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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:

`15-15`

`15-15`

`15-15`

`14-15`

`16-15`

`0`

`0`

`0`

`-1`

`1`


iii.  Square the result:

`0^2=0`

`0^2=0`

`0^2=0`

 `-1^2=1`

 `1^2=1`

 

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:

`2-15` `7-15` `14-15` `22-15` `30-15`
`-13` `-8` `-1` `7` `-15`

 

 

 iii.  Square the result:

`-13^2=169` `-8^2=64` `-1^2=1` `7^2=49` `15^2=225`

 

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=11.2`

 

Answer:

The standard deviation for the above set of data = 11.2

That is 68% of all the data is within 11.2 of 15.

 

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