Download Ch3-Sec3.3

Chapter
3
Numerically
Summarizing Data
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Section
3.3
Measures of
Central Tendency
and
Dispersion from
Grouped Data
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objectives
1. Approximate the mean of a variable from
grouped data
2. Compute the weighted mean
3. Approximate the standard deviation of a
variable from grouped data
3-3
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objective 1
• Approximate the Mean of a Variable from
Grouped Data
3-4
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
We have discussed how to compute descriptive
statistics from raw data, but often the only
available data have already been summarized in
frequency distributions (grouped data).
Although we cannot find exact values of the
mean or standard deviation without raw data, we
can approximate these measures using the
techniques discussed in this section.
3-5
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Approximate the Mean of a Variable from a
Frequency Distribution
Population Mean
Sample Mean
xf


f
xf

x
f
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
x1 f1  x2 f2  ...  xn fn

f1  f2  ...  fn
i i
i
i i
i
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
n is the number of classes
3-6
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE Approximating the Mean from a Relative
Frequency Distribution
The National Survey of Student Engagement is a survey
that (among other things) asked first year students at
liberal arts colleges how much time they spend
preparing for class each week. The results from the 2007
survey are summarized below. Approximate the mean
number of hours spent preparing for class each week.
Hours
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
3-7
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Time Frequency
0
0
1-5
130
6 - 10
250
11 - 15
230
16 - 20
180
21 - 25
100
26 – 30
60
31 – 35
50
 fi  1000
3-8
xi
xi fi
0
0
3.5
455
8.5
2125
13.5
3105
18.5
3330
23.5
2350
28.5
1710
33.5
1675
 xi fi  14,750
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
xf

x
f
i i
i
14,750

1000
 14.75
Objective 2
• Compute the Weighted Mean
3-9
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
The weighted mean, xw , of a variable is found by
multiplying each value of the variable by its
corresponding weight, adding these products, and
dividing this sum by the sum of the weights. It
can be expressed using the formula
xw
wx


w
i i
i
w1 x1  w2 x2  ...  wn xn

w1  w2  ...  wn
where w is the weight of the ith observation
xi is the value of the ith observation
3-10
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
EXAMPLE Computed a Weighted Mean
Bob goes to the “Buy the Weigh” Nut store and creates
his own bridge mix. He combines 1 pound of raisins, 2
pounds of chocolate covered peanuts, and 1.5 pounds
of cashews. The raisins cost $1.25 per pound, the
chocolate covered peanuts cost $3.25 per pound, and
the cashews cost $5.40 per pound. What is the cost per
pound of this mix?
1($1.25)  2($3.25)  1.5($5.40)
xw 
1  2  1.5
$15.85

 $3.52
4.5
3-11
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Objective 3
• Approximate the Standard Deviation of a
Variable from Grouped Data
3-12
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Approximate the Standard Deviation of a
Variable from a Frequency Distribution
Population
Standard Deviation
 x   
f
2

i
fi
Sample
Standard Deviation
 x  x  f
 f  1
2
s
i
i
i
i
where xi is the midpoint or value of the ith class
fi is the frequency of the ith class
3-13
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
An algebraically equivalent formula for the
population standard deviation is
x f


f 
2
x
i
2
i i
f
f
i
3-14
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
i
EXAMPLE Approximating the Standard Deviation
from a Relative Frequency Distribution
The National Survey of Student Engagement is a survey
that (among other things) asked first year students at
liberal arts colleges how much time they spend
preparing for class each week. The results from the 2007
survey are summarized below. Approximate the standard
deviation number of hours spent preparing for class each
week.
Hours
0
1-5 6-10 11-15 16-20 21-25 26-30 31-35
Frequency
0 130 250
230
180
100
60
50
Source:http://nsse.iub.edu/NSSE_2007_Annual_Report/docs/withhold/NSSE_2007_Annual_Report.pdf
3-15
Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
Time
0
1-5
6 - 10
11 - 15
16 - 20
21 - 25
26 – 30
31 – 35
3-16
Frequ
ency
xi
0
0
130
3.5
250
8.5
230 13.5
180 18.5
100 23.5
60
28.5
50
33.5
 fi  1000
xi  x
0
–11.25
–6.25
–1.25
3.75
8.75
13.75
18.75
 
 
xi  x f i s 2   x i  x f i
0
 fi  1
16,453.125
65,687.5

9765.625
1000  1
 65.8
359.375
2531.25
7656.25 s  s 2  65.8
11,343.75  8.1 hours
17,578.125
2
 xi  x fi  65,687.5

2

2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc.
