deviation of 5 is large for values around 10 but small for values around 1,000. For the example as a population, 2 ÷ 5 = 40%.
Standard Deviation in Practice
In finance, the standard deviation of returns measures an investment's volatility. In manufacturing, it measures how consistently parts meet their target size. In education, it describes how spread out test scores are. In science, it summarises the variability of repeated measurements and underlies error bars and confidence intervals.
Standard Deviation and Outliers
Because the deviations are squared, values far from the mean have an outsized effect on the standard deviation. Adding a single value of 20 to the example data raises the population standard deviation from 2 to about 5.1. Before reporting a standard deviation, it is worth looking at the data for any extreme values and deciding whether they are genuine observations or errors such as a misplaced decimal point. When outliers are genuine and important, it can help to report a robust measure such as the interquartile range alongside the standard deviation, so readers can see both the typical spread and the influence of the extremes.
Understanding Your Result
The headline is the standard deviation, marked s for a sample or σ for a population.
The mean and variance lines show the intermediate results.
The other formula line gives the standard deviation if the data were treated the other way.
The within one SD line counts the values between the mean minus and plus one standard deviation.
The coefficient of variation line gives the spread as a percentage of the mean.
When Should You Use This Calculator?
Use it for statistics homework and exams.
Use it to measure the consistency of test scores, measurements or results.
Use it to gauge the volatility of prices or investment returns.
Use it before calculating z-scores, standard errors and confidence intervals.
Common Mistakes
Using the wrong divisor. Divide by n − 1 for a sample, n for a population.
Forgetting to square the deviations. Unsquared deviations always sum to zero.
Forgetting the square root. The variance is not the standard deviation.
Confusing standard deviation with standard error. The standard error is the SD divided by √n.
Applying the 68–95–99.7 rule to skewed data. It holds only for roughly normal data, so check the shape of the data before relying on it.