Aurora State College of Technology

GE4: Mathematics in the Modern Word

MODULE 4: Measures of Central Tendency

Number of Hours: 1.5Hrs
LEARNING OUTCOME(S): •
a. Use a variety of statistical tools to process and manage numerical data
b. Use methods of linear regression and correlations to predict the value of a variable given
certain conditions
c. Advocate the use of statistical data in making important decisions
LESSON OUTLINE:
• The Arithmetic Mean
• The Median
• The Mode
• The Weighted Mean

THE ARITHMETIC MEAN

Statistics involves the collection, organization, summarization, presentation, and
interpretation of data. The branch of statistics that involves the collection, organization,
summarization, and presentation of data is called descriptive statistics. The branch that interprets
and draws conclusions from the data is called inferential statistics.
One of the most basic statistical concepts involves finding measures of central tendency of a
set of numerical data. It is often helpful to find numerical values that locate, in some sense, the
center of a set of data. Suppose Elle is a senior at a university. In a few months she plans to graduate
and start a career as a landscape architect. A survey of five landscape architects from last year’s
senior class shows that they received job offers with the following monthly salaries.
₱43,750 ₱39,500 ₱38,00 ₱41,250 ₱44,000

Before Elle interviews for a job, she wishes to determine an average of these 5 salaries.
This average should be a “central” number around which the salaries cluster. We will consider
three types of averages, known as the arithmetic mean, the median, and the mode. Each of these
averages is a measure of central tendency for the numerical data.
The arithmetic mean is the most commonly used measure of central tendency. The
arithmetic mean of a set of numbers is often referred to as simply the mean. To find the mean for a
set of data, find the sum of the data values and divide by the number of data values. For instance,
to find the mean of the 5 salaries listed above, Elle would divide the sum of the salaries by 5.
₱43,750+₱39,500+₱38,000+₱41,260+₱44,000
Mean= 5

₱206,500
= = ₱41,300
5

The mean suggests that Elle can reasonably expect a job offer at a salary of about ₱41,300.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

In statistics it is often necessary to find the sum of a set of numbers. The traditional symbol
used to indicate a summation is the Greek letter sigma, Σ. Thus, the notation Σ𝑥 called summation
notation, denotes the sum of all the numbers in a given set. We can define the mean using

MEAN
The mean of n numbers is the sum of the numbers divided by n.
𝛴𝑥
Mean = 𝑛

summation notation.
Statisticians often collect data from small portions of a large group in order to determine information

about the group. In such situations the entire group under consideration is known as the population,

and any subset of the population is called a sample. It is traditional to denote the mean of a sample

by 𝑥̅ (which is read as “x bar”) and to denote the mean of a population by the Greek letter μ

(lowercase mu).

Example 1. Find the Mean

Six friends in a biology class of 20 students received test grades of
92, 84, 65, 76, and 90.

Find the mean score of these test scores.

Solution.
The 6 friends are a sample of the population of 20 students. Use 𝑥̅ to represent the mean.
Σ𝑥 92 + 84 + 65 + 76 + 88 + 90 495
𝑥̅ = = = = 82.5
𝑛 6 6

The mean of these test scores is 82.5.

TRY THIS!! A doctor ordered 4 separate blood tests to measure a patient’s total blood cholesterol

levels. The test results were

245, 235, 220, and 210.

Find the mean of the blood cholesterol levels.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

THE MEDIAN

Another type of average is the median. Essentially, the median is the middle number or the mean
of the two middle numbers in a list of numbers that have been arranged in numerical order from
smallest to largest or largest to smallest. Any list of numbers that is arranged in numerical order
from smallest to largest or largest to smallest is a ranked list.

Median
The median of a ranked list of n numbers is
▪ The middle number if n is odd.
▪ The mean of the two middle numbers if n is even.

