Download MEASURES OF VARIATION OR DISPERSION

MEASURES OF VARIATION
OR DISPERSION
THE SPREAD OF A DATA SET
3 MEASURES OF VARIATION
1) Range: (R)
highest value – lowest value
2) Variance: (s2, 2)
the average of the squares of the
distance from the actual mean
3) Standard Deviation: (s, )
the average distance from the actual
mean of the data set.
Calculating Variance and Standard
Deviation of Listed Data
Example 1: The following data represent the
high temperatures recorded over the past
week. Find the range, variance and
standard deviation.
35, 45, 30, 40, 25, 33, 38
Example 1: Answer
Range: R = 45 – 25
R = 20
*Since variance and standard deviation both
represent distances from the mean we must
first find the mean of the data set.
35  45  30  40  25  33  38

7
246

7
  35.1
Calculating the variance
1. Set up your values
in a table.
Temp (x)
25
30
33
35
38
40
45
Calculating the variance
2. Subtract the mean
from each value.
(reminder: variance and
st. dev. are a
difference from the
mean)
Temp (x)
25
30
33
35
38
40
45
x-
-10.1
-5.1
-2.1
-.1
2.9
4.9
9.9
Calculating the variance
3. Square each
difference from step
2.
(reminder: variance is
the squared
difference from the
mean)
Temp
(x)
25
30
33
35
38
40
45
x-
(x - )2
-10.1
-5.1
-2.1
-.1
2.9
4.9
9.9
102.01
26.01
4.41
.01
8.41
24.01
98.01
Calculating the variance
4. Find the mean of
the squares in step
3. (variance)
 x   

2
2
x  


n
2
 262.87
262.87

 37.6
7
(x - )2
102.01
26.01
4.41
.01
8.41
24.01
98.01
Calculating the Standard Deviation
• Standard deviation is
merely the square
root of the variance.
  37.6
2
  6.13
Calculating Variance of Grouped
Data
Ex 2: The following data
represent scores on a
75 point final exam.
Find the mean,
variance and
standard deviation of
the data set.
Scores (x)
Freq.
10 – 20
2
21 – 31
8
32 – 42
15
43 – 53
7
54 – 64
10
65 – 75
3
Ex 2: answer
1. Find the mean.
Scores (x)
10 – 20
21 – 31
32 – 42
43 – 53
54 – 64
65 – 75
Freq.
2
8
15
7
10
3
mean:   42.9
xm
fxm
15
26
37
48
59
70
30
208
555
336
590
210
Ex 2: answer
2. Subtract mean from
each midpoint.
xm
xm  
15
26
37
48
59
70
-27.9
-16.9
-5.9
5.1
16.1
27.1
Ex 2: answer
3. Square each
difference from step
2.
xm  
 xm   
-27.9
-16.9
-5.9
5.1
16.1
27.1
778.41
285.61
34.81
26.01
259.21
734.41
2
Ex 2: answer
4. Multiply each
squared value by
the frequency of
that class.
 xm   
2
778.41
285.61
34.81
26.01
259.21
734.41
f
 xm   
1556.82
2284.88
522.15
182.07
2592.10
2203.23
2
Ex 2: answer
5. Sum up last column and then divide by the
total frequency (n).

f ( xm   ) 2  9341.25
9341.25
 
 207.6
45
2
Ex 2: Answer
6. Standard deviation
is merely the square
root of the variance.
  207.6
2
  14.4
Samples vs. Populations
In most cases, the variability of a sample
will be significantly less than that of the
corresponding population….why?
Since samples are often used to represent
the variability of an entire population, we
must be sure to correct for this bias.
Dividing by n-1 gives us an unbiased
estimate of the true population standard
deviation or variance.
Calculating Variance of Grouped
Data (Sample)
• Formula for grouped:
s
2
f x


m
x

2
n 1
• This does not alter your calculation for
the mean! (still divide by ‘n’)
Comparing Standard Deviations
• Whenever two samples have the same units of
measure, the variance and standard deviation
for each can be compared directly. bus
• Ex: A car dealer wanted to compare miles driven
on trade-ins:
• Buicks
s = 422 miles
• Cadillacs s = 350 miles
• Variation in mileage was greater for Buicks!
Comparing Standard Deviations
• However if two samples have different units of
measure, the variance and standard deviation
must be compared using the coefficient of
variation.
• The coefficient of variation is the standard
deviation divided by the mean. (x 100%)
For Samples:
For Populations:
s
CVar  100%
x
CVar 

100%

Comparing Standard Deviations
Ex 1: The mean of the number of sales of cars
over a 3-month period is 87, and the standard
deviation is 5. The mean of the commissions
is $5225, and the standard deviation is $773.
Compare the variability of the two.
Solution: Find the coefficient of variation of each
and whichever number is larger, that set is
more variable.
• Coefficient of Variation for sales:
5
C var 
100%  5.7%
87
• Coefficient of Variation for commissions:
773
C var 
100%  14.8%
5225
• Commissions are more variable than the
sales!
Comparing Standard Deviations
Ex 2: The mean for the number of pages of a
sample of women’s fitness magazines is 132,
with a variance of 23; the mean for the number
of advertisements of a sample of women’s
fitness magazine is 182, with a variance of 62.
Compare the variations.
Solution: Find the coefficient of variation of each
and whichever number is larger, that set is
more variable.
• Coefficient of Variation for pages:
23
C var 
100%  3.6%
132
• Coefficient of Variation for advertisements:
62
C var 
100%  4.3%
182
• Advertisements are more variable than
the number of pages!