Find the standard deviation of a set of numbers.
Standard deviation is simply the square root of variance, bringing the units back to match the original data.
Standard deviation: √Variance
A standard deviation calculator measures the typical distance of values from their mean, in the same units as the data. It computes the population variance by dividing by n and then takes the square root. It also shows the mean and the variance so you can follow the working.
Standard deviation is the most widely reported measure of spread. A food company checks that its 500 g packets do not stray far from 500 g. Mutual fund factsheets in India list the standard deviation of returns as a risk measure. Teachers use it to see whether a class's marks are tightly bunched or widely scattered. The question it answers is: how far, typically, is a value from the average?
1. Find the mean: x̄ = Σx ÷ n.
2. Find each deviation from the mean: x − x̄.
3. Square each deviation and add them: Σ(x − x̄)².
4. Divide by n to get the population variance σ².
5. Take the square root: σ = √(Σ(x − x̄)² ÷ n).
6. Interpret it alongside the mean, for example as mean ± σ.
Variance is measured in squared units, grams squared for packet weights, which is hard to interpret. Taking the square root returns the measure to grams, so you can compare it directly with the data. A standard deviation of 2.3 g around a mean of 500 g says a typical packet is within a couple of grams of the target. Because the square root is taken after averaging, standard deviation is not the average absolute deviation, though the two are usually of similar size.
For data that follows a normal, bell-shaped distribution, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. This makes σ a practical yardstick: a packet weighing 509 g when σ is 2.3 g is nearly four standard deviations high and worth investigating. The rule does not hold for skewed data such as incomes, and it needs a reasonably large data set to be reliable.
This calculator divides by n, the population convention. If your values are a sample used to estimate the spread of a larger group, divide by n − 1 instead; the separate sample standard deviation calculator does that. To compare spread across data with different means, use the coefficient of variation, σ ÷ mean × 100. A 2 g spread matters more in a 50 g packet than in a 500 g packet, and the coefficient of variation captures that.
At a spice packing unit in Guntur, Sunita weighs five packets labelled 500 g from the line and records 498, 502, 500, 497 and 503 grams.
Find the mean (average): 498 + 502 + 500 + 497 + 503 = 2500 Mean = 2500 ÷ 5 = 500
Subtract the mean from each number and square the result: (498 − 500)² = 4 (502 − 500)² = 4 (500 − 500)² = 0 (497 − 500)² = 9 (503 − 500)² = 9
Add up the squares: Sum of squares = 26
Divide by the count (N) to get the variance: 26 ÷ 5 = 5.2
Take the square root: σ = √5.2 = 2.28035085
Answer: Standard deviation (σ) 2.280351; Variance (σ²) 5.2; Mean 500
Taking the square root of each deviation before adding, instead of the square root at the end.
Using the population formula on a small sample when the question asks for the sample standard deviation.
Applying the 68-95-99.7 rule to data that is clearly skewed.
Comparing standard deviations of data sets with very different means without using the coefficient of variation.
Checking fill weights in food and pharmaceutical packing.
Reading risk figures on mutual fund factsheets.
Summarising the spread of marks in a class or exam.
Setting control limits on manufacturing processes.
Physics and chemistry practicals, where repeated readings are summarised as mean ± standard deviation.
Why take the square root of variance?
Variance is measured in squared units (like squared rupees), which is hard to picture. Taking the square root brings the spread back to the normal units, making it easier to understand.