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Standard Deviation Calculator


  • Standard Deviation

How does Standard Deviation Calculator work?

  • Enter the data set of x In The Input Box.
  • Hit The Calculate Button.
  • Use The Reset Button To calculate New Values.




Standard deviation calculator finds the standard deviation of sample and population data. This tool also provides variance of sample and population data, mean, differences, the sum of differences, and the square of differences.


What is the standard deviation?


In statistics, standard deviation measures the spread of data distribution. It measures the typical distance between each data point and the mean. In other words, it shows how the data spread around the mean or average.

Standard deviation is the square root of the variance. It is denoted by s for sample data and sigma (σ) for population data.


Standard deviation formula


The formula for sample standard deviation is:

standard deviation formula sample

The formula for population standard deviation is:

population standard deviation


How to calculate the standard deviation?


Below are a few examples of standard deviation solved by our mean and standard deviation calculator.

Example 1: For sample data

Find the standard deviation of the given sample data,
1, 5, 12, 18, 24

Solution:

Step 1: Take the given information and find the mean of the sample data.
Sample data = 1, 5, 12, 18, 24
Total observation = N = 5
Mean of sample data = Σxi/N = (1 + 5 + 12 + 18 + 24)/5
Mean of sample data = x̅ = 60/5 = 12

Step 2: Find the difference and square of difference of each data value.

xi (xi - x̅) (xi - x̅)2
1 (1 - 12) = -11 (-11)2 = 121
5 (5 - 12) = -7 (-7)2 = 49
12 (12 - 12) = 0 (0)2 = 0
18 (18 - 12) = 6 (6)2 = 36
24 (24 - 12) = 12 (12)2 = 144
    Σ(xi - x̅)2 = 350

Step 3: Substitute the calculated terms in the formula of sample standard deviation.
s = √(350/(5-1))
s = √(350/4)
s = √(87.5)
s = 9.3541

Example 2: For population data

Find the standard deviation of the given population data,
11, 24, 26, 31, 36, 40

Solution:

Step 1: Take the given information and find the mean of the sample data.
Population data = 11, 24, 26, 31, 36, 40
Total observation = N = 6
Mean of population data = Σxi/N = (11 + 24 + 26 + 31 + 36 + 40)/5
Mean of population data = µ = 168/6 = 28

Step 2: Find the difference and square of difference of each data value.

xi (xi - µ) (xi - µ)2
11 (11 - 28) = -17 (-17)2 = 289
24 (24 - 28) = -4 (-4)2 = 16
26 (26 - 28) = -2 (-2)2 = 4
31 (31 - 28) = 3 (3)2 = 9
36 (36 - 28) = 8 (8)2 = 64
40 (40 - 28) = 12 (12)2 = 144
    Σ(xi - µ)2 = 526

Step 3: Substitute the calculated terms in the formula of sample standard deviation.
σ = √(526/6)
σ = √(87.6667)
σ = 9.363


References:


Khan Academy. (n.d.). What is the standard deviation? | Khan Academy.