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Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 1
Chapter 13
Statistics
Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 2
WHAT YOU WILL LEARN
• Mode, median, mean, and
midrange
• Percentiles and quartiles
• Range and standard deviation
Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 3
Section 5
Measures of Central Tendency
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Chapter 13 Section 5 - Slide 4
Definitions

An average is a number that is representative
of a group of data.

The arithmetic mean, or simply the mean, is
symbolized by x , when it is a sample of a
population or by the Greek letter mu, , when it
is the entire population.
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Chapter 13 Section 5 - Slide 5
Mean

The mean, x , is the sum of the data divided by
the number of pieces of data. The formula for
calculating the mean is
x

x
n

where  x represents the sum of all the data
and n represents the number of pieces of data.
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Chapter 13 Section 5 - Slide 6
Example-find the mean

Find the mean amount of money parents spent
on new school supplies and clothes if 5 parents
randomly surveyed replied as follows:
$327 $465 $672 $150 $230
x $327  $465  $672  $150  $230

x

n
$1844

 $368.80
5
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5
Chapter 13 Section 5 - Slide 7
Median


The median is the value in the middle of a set
of ranked data.
Example: Determine the median of
$327 $465 $672 $150 $230.
Rank the data from smallest to largest.
$150 $230 $327 $465 $672
middle value
(median)
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Chapter 13 Section 5 - Slide 8
Example: Median (even data)

Determine the median of the following set of
data: 8, 15, 9, 3, 4, 7, 11, 12, 6, 4.
Rank the data:
3 4 4 6 7 8 9 11 12 15
There are 10 pieces of data so the median will
lie halfway between the two middle pieces (the
7 and 8). The median is (7 + 8)/2 = 7.5
3 4 4 6 7 8 9 11 12 15
Median = 7.5
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Chapter 13 Section 5 - Slide 9
Mode

The mode is the piece of data that occurs most
frequently.

Example: Determine the mode of the data set:
3, 4, 4, 6, 7, 8, 9, 11, 12, 15.

The mode is 4 since it occurs twice and the
other values only occur once.
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Chapter 13 Section 5 - Slide 10
Midrange

The midrange is the value halfway between the
lowest (L) and highest (H) values in a set of
data.
lowest value + highest value
Midrange =
2

Example: Find the midrange of the data set
$327, $465, $672, $150, $230.
$150 + $672
Midrange =
 $411
2
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Chapter 13 Section 5 - Slide 11
Example

The weights of eight Labrador retrievers
rounded to the nearest pound are 85, 92, 88,
75, 94, 88, 84, and 101. Determine the
a) mean
b) median
c) mode
d) midrange
e) rank the measures of central tendency
from lowest to highest.
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Chapter 13 Section 5 - Slide 12
Example--dog weights 85, 92, 88, 75,
94, 88, 84, 101 (continued)
a. Mean
85  92  88  75  94  88  84  101
x
8
707

 88.375
8
b. Median-rank the data
75, 84, 85, 88, 88, 92, 94, 101
The median is 88.
Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 13
Example--dog weights 85, 92, 88, 75,
94, 88, 84, 101
c. Mode-the number that occurs most frequently.
The mode is 88.
d. Midrange = (L + H)/2
= (75 + 101)/2 = 88
e. Rank the measures, lowest to highest
88, 88, 88, 88.375
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Chapter 13 Section 5 - Slide 14
Measures of Position


Measures of position are often used to make
comparisons.
Two measures of position are percentiles and
quartiles.
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Chapter 13 Section 5 - Slide 15
To Find the Quartiles of a Set of Data
1. Order the data from smallest to largest.
2. Find the median, or 2nd quartile, of the set of
data. If there are an odd number of pieces of
data, the median is the middle value. If there
are an even number of pieces of data, the
median will be halfway between the two middle
pieces of data.
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Chapter 13 Section 5 - Slide 16
To Find the Quartiles of a Set of Data
(continued)
3. The first quartile, Q1, is the median of the lower
half of the data; that is, Q1, is the median of the
data less than Q2.
4. The third quartile, Q3, is the median of the
upper half of the data; that is, Q3 is the median
of the data greater than Q2.
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Chapter 13 Section 5 - Slide 17
Example: Quartiles

The weekly grocery bills for 23 families are as
follows. Determine Q1, Q2, and Q3.
170
330
225
75
95
210
80
225
160
172
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270
170
215
130
190
270
240
310
74
280
270
50
81
Chapter 13 Section 5 - Slide 18
Example: Quartiles (continued)

Order the data:
50 74 75 80 81 95 130
160 170 170 172 190 210 215
225 225 240 270 270 270 280
310 330
Q2 is the median of the entire data set which is
190.
Q1 is the median of the numbers from 50 to 172
which is 95.
Q3 is the median of the numbers from 210 to
330 which is 270.
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Chapter 13 Section 5 - Slide 19
Section 6
Measures of Dispersion
Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 20
Measures of Dispersion

Measures of dispersion are used to indicate the
spread of the data.

The range is the difference between the highest
and lowest values; it indicates the total spread
of the data.
Range = highest value – lowest value
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Chapter 13 Section 5 - Slide 21
Example: Range


Nine different employees were selected and the
amount of their salary was recorded. Find the
range of the salaries.
$24,000
$32,000 $26,500
$56,000 $48,000 $27,000
$28,500 $34,500 $56,750
Range = $56,750  $24,000 = $32,750
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Chapter 13 Section 5 - Slide 22
Standard Deviation

The standard deviation measures how much
the data differ from the mean. It is symbolized
with s when it is calculated for a sample, and
with  (Greek letter sigma) when it is calculated
for a population.
 x  x 
2
s
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n 1
Chapter 13 Section 5 - Slide 23
To Find the Standard Deviation of a
Set of Data
1. Find the mean of the set of data.
2. Make a chart having three columns:
Data
Data  Mean
(Data  Mean)2
3. List the data vertically under the column
marked Data.
4. Subtract the mean from each piece of data and
place the difference in the Data  Mean
column.
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Chapter 13 Section 5 - Slide 24
To Find the Standard Deviation of a
Set of Data (continued)
5. Square the values obtained in the Data  Mean
column and record these values in the
(Data  Mean)2 column.
6. Determine the sum of the values in the
(Data  Mean)2 column.
7. Divide the sum obtained in step 6 by n  1,
where n is the number of pieces of data.
8. Determine the square root of the number
obtained in step 7. This number is the standard
deviation of the set of data.
Copyright © 2009 Pearson Education, Inc.
Chapter 13 Section 5 - Slide 25
Example

Find the standard deviation of the following
prices of selected washing machines:
$280, $217, $665, $684, $939, $299
Find the mean.
x 280  217  665  684  939  299

x

n
3084

 514
6
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6
Chapter 13 Section 5 - Slide 26
Example (continued), mean = 514
Data
217
280
299
665
684
939
Data  Mean
297
234
215
151
170
425
0
Copyright © 2009 Pearson Education, Inc.
(Data  Mean)2
(297)2 = 88,209
54,756
46,225
22,801
28,900
180,625
421,516
Chapter 13 Section 5 - Slide 27
Example (continued), mean = 514


s=
421,516
6- 1
s=
421,516
» 290.35
5
The standard deviation is $290.35.
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Chapter 13 Section 5 - Slide 28