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Section 13.4
Measures of
Central
Tendency
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Averages
Mean
Median
Mode
Midrange
Quartiles
13.4-2
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Measures of Central Tendency
An average is a number that is
representative of a group of data.
There are at least four different
averages: the mean, the median, the
mode, and the midrange.
Each is calculated differently and may
yield different results for the same set
of data.
13.4-3
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Measures of Central Tendency
Each will result in a number near the
center of the data; for this reason,
averages are commonly referred to as
measures of central tendency.
13.4-4
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Mean (or Arithmetic 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 Σx represents the sum of all the
data and n represents the number of
pieces of data.
13.4-5
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Example 1: Determine the Mean
Determine the mean age of a group of
patients at a doctor’s office if the ages
of the individuals are 28, 19, 49, 35,
and 49.
Solution
x 28  19  49  35  49
x

n
5
180
 36

5
13.4-6
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Median
The median is the value in the middle
of a set of ranked data.
13.4-7
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Example 2: Determine the Median
Determine the median age of a group
of patients at a doctor’s office if the
ages of the individuals are 28, 19, 49,
35, and 49.
Solution
Rank the data from smallest to largest.
19 28 35 49 49
35 is in the middle, 35 is the median.
13.4-8
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Example 3: Determine the
Median of an Even Number of
Pieces of Data
Determine the median of the following
sets of data.
a) 9, 14, 16, 17, 11, 16, 11, 12
b) 7, 8, 8, 8, 9, 10
13.4-9
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Example 3: Determine the
Median of an Even Number of
Pieces of Data
Solution
9, 11, 11, 12, 14, 16, 16, 17
8 pieces of data
Median is half way between middle two
data points 12 and 14
(12 + 14)÷2 = 26 ÷ 2 = 13
13.4-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Determine the
Median of an Even Number of
Pieces of Data
Solution
7, 8, 8, 8, 9, 10
6 pieces of data
Median is half way between middle two
data points 8 and 8
(8 + 8)÷2 = 16 ÷ 2 = 8
13.4-11
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Mode
The mode is the piece of data that
occurs most frequently.
13.4-12
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Example 4: Determine the Mode
Determine the mean age of a group of
patients at a doctor’s office if the ages
of the individuals are 28, 19, 49, 35,
and 49.
Solution
The age 49 is the mode because it
occurs twice and the other values
occur only once.
13.4-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
13.4-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 5: Determine the
Midrange
Determine the midrange age of a
group of patients at a doctor’s office if
the ages of the individuals are 28, 19,
49, 35, and 49.
Solution
68
19  49
 34

Midrange 
2
2
13.4-15
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Measures of Position
Measures of position are often used to
make comparisons.
Two measures of position are
percentiles and quartiles.
13.4-16
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Percentiles
There are 99 percentiles dividing a set
of data into 100 equal parts.
13.4-17
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Percentiles
A score in the nth percentile means
that you out-performed about n% of
the population who took the test and
that (100 – n)% of the people taking
the test performed better than you did.
13.4-18
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Quartiles
Quartiles divide data into four equal
parts:
The first quartile is the value that is
higher than about 1/4, or 25%, of the
population. It is the same as the 25th
percentile.
13.4-19
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Quartiles
The second quartile is the value that is
higher than about 1/2 the population
and is the same as the 50th percentile,
or the median.
The third quartile is the value that is
higher than about 3/4 of the population
and is the same as the 75th percentile.
13.4-20
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Quartiles
13.4-21
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To Determine the Quartiles of a
Set of Data
1. Order the data from smallest to
largest.
13.4-22
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To Determine the Quartiles of a
Set of Data
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.
13.4-23
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To Determine the Quartiles of a
Set of Data
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.
13.4-24
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To Determine the Quartiles of a
Set of Data
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.
13.4-25
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Example 8: Finding Quartiles
Electronics World is concerned about
the high turnover of its sales staff. A
survey was done to determine how
long (in months) the sales staff had
been in their current positions. The
responses of 27 sales staff follow.
Determine Q1, Q2, and Q3.
13.4-26
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Example 8: Finding Quartiles
25
3 7 15 31 36 17 21 2
11 42 16 23 16 21 9 20 5
8 12 27 14 39 24 18 6 10
Solution
List data from
2 3 5
12 14 15
21 23 24
13.4-27
smallest to largest.
6 7 8 9 10 11
16 17 18 19 20 21
25 27 31 36 39 42
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Example 8: Finding Quartiles
Solution
2 3 5 6 7 8 9
12 14 15 16 17 18 19
21 23 24 25 27 31 36
The median, or middle of the
points is Q2 = 17.
The median, or middle of the
pieces of data is Q1 = 9.
The median, or middle of the
pieces of data is Q3 = 24.
13.4-28
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10
20
39
27
11
21
42
data
lower 13
upper 13
Section 13.5
Measures of
Dispersion
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What You Will Learn
Range
Standard Deviation
13.5-30
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Measures of Dispersion
Measures of dispersion are used to
indicate the spread of the data.
13.5-31
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Range
The range is the difference between
the highest and lowest values; it
indicates the total spread of the data.
Range = highest value – lowest value
13.5-32
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Example 1: Determine the Range
The amount of caffeine, in milligrams,
of 10 different soft drinks is given
below. Determine the range of these
data.
38, 43, 26, 80, 55, 34, 40, 30, 35, 43
13.5-33
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Determine the Range
Solution
38, 43, 26, 80, 55, 34, 40, 30, 35, 43
Range = highest value – lowest value
= 80 – 26 = 54
The range of the amounts of caffeine is
54 milligrams.
13.5-34
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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
13.5-35
n 1
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Standard Deviation
The standard deviation, s, of a set of
data can be calculated using the
following formula.
 x  x 
2
s
13.5-36
n 1
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
13.5-37
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To Find the Standard Deviation of
a Set of Data
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.
13.5-38
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To Find the Standard Deviation of
a Set of Data
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.
13.5-39
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Example 3: Determine the
Standard Deviation of Stock Prices
The following are the prices of nine
stocks on the New York Stock
Exchange. Determine the standard
deviation of the prices.
$17, $28, $32, $36, $50, $52, $66,
$74, $104
13.5-40
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Example 3: Determine the
Standard Deviation of Stock Prices
Solution
The mean x is
x

x
n
17  28  32  36  50  52  66  74  104

9
459

13.5-41
9
 51
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Example 3: Determine the
Standard Deviation of Stock Prices
13.5-42
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Example 3: Determine the
Standard Deviation of Stock Prices
Solution
Use the formula
 x  x 
2
s
n 1

5836
 729.5  27.01
9 1
The standard deviation, to the nearest
tenth, is $27.01.
13.5-43
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