1.4. The Mode

The value that occurs most often in a data set is called the mode.

Example:

The following data give the GPA of 7 students

3.6; 3.2; 2.8; 3.6; 3.8; 3.6; 2.9

Find the mode.

Solution:

It is helpful to arrange the data in order, although it is not necessary

2.8; 2.9; 3.2; 3.6; 3.6; 3.6; 3.8

Since 3.6 occurs three times and 3.6 has a frequency larger than any

other number- the mode for the data set is 3.6.

A data set can have more than one mode or no mode at all, whereas it

will have only one mean and only one median.

A data set with each value occurring only once has no mode.

A data set with two (or more) values occurring with the same (highest)

frequency has two (or more) modes.

Example:

Find the mode of the set data set:

2, 3, 3, 1, 2, 7, 7, 3, 1, 2

Solution: Since 2 and 3 both occur three times, the modes are 2 and 3.

This data set is said to be bimodal.

Example:

Last year’s income of six randomly selected families were

210.000, 300.000, 325.000, 280.000, 315.000, 410.000

Find the mode.

Solution:

Since each value occurs only once, there is no mode.

Remark: One advantage of the mode is that it can be calculated for

both kinds of data, quantitive and qualitaive, whereas the mean and

median can be calculated only for quantitive data.

Example:

6 students are selected at random. Their statuses are:

Senior, sophomore, senior, junior, senior, sophomore.

Find the mode.

Solution:

Since senior occurs more frequently than the other categories, it is the

mode for this data set. However, we can not calculate the mean and

median for this data set.

For a data set, the mean, median, and mode can be quite different.

Consider the following example.

Example:

The number of days the first six heart transplant patients survived after their operations were

15; 3; 46; 623; 126; 64

Find the mean, median, and mode.

Solution:

The sample mean is

To find median, first we rank the data. The ordered values are

3, 15, 46, 64, 126, 623

.

Since each value occurs only once, there is no mode.

In this example, the mean is much higher than the median. Only 1 out of 6 patients survived longer than days. It is because of only large survival time that greatly inflates the mean. In this and similar situations, the median should be used as the measure of central tendency.

To sum up, we can not conclude which of the three measures of central tendency is a better measure overall. Each of them may be better under different situations. But the mean is the most used measure of central tendency. The advantage of the mean is that its calculation includes each value of the data set.

Exercises

1. Calculate the mean and median for each of the following data set.

a) 6, 10, 7, 14, 8

b) 2, 1, 4, 2, 1

2. The following data set belongs to a population

5, -7, 2, 0, -9, 12, 10, 7

Calculate the mean, median and mode.

3. Find the mode of each of the following samples

a) 5, 8, 11, 9, 8, 6, 8

b) 7, 12, 8, 7, 10, 11, 8, 6, 10, 13, 7, 8

4. Twelve secretaries were given a typing test, and the time (in minutes) to complete it were as follows:

8, 12, 15, 9, 6, 8, 10, 9, 8, 6, 7, 8

Find the mean, median, and mode.

5. The grade point average (GPA) of 10 students who applied for financial aid are shown below

3.15, 3.62, 2.54, 2.81, 3.97, 1.85, 1.93, 2.63, 2.50, 2.80

Find the mean, median, and mode.

6. During a year, the major earthquakes had Richter magnitudes as shown below

7.0, 6.2, 7.7, 8.0, 6.4, 6.2, 7.2, 5.4, 6.4, 6.5, 7.2, 5.4

Find the mean, median, and mode.

7. Eight participants in a bike race had following finishing times in minutes

28, 22, 26, 33, 21, 23, 37, 24

Find the mean and median for the finishing times.

8. The monthly income in dollars for seven families are

950, 775, 925, 2500, 1150, 850, 975

a) Calculate the mean and median income

b) Which of the two is preferable as a measure of center,

and why?

9. The numbers of defective parts observed on 16 different days are shown below:

11, 14, 18, 14, 21, 17, 13, 21, 25, 19, 17, 13, 28, 13, 17, 18

What statistical measures of central tendency would help summarize this data set?

What information would you report to the production manager?

10. Consider the following two data sets:

Data set I: 10, 12, 17, 8, 15

Data set II: 13, 15, 20, 11, 18

Notice that each value of the second data set is obtained by adding 3 to the corresponding value of the first data set. Calculate the mean for each of these data sets. Comment on the relationship between the two means.

11. Consider the following two data sets:

Data set I: 2, 5, 7, 9, 8

Data set II: 6, 15, 21, 27, 24

Notice that each value of the second data set is obtained by multiplying the corresponding value of the first data set by 3. Calculate the mean for each of these data sets. Comment on the relationship between the two means.

Answers

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