Example 2. Find the median of the data in the following lists

a. 4, 8, 1, 14, 9, 21, 12 b. 46, 23, 92, 89, 77, 108

Solution.
a. The list 4, 8, 1, 14, 9, 21, 12 contains 7 numbers. The median of a list with an oddnumber
of entries is found by ranking the numbers and fnding the middle number. Ranking the
numbers from smallest to largest gives
1, 4, 8, 9, 12, 14, 21
The middle number is 9. Thus 9 is the median.
b. The list 46, 23, 92, 89, 77, 108 contains 6 numbers. The median of a list of data with an
even number of entries is found by ranking the numbers and computing the mean of the
two middle numbers. Ranking the numbers from smallest to largest give
23, 46, 77, 89, 92, 108
The two middle numbers are 77 and 89. The mean of 77 and 89 is 83. Thus 83 is the
median of the data.
TRY THIS!!! Find the median of the data in the following lists.
a. 14, 27, 3, 82, 64, 34, 8, 51
b. 21.3, 37.4, 11.6, 82.5, 17.2

THE MODE
A third type of average is the mode.

Mode
The mode of a list of numbers is the number that occurs most frequently.

Some lists of numbers do not have a mode. For instance, in the list 1, 6, 8, 10, 32, 15, 49,
each number occurs exactly once. Because no number occurs more often than the other numbers,
there is no mode.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

A list of numerical data can have more than one mode. For instance, in the list 4, 2, 6, 2, 7,
9, 2, 4, 9, 8, 9, 7, the number 2 occurs three times and the number 9 occur three times. Each of the
other numbers occurs less than three times. Thus 2 and 9 are both modes for the data.
Example 3. Find a Mode

Find the mode of the data in the following lists.

a. 18, 15, 21, 16, 15, 14, 15, 21 b. 2, 5, 8, 9, 11, 4, 7, 23

Solution
a. In the list 18, 15, 21, 16, 15, 14, 15, 21, the number 15 occurs more often than the
other numbers. Thus 15 is the mode.
b. Each number in the list 2, 5, 8, 9, 11, 4, 7, 23 occurs only once. Because no number
occurs more often than the others, there is no mode.
TRY THIS!!! Find the mode of the data in the following lists.
a. 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 8 b. 12, 34, 12, 71, 48, 93, 71
The mean, the median, and the mode are all averages; however, they are generally not equal.
The mean of a set of data is the most sensitive of the averages. A change in any of the numbers
changes the mean, and the mean can be changed drastically by changing an extreme value.
In contrast, the median and the mode of a set of data are usually not changed by changing
an extreme value.
When a data set has one or more extreme values that are very different from the majority of
data values, the mean will not necessarily be a good indicator of an average value. In the following
example, we compare the mean, median, and mode for the salaries of 5 employees of a small
company.

Salaries. ₱370,000 ₱60,000 ₱36,000 ₱20,000 ₱20,000

The sum of the 5 salaries is ₱506,000. Hence the mean is
506,000
= 101,200
5
The median is the middle number, ₱36,000. Because the ₱20,000 salary occurs the most, the mode
is ₱20,000. The data contain one extreme value that is much larger than the other values. This
extreme value makes the mean considerably larger than the median. Most of the employees of this
company would probably agree that the median of $36,000 better represents the average of the
salaries than does either the mean or the mode.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

A SIIMPLE APPLICATION. Average Rate for a Round Trip

Suppose you average 60 mph on a one-way trip of 60 mi. On the return trip you aver- age 30 mph. You
might be tempted to think that the average of 60 mph and 30 mph, which is 45 mph, is the average rate for
the entire trip. However, this is not the case. Because you were traveling more slowly on the return trip, the
return trip took longer than the original trip to your destination. More time was spent traveling at the
slower speed. Thus the average rate for the round trip is less than the average (mean) of 60 mph and 30
mph.
To find the actual average rate for the round trip, use the formula
total distance;
Average rate=
total time

The total round-trip distance is 120 mi. The time spent traveling to your destination was 1 h, and
the time spent on the return trip was 2 h. The total time for the round trip was 3 h. Thus,
total distance 120
Average rate = = = 40mph
total time 3

