Standard Deviation Calculator
Enter your numbers to get the mean, variance and standard deviation (σ and s), with each calculation step.
Standard Deviation and Variance Calculator
Standard deviation measures how far the numbers in a data set sit from the arithmetic mean. It is a basic statistical measure of how close together (consistent) or spread out the data is.
What Is Standard Deviation and What Does It Represent?
Standard deviation describes how far the data sits from the mean. A low standard deviation means the data is close to the average and consistent. A high standard deviation means the data is spread over a wide range, with large differences within the group.
Population vs. Sample Standard Deviation
There are two methods: 'population', which covers all the data, and 'sample', which stands for a larger group. The sample formula divides by (n-1) to make up for the margin of error.
Standard Deviation Formula
First find the arithmetic mean. Take each number's difference from the mean and add up the squares of these differences. Divide the total by the number of data points to get the variance. The square root of the variance is the standard deviation.
σ (population) = √[ Σ(xᵢ − x̄)² / n ]
s (sample) = √[ Σ(xᵢ − x̄)² / (n − 1) ]
x̄ = mean, xᵢ = each value, n = count of data points. The (n−1) divisor in the sample formula is known as "Bessel's correction" and ensures the sample variance is an unbiased estimate.
Where Is Standard Deviation Used?
- Finance: Return volatility of stocks or investment funds; the larger σ is, the higher the risk.
- Quality control: Checking whether product dimensions in manufacturing fall within tolerance limits.
- Education: Distribution of class grades; low σ means scores are clustered, high σ means they are spread widely.
- Scientific research: Measurement uncertainty and experimental error analysis.
How to Interpret Standard Deviation in Practice
Standard deviation shows how far a dataset strays from its mean and is the basic way to measure variability. A low value means the data clusters near the average; a high value means it is spread over a wide range. It is used often in finance to assess investment risk and in quality control to check production line consistency.
Steps for Calculating Standard Deviation
The steps are: find the arithmetic mean of the dataset, square each value's difference from the mean, add the squares and divide by the number of observations (n for a population, n-1 for a sample), then take the square root. The calculator runs these steps for you and shows both the population and sample standard deviation.
Standard Deviation and the Normal Distribution
In a normal distribution, 68.2% of the data lies within one standard deviation of the mean, 95.4% within two and 99.7% within three. This rule (the 68-95-99.7 rule, or empirical rule) is widely used in quality control, medicine and engineering to spot outliers quickly.