Transcript Anatomy: Variance & SD (Sample)

ANATOMY OF A SAMPLE VARIANCE (s2) AND STANDARD DEVIATION (s)
The Sample Variance and Standard Deviation represent measures of distance of sample values from the sample’s mean.
The Variance represents the sum of the squared deviations of values from the mean divided by n-1.
The Standard Deviation represents the square root of the summed deviation values divided by n-1 (or the square root of the variance.)
The definition formula used to calculate
the Sample Variance looks like this:
s2 
The definition formula for a Sample
Standard Deviation is calculated by
taking the square root of the Sample
Variance. The formula looks like this:
s
( x  x ) 2
n 1
( x  x)
n 1
2
21
18
23
21
17
20
( x  x)
1
-2
3
1
-3
0
( x  x)2
1
4
9
1
9
0
24
Don’t panic…we just
need an example here.
NOTE: The sum of
the deviation scores
( x  x)
will ALWAYS = 0.
 ( x  x)2
Then the Sample Variance is:
( x  x) 2 24
s2 

 4.80 min.2
n 1
5
s
( x  x)
24

 4.80  2.19 min.
n 1
5
( x ) 2
n
n 1
x 2 
x
2

( x ) 2
n 1
n
Determining the Variance and Standard
Deviation via the computation formulas.
x2
x
21
18
23
21
17
20
120 =  x
THE DATA:
The following data give the
time (in minutes) taken by a
student to travel from his
home to school for a sample
of six days:
21, 18, 23, 21, 17, 20
441
324
529
441
289
400
2424 = x2
Then the Sample Variance is:
The mean for this sample is:
x  20
n, the number of data values
is 6.
s2 
( x ) 2
n
n 1
x 2 

(120)2
6  24  4.80
5
5
min.2
2424
And the Sample Standard Deviation is:
And the Sample Standard Deviation is:
2
s2 
The computation formula for a Sample
Standard Deviation takes the square
root of the Sample Variance. The
formula looks like this:
s
Given the following small
data set, we will calculate the
Sample Variance, and the
Sample Standard Deviation.
Determining the Variance and Standard
Deviation via the definition formulas.
x
The computation formula for the
Sample Variance looks like this:
s
(120) 2
6

5
2424
24
 4.80  2.19 min.
5