What is standard deviation and how does it work for non-grouped data?

Standard deviation is the spread of data around the mean value. It is useful for evaluating the spread of a selection of data, and is used to compare multiple data sets. Ideally a data set would have a low standard deviation.

 

For NON-GROUPED DATA

This is the standard deviation for a list of numbers (there are no frequencies)

 

 s² = (∑ [ x - x̄ ]²)/n

 

s = Standard deviation

x̄ = the mean of the data set

x = number in the data set

n = the amount of numbers in the data set

∑ = the sum of

 

Example

Find out the standard deviation of 4, 9, 11, 12, 17, 5, 8, 12, 14

 

1) Work out n

There are 9 numbers in the data set

n = 9

 

2) Work out the mean of the data set (the sum of the numbers divided by the amount of numbers)

x̄  = (∑x)/n

x̄ = (4+9+11+12+17+5+8+12+14)/9 = 10.222 (3dp)

 

3) Work out ∑( x - x̄ )²

Subtract x̄ from each value to get x - x̄

The square the result (to 3dp) to get ( x - x̄ )²

Then add all of the last column together (sum) to get the total (to 3 decimal places) = (∑ [ x - x̄ ]²)

 

x              x- x̄                         ( x - x̄ )²

4              -6.222                    38.713

9              -1.222                    1.493

11           0.778                     0.605

12           1.778                     3.161

17           6.778                     45.941

5              -5.222                    27.269

8              -2.222                    4.937

12           1.778                     3.161

14           3.778                     14.273

 

∑ ( x - x̄ )² =                          139.553

 

4) Work out the standard deviation

Divide the answer from 3 by the answer from 1

Square root the result (to 3dp)

 

s² = (∑ [ x - x̄ ]²)/n

s = √{(∑ [ x - x̄ ]²)/n}

 

s² = 139.553/9

s = 3.938 (3dp)

 

 

So the standard deviation for this data set it 3.938 (3dp)