Download 2.4 Measures of Variation

2.4 Measures of Variation
Range and Deviation

Range - The difference
between the maximum
and the minimum data
entries in the set
Range = (Max – Min)

Deviation – the
difference between the
entry and the mean 
of the data set.
Deviation of
x  x
Variance and Standard Deviation




Population Variance:
2

(
x


)
2 
N
Population Standard Deviation:
Sample Variance:
( x   ) 2

N
2

(
x

x
)
2
s 
n 1
Sample Standard Deviation:
( x  x ) 2
s
n 1
Heights (in
inches)
Deviation
70
-.3
.09
72
1.7
2.89
71
.7
.49
70
-.3
.09
69
-1.3
1.69
73
2.7
7.29
69
-1.3
1.69
68
-2.3
5.29
70
-.3
.09
.7
.49
71
Mean = 70.3
x

Squares
( x   )2
SSx=
  2.01
2

2
s 
s
1.418
2.233
1.494
Empirical Rule
About 68% of the data lies
within 1 standard deviation of
the mean
About 95% of the data lies within 2
standard deviation of the mean
About 99.7 of the data lies within 3
standard deviation of the mean
    68%
  2  95%
  3  99.7
Examples
Heights of Women in the U.S. have a mean of 64 with a
standard deviation of 2.75. Use the empirical rule to
estimate:
 The percent of the heights that are between 61.25 and 64
inches.
ANS: 34%
•
Between what two values does about 95% of the data lie?
ANS: (58.5, 69.5)
Chebychev’s Theorem

The portion of any data set
lying within k standard
deviations (k>1) of the
mean is at least
Example: k = 2 , 75% of the
data is within 2 standard
deviations of the mean
k = 3; 88.9% of the data lies
within 3 standard deviations
of the mean.
1
1 2
k
Sample Standard deviation for grouped
data
( x  x ) f
s
n 1
2