The Weighted Mean

A value called the weighted mean is often used when some data values are more important than
others. For instance, many professors determine a student’s course grade from the student’s tests
and the final examination. Consider the situation in which a professor counts the final examination
score as 2 test scores. To find the weighted mean of the student’s scores, the professor first assigns
a weight to each score. In this case the professor could assign each of the test scores a weight of 1
and the final exam score a weight of 2.A student with test scores of 65, 70, and 75 and a final
examination score of 90 has a weighted mean of

(65 × 1) + (70 × 1) + (75 × 1) + (90 × 2) 390

= = 78
5 5
Note that the numerator of the weighted mean above is the sum of the products of each test
score and its corresponding weight. The number 5 in the denominator is the sum of all the weights
(1 + 1 + 1 + 2 = 5). The procedure for finding the weighted mean can be generalized as follows.

The Weighted Mean

The weighted mean of the n numbers 𝒙𝟏 , 𝒙𝟐 , 𝒙𝟑 , … , 𝒙𝒏 with the respective assigned
weights 𝒘𝟏 , 𝒘𝟐 , 𝒘𝟑 , … , 𝒘𝒏 is
Σ(𝑥⋅𝑤)
Weighted mean =
Σ𝑤
where Σ(𝑥 ⋅ w) is the sum of the products formed by multiplying each number by its
assigned weight, and Σ𝑤 is the sum of all the weight.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

Example 4 Find a Weighted Mean

Table 13.1 shows Dillon’s fall semester course grades. Use

the weighted mean formula to find Dillon’s GPA for the fall
semester.
Solution:

The B is worth 3 points, with a weight of 4; the A is worth 4

points with a weight of 3; the D is worth 1 point, with a weight
of 3; and the C is worth 2 points, with a weight of 4. The sum
of all the weights is 4 + 3 + 3 + 4, or 14.
(3×4)+(4×3)+(1×3)+(2×4)
Weighted mean=
14
35
= = 12.5
14
Dillon’s GPA for the fall semester is 2.5.

TRY THIS!!! Table 13.2 shows Janet’s spring semester

course grades. Use the weighted mean formula to find
Janet’s GPA for the spring semester. Round to the nearest
hundredth.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

Frequency Distribution

Data that have not been organized

or manipulated in any manner are called raw
data. A large collection of raw data may not
provide much readily observable information. A
frequency distribution, which is a table that lists
observed events and the frequency of
occurrence of each observed event, is often
used to organize raw data. For instance,
consider the following table, which lists the
number of laptop computers owned by families
in each of 40 homes in a subdivision.

The frequency distribution in Table 13.4

below was constructed using the data from Table
13.3. The first column of the frequency
distribution consists of the numbers 0, 1, 2, 3, 4,
5, 6, and 7. The corresponding frequency of
occurrence, f, of each of the numbers in the first
column is listed in the second column.

The formula for a weighted mean can be used to find the mean of the data in a frequency distribution.
The only change is that the weights 𝑤1 , 𝑤2 , 𝑤3 , … , 𝑤𝑛 are replaced with the frequencies 𝑓1 , 𝑓2 , 𝑓3 , … , 𝑓𝑛 .
This procedure is illustrated in the next example.
Example 5. Find the Mean of Data Displayed in a Frequency Distribution
Find the mean of the data in Table 13.4.
Solution
The numbers in the right-hand column of Table 13.4 are the frequencies f for the numbers in the
first column. The sum of all the frequencies is 40.
Σ(𝑥⋅𝑓 ) (0⋅5)+(1⋅12)+(2⋅14)+(3⋅3)+(4⋅2)+(5⋅3)+(6⋅0)+(7⋅1) 79
Mean = = = 40 = 1.975
Σ𝑓 40

The mean number of laptop computers per household for the homes in the sub division is
1.975.

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Instructor: Neil B. Montero
 Aurora State College of Technology
GE4: Mathematics in the Modern Word

TRY THIS!!! A housing division consists of 45 homes. The

following frequency distribution shows the number of homes in

the subdivision that are two-bedroom homes, the number that

are three-bedroom homes, the number that are four-bedroom

homes, and the number that are five-bedroom homes. Find the

mean number of bedrooms for the 45 homes.

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Instructor: Neil B. Montero