Standard deviation and variance calculator
Sample and population deviation, variance and coefficient of variation.
Calculator inputs
Results
Enter your values and press “Calculate” to see the result.
In short
- What it calculates
- Sample and population deviation, variance and coefficient of variation.
- Formula used
s = √( Σ(xᵢ − x̄)² / (n − 1) ) (sample)- Example
- Data: 12, 15, 11, 18, 14, 16
Standard deviation calculator
Standard deviation measures how far the data spread from their mean. Two sets can share a mean and behave completely differently.
Both versions, sample and population, are shown because mixing them up is a common mistake.
How it works
The sample version divides by n−1 and is what you use when your data is a sample of something larger. The population version divides by n and only applies when you have every case.
Formula
s = √( Σ(xᵢ − x̄)² / (n − 1) ) (sample)
σ = √( Σ(xᵢ − x̄)² / n ) (population)
CV = s / |x̄| × 100
Standard error = s / √n
Worked example
Data: 12, 15, 11, 18, 14, 16
Mean: 14.33
Sample deviation: 2.58
Population deviation: 2.36
Coefficient of variation: 18%
Explanation
The mean alone misleads
Two groups can have exactly the same mean and behave in opposite ways. Marks of 5, 5 and 5 average 5; marks of 0, 5 and 10 also average 5. The standard deviation is what separates a homogeneous group from one where anything goes, which is why no mean should be published without it.
The 68, 95 and 99.7 rule
In an approximately normal distribution, two thirds of the data fall within one standard deviation of the mean, ninety-five per cent within two and virtually all within three. It is the rule that lets you judge at a glance whether a value is unusual or expected.
Be careful when data are not normal
In highly skewed distributions, such as salaries or house prices, the standard deviation describes reality poorly because a few extreme values inflate it. There it is better to look at the median and the quartiles instead of mean and deviation.
Standard deviation and standard error are not the same
The standard deviation describes the spread of the data. The standard error describes the precision of the estimated mean, and shrinks as the sample grows. Confusing them makes results look far more precise than they are.
Frequently asked questions
Which one should I use?
The sample version (n−1) in almost every practical case, unless you hold the entire population.
What is the coefficient of variation for?
Comparing dispersion across sets with very different means, since it expresses the deviation as a percentage of the mean.
What is the difference between standard deviation and standard error?
The deviation describes how much the data vary; the standard error, how much uncertainty the calculated mean carries. The latter shrinks as the sample grows.
Does it work with highly skewed data?
It describes such cases poorly. With salaries or house prices the median and quartiles are preferable, being less sensitive to extreme values.
Need to calculate something else?
These tools are often used alongside this calculator.
Average
Paste your list of numbers and get the mean, median, mode, range and standard deviation.
Confidence interval
The range the true mean is likely to fall in at your chosen confidence level.
Z score
How many standard deviations a value sits from the mean and which percentile it occupies.
Cohen’s d
Effect size between two groups or before and after, with confidence intervals, plain-language equivalents…
Sample size
How many responses you need to estimate a percentage or a mean, with finite population, design effect and…
Pearson correlation
Pearson’s r between two variables, with p value, confidence interval, regression line and scatter plot.