Using the standard deviation formula

The standard deviation formula is:

Standard deviation `=sigma=sqrt((Sigma(x-bar x)^2)/(n-1))`

For any set of values to find the standard deviation you should:

1.  Find the mean (of all the numbers) `= barx`

2.  Subtract the mean (from each number) `= x-barx`

3.  Square the result (of each of the above) `= (x-barx)^2`

4.  Add the results up (Add) `=sum(x-barx)^2`

5.  Divide (the result) by the number of data values minus one `=(sum(x-barx)^2)/(n-1)`

6.  Take the square root of the result `=sqrt((sum(x-barx)^2)/(n-1))`

Example 1

What is the standard deviation for the following set of data?

15, 15, 15, 14, 16

The process is as follows:

i.  Find the mean:  `(15+15+15+14+16)/5=75/5=15`

ii  Subtract the mean from each of the data points:

iii.  Square the result:

iv.  Add the results up:

      `0+0+0+1+1=2`

v.  Divide by the number of data points less one. 

Therefore  `2/(5-1)=2/4=0.5`

vi.  Square root the result:

`sqrt0.5=0.707`

Answer:

The standard deviation for the above data = 0.707

That is 68% of all the data is within 0.707 of 15.

Example 2

What is the standard deviation for the following set of data?

     2, 7, 14, 22, 30.

The process is as follows:

i.   Find the mean:  `(2+7+14+22+30)/5=75/5=15`

ii.  Subtract the mean from each of the data points:

 iii.  Square the result:

iv.  Add the results up:  

`169 + 64 + 1 + 49 + 225 = 508`

v.   Divide by the number of data points less one.

`508/(5-1)=508/4=127`

vi.  Square root the result:  

`sqrt127=11.3`

Answer:

The standard deviation for the above set of data = 11.3

That is 68% of all the data is within 11.3 of 15.