Measures of Central Location
(Averages) and Percentiles
BUSA 2100, Section 3.1
Introduction
Values of a variable tend to cluster
around a central point.
 A measure of central location indicates
a center, average, or typical value.
 A measure of central location (average)
helps summarize data concisely with
one value.

1st Type of Average: Mode
Definition: The mode is the item that
occurs most often.
 Example 1 – Consider this data set:
{96,91,91,87,84, 82,79,75,72,69,62}.
 The mode is 91 since it occurs twice.
 But 91 isn’t a “central” or “typical” value.
 Another problem: some data sets have
no mode, e.g. set above with one 91
removed.

Mode (Page 2)
Some data sets have more than one
mode. Example: Height of adults is
bimodal (two modes). Why?
 So the mode is not very useful and not
reliable except for categorical data.
 Example 2: (categorical data) Ask
students to state their favorite kind of
pie.

2nd Type of Average: Median
Definition: The median is the middle
item of a ranked set of data.
 It is the (n+1)/2 th item in a ranked set
of n items.
 Example 1: Find the median of this set
of 7 items. {75,64,82,96,72,47,59}

Median (Page 2)

Example 2: Add 89 to the previous set.
{89,75,64,82,96,72,47,59}
3rd Type of Average: Mean
 Definition:
The sample mean, X-bar =
Sum of the X’s divided by n, where
n = number of items in the data set
(sample).
 Example 1: {89,75,64,82,96,72,47,59}
Mean (Page 2)
Mean is the most widely used and best
measure of central location except in
one situation (to be discussed later).
 Advantages of the mean: (1) More
comprehensive because it uses all of
the data (not just the center item(s)).
 (2) Combined or weighted means can
be calculated.

Mean (Page 3)
Example 2 (combined mean): Class #1
had a mean test score of 80; Class #2
had a mean test score of 60.
 What is the overall mean for both
classes combined?
 Is it 70, the average of 80 and 60?

Mean (Page 4)

Class #1 has 40 students; Class #2 has
10 students.
Mean (Page 5)
Example 3 (weighted mean): In a
course, a professor gives 3 tests and a
final exam, and requires a project.
 The final exam counts 1 1/2 times as
much as each test and the project
counts twice as much as each test.
 Charles Malone made 80, 74, 67, 86,
and 90. What is his course average?

Mean (Page 6)

Note: The unweighted mean is 79.4.
Mean (Page 7)
The median is preferred to the mean if
there are extreme values present.
 Example 4: Incomes for 5 families are:
{$30K, $40K, $50K, $60K, $820K}
 Mean = $1,000,000 / 5 = $200,000, but
this is not a “center” or “typical” value.
 Median = $50,000 (more accurate)

Percentiles
Definition: The pth percentile is a value
that is > p percent of the values in a
data distribution.
 Values for p: 0, 1, 2,...,98, 99, 99.5, 99.9
 Example: If you were in the 86th
percentile on a test, what does that
mean?

Percentiles (Page 2)
Three steps for calculating percentiles:
 (1) Arrange data in ascending order
 (2) Calculate index (rank): i = (p/100)* n.

(a) If i is not a whole number, round
up to the next whole number.

(b) If i is a whole number, use i + .5 .
 (3) Identify the answer.

Percentiles (Page 3)



Example: {2450, 2500, 2650, 2430, 2355,
2260, 2490, 2680, 2540, 2775, 2525, 2465,
2610, 2390} are monthly salaries for 14
business graduates.
Find the 67th percentile.
Step 1: Arrange in ascending order: {2260,
2355, 2390, 2430, 2450, 2465, 2490, 2500,
2525, 2540, 2610, 2650, 2680, 2775}
Percentiles (Page 4)

Find the 50th percentile.
Percentiles (Page 5)

The 50th percentile is the median.
The 25th percentile is the 1st quartile.
 The 75th percentile is the 3rd quartile.
