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Standard Deviation Calculator

Enter the comma separated values in the box to find standard deviation using standard deviation calculator.

Why do we use it?

Finding standard deviation was never so simple, thanks to the sample standard deviation calculator. The calculations offered by the sample mean calculator are:

Sample Variance
Sample Standard Deviation
Population Variance
Population Standard Deviation
Differences
Square of Differences

Let’s explore a few key points such as standard deviation definition, how to find standard deviation without using a relative standard deviation calculator, and the standard deviation equation.

What is the standard deviation?

A measure of the extent to which numbers are spread out.
Standard Deviation is a statistic that measures the dispersion of a dataset relative to its mean and is calculated as the square root of the variance. It is denoted by the Greek symbol sigma σ.

Below, you can find the plot of a normal distribution with a width of 1 band.

standard deviation graph

Standard Deviation Formula

The standard deviation formula can be expressed by taking the square root of the variance.

Sample Standard Deviation Formula

s refers to sample standard deviation
N is the number of observations
x_i is the observed values of the sample item, and
x̄ is the mean value of the sample

Population Standard Deviation Formula

σ refers to population standard deviation
N is the size of the population
x_i is the observed values from the population, and
μ is the population mean

How To Calculate Standard Deviation?

Standard deviation calculation can be carried out using the mean and standard deviation calculator above. However, we will explain the method to calculate SD with examples.

Example 1: For sample variance

Find the standard deviation of the given sample:
30, 20, 28, 24, 11, 17

Solution

Step 1: Calculate the mean value of sample data:

N = 6

Step 2: Calculate (x_i - x̄) by subtracting the mean value from each value of the data set and calculate the square of differences to make them positive.

Data Values (x_i)	x_i - x̅	(x_i - x̅ )²
30	30 - 21.67 = 8.33	(8.332)² = 69.4
20	20 - 21.67 = -1.67	(-1.672)² = 2.78
28	28 - 21.67 = 6.33	(6.332)² = 40
24	24 - 21.67 = 2.33	(2.332)² = 5.43
11	11 - 21.67 = -10.67	(-10.672)² = 113.85
17	17 - 21.67 = -4.67	(-4.672)² = 21.80

Step 3: Get the sum of all values for (x_i - x̅)².

∑(x_i - x̅)²= 69.4 + 2.78 + 40 + 5.43 + 113.85 + 21.80

∑(x_i - x̅)² = 253.26

Step 4: Divide ∑(x_i - x̅)² with (N-1).

variance

Variance = 50.65

Step 5:Take the square root of ∑(x_i - x̅)²/(N-1) to get the standard deviation.

standard deviation

s = 7.1169

The above sample standard deviation calculator can be used to check the result of this example.

Example 2: For population variance

Find standard deviation of the given population data:
10, 12, 18, 14, 21, 27

Solution

Step 1: Calculate the expected value of population data:

N = 6

Step 2: Calculate (x_i - µ) by subtracting the mean value from each value of the data set and calculate the square of differences to make them positive.

Data Values (x_i)	x_i - µ	(x_i - µ)²
10	10 - 17 = -7	(-7)² = 49
12	12 - 17 = -5	(-5)² = 25
18	18 - 17 = 1	(1)² = 1
14	14 - 17 = -3	(-3)² = 9
21	21 - 17 = 4	(4)² = 16
27	27 - 17 = 10	(10)² = 100

Step 3: Get the sum of all values for (x_i - µ)².

∑(x_i - µ)²= 49 + 25 + 1 + 9 + 16 + 100

∑(x_i - µ)² = 200

Step 4: Divide ∑(x_i - µ)² with (N).

variance

Variance = 33.333

Step 5: Take the square root of ∑(x_i - µ)²/N to get the standard deviation.