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:

$0^2=0$

$-1^2=1$

$1^2=1$

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:

$sqrt127=11.3$

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.