Standard Deviation Calculator
Calculate mean, median, mode, range, variance, and population & sample standard deviation.
What Is Standard Deviation?
Standard deviation is a measure of how spread out the numbers in a data set are. A low standard deviation means the values are clustered close to the mean, while a high standard deviation means they are spread out over a wider range. It is one of the most widely used measures of dispersion in statistics.
How Standard Deviation Is Calculated
The calculation has four steps:
- Find the mean: Add all the values and divide by the count.
- Find the deviations: Subtract the mean from each value.
- Square and average: Square each deviation, sum them, and divide by N (population) or N − 1 (sample). This gives the variance.
- Take the square root: The square root of the variance is the standard deviation.
Population vs Sample Standard Deviation
Use population standard deviation (σ) when your data represents the entire group you're studying — divide the squared deviations by N. Use sample standard deviation (s) when your data is a sample drawn from a larger population — divide by N − 1 (Bessel's correction). The sample formula gives an unbiased estimate of the population standard deviation. This calculator reports both so you can choose the right one for your context.
Mean, Median, and Mode
These three measures of central tendency describe the "center" of a data set in different ways. The mean is the arithmetic average. The median is the middle value when the data is sorted. The mode is the value that appears most often. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode at all if every value is unique.
Range and Variance
The range is the difference between the largest and smallest values — a simple but crude measure of spread. The variance is the average of the squared deviations from the mean. It is measured in squared units, which makes it harder to interpret than standard deviation (the square root of variance, measured in the original units). The calculator shows both so you can see how they relate.
Real-World Uses of Standard Deviation
- Finance: Measure investment risk and price volatility.
- Quality control: Monitor manufacturing tolerances and process consistency.
- Science: Report the precision and reproducibility of measurements.
- Education: Compare the spread of test scores across classes or schools.
- Sports: Analyze consistency in athletic performance over time.
A Worked Example
For the data set 4, 8, 15, 16, 23, 42: the mean is (4 + 8 + 15 + 16 + 23 + 42) ÷ 6 = 18. The deviations are −14, −10, −3, −2, 5, 24. Squaring them gives 196, 100, 9, 4, 25, 576, which sum to 910. The population variance is 910 ÷ 6 ≈ 151.67 and the population standard deviation is √151.67 ≈ 12.32. The sample variance is 910 ÷ 5 = 182 and the sample standard deviation is √182 ≈ 13.49. The median is the average of the two middle values, (15 + 16) ÷ 2 = 15.5, and there is no mode because every value is unique.
Frequently Asked Questions
Population standard deviation divides the squared deviations by N (the total count) and is used when your data represents an entire population. Sample standard deviation divides by N − 1 (Bessel's correction) and is used when your data is a sample of a larger population, giving an unbiased estimate of the population standard deviation.
First find the mean. Then subtract the mean from each value and square the result. Sum all the squared deviations, divide by N (population) or N − 1 (sample), and take the square root. The calculator shows each of these values.
Variance is the average of the squared deviations from the mean, measured in squared units. Standard deviation is the square root of the variance, measured in the same units as the original data. Standard deviation is easier to interpret because it shares the data's units.