 # 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

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

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

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