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Standard deviation calculator

Enter sample or population data in the given input box to find standard deviation using standard deviation calculator.

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Standard deviation calculator 

Standard deviation calculator is used to find the standard deviation, variance, mean, and the statistical sum of squares. This deviation calculator takes sample and population set of data values.

What is standard deviation?

The measure of the spread of a data distribution is known as standard deviation. It measures the distance among each data observation and the mean. The standard deviation is of two types:

  • Sample standard deviation 
  • Population standard deviation

If the given data is population of its own, then divide the sum of squares by N.
If the given data is the sample from a larger population, then the sum of squares must be divided by n – 1. 

Standard deviation formula

The formula for the population standard deviation is:

Population standard deviation formula
 
The formula for the sample standard deviation is: 

sample standard deviation formula

How to calculate standard deviation?

Below are a few solved examples of standard deviation.

Example 1: For population standard deviation

Find the population standard deviation of 8, 22, 26, 25, 30, 33?

Solution

Step I: Find the mean of the given population data.

Mean of population data = Σx/n
                                              = [8 + 22 + 26 + 25 + 30 + 33]/6
                                              = 144/6
                                              = 24

Step II: Now find the typical distance of each data point & the mean and square of each deviation. 

Data values (xi xi - µ  (xi - µ)2
8 8 – 24 = -16 (-16)2 = 256
22 22 – 24 = -2  (-2)2 = 4
26 26 – 24 = 2   (2)2 = 4
25 25 – 24 = 1 (1)2 = 1
30 30 – 24 = 6 (6)2 = 36
33 33 – 24= 9 (9)2 = 81

Step III: Add the deviations to find statistical sum of squares.

Σ(xi - µ)2 = 256 + 4 + 4 + 1 + 36 + 81

Σ(xi - µ)2 =  382

Step IV: Now divide the sum of squares by n.

Σ(xi - µ)2/n = 382/6

Σ(xi - µ)2/n = 63.667

Step V: Take the square root.

√[Σ(xi - µ)2/n] = √63.667

√[Σ(xi - µ)2/n] = 7.979

Example 2: For sample standard deviation

Find the sample standard deviation of 12, 15, 18, 20, 25?

Solution 

Step I: Find the mean of the given sample data.

Mean of sample data = Σx/n
                                       = [12 + 15 + 18 + 20 + 25]/5
                                       = 90/5
                                       = 18

Step II: Now find the typical distance of each data point & the mean and square of each deviation. 

Data values (xi)    xi - x̅   (xi - x̅)2
12 12 – 18 = -6    (-6)2 = 36
15 15 – 18 = -3   (-3)2 = 9
18 18 – 18 = 0 (0)2 = 0
20 20 – 18 = 2  (2)2 = 4
25 25 – 18 = 7   (7)2 = 49

Step III: Add the deviations to find statistical sum of squares.

Σ(xi - x̅)2 = 36 + 9 + 0 + 4 + 49

Σ(xi - x̅)2 =  98

Step IV: Now divide the sum of squares by n – 1.

Σ(xi - x̅)2 / n-1 = 98/5-1

Σ(xi - x̅)2 / n-1 = 98/4

Σ(xi - x̅)2 / n-1 = 24.5

Step V: Take the square root.

√[Σ(xi - x̅)2 /n-1] = √24.5

√[Σ(xi - x̅)2 /n-1] = 4.95

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