6.3: The Mode
Learning Objectives
- Understand the concept of the mode.
- Identify the mode of a set of given data.
- Identify the mode of a set of data given in different representations.
Introduction
Do you remember the problem presented in the lesson on mean that dealt with the hand spans of students in a classroom? If you were making gloves for the winter Olympics, what measurement would be of interest to you?
The mode of a set of data is simply the number that appears most frequently in the set. If two or more values appear with the same greatest frequency, each is a mode. When no value is repeated, there is no mode. The word ‘modal’ is often used when referring to the mode of a data set. An example would be the response to the question “What is the mode of the numbers?” The response may be writes as “The modal number is 4.” Observation, rather than calculation, is necessary when determining the mode of a data set.
Example 1: What is the mode of the numbers?
a) 1, 2, 2, 4, 5, 5, 5, 7, 8?
b) 1, 3, 5, 6, 7, 8, 9
Solution:
a) The modes of the above numbers are 2 and 5, since both numbers appear twice and no other number is repeated.
b) There is no mode for these values since none of the values is repeated.
Example 2: The life of a new type of battery was measured (in hours) for a sample of 24 batteries with the following results:
\begin{align*}& 34, 28, 36, 30, 33, 32, 35, 31, 28, 29, 30, 27 \\
& 31, 25, 32, 30, 32, 30, 29, 34, 31, 33, 35, 29\end{align*}
What is the modal number of hours for the tested batteries?
Solution:
It is not necessary, but you may find it easier to determine the mode if the data was organized – arranged from smallest to largest.
\begin{align*}& 25, 27, 28, 28, 29, 29, 29, \fbox{30, 30, 30, 30}, 31 \\ & 31, 31, 32, 32, 32, 33, 33, \ 34, \ 34, \ 35, 35, 36\end{align*}
The mode of the number of hours for the tested batteries is 30 since it is repeated 4 times. If the data set contains a large number of data, the mode can be readily seen if the values are represented in a tally chart. Creating a tally chart is less time consuming than creating a frequency chart – you don’t have to constantly review the numbers. The tally can be placed beside the number when you come to it in the data set.
Example 3: Find the mode of the following:
8, 7, 6, 5, 8, 7, 7, 6, 5, 7, 8, 6, 7, 8, 7, 7, 6, 6, 6, 7, 8, 6, 7, 7, 5, 8, 5, 5, 6, 8, 6, 5, 5, 7, 7
Solution:
Number | Tally | Frequency |
---|---|---|
5 | \begin{align*}\bcancel{/ / / /}\ / /\end{align*} | 7 |
6 | \begin{align*}\bcancel{/ / / /}\ / / / /\end{align*} | 9 |
7 | \begin{align*}\bcancel{/ / / /}\ \bcancel{/ / / /}\ / /\end{align*} | 12 |
8 | \begin{align*}\bcancel{/ / / /}\ / /\end{align*} | 7 |
The mode of the numbers is obvious from the tally chart. The mode of the data is 7 since it is repeated the most. If we return to the problem about hand spans, a person making gloves for the winter Olympics would be interested in the measure of \begin{align*}7 \frac{3}{4}\end{align*} inches since it is the most common measurement of the group.
Lesson Summary
Although there are no mathematical calculations involved in determining the mode of a data set, it is still an important measure of central tendency. The mode is often used in everyday life by businesses and people who are concerned about the most popular or most common item in a data set. If you are operating a deli and you offer ten different sandwiches, you will make sure that you have all the ingredients for the one that you sell the most. Clothing stores also operate their business to include the most popular apparel. The mode helps many people in many walks of life to be successful – all based on the one that appears the most often.
Points to Consider
- Is the mode referred to in any other area of statistics?
Review Questions
- A class of students recorded their shoe sizes and the results are as follows: \begin{align*}8, 5, 8, 5, 7, 6, 7, 7, 5, 7, 5, 5, 6, 6, 9, 8, 9, 7, 9, 9, 6, 8, 6, 6, 7, 8, 7, 9, 5, 6\end{align*} What size represents the mode?
- In a local hockey league, the goals scored by all the teams during a weekend tournament were: \begin{align*}4, 1, 0, 7, 6, 3, 2, 2, 1, 7, 4, 0, 2, 5, 6, 6, 0, 3, 6, 5, 2, 7, 5, 3, 2, 3, 6, 6\end{align*} What is the mode for the goals scored during the tournament?
- Two dice are thrown together 20 times and the results are shown below:
Score of The Roll | Frequency |
---|---|
2 | 1 |
3 | 1 |
4 | 3 |
5 | 1 |
6 | 3 |
7 | 3 |
8 | 4 |
9 | 1 |
10 | 1 |
11 | 1 |
12 | 1 |
What is the modal score?
- The time (in minutes) taken by a man riding his bicycle to work were \begin{align*}54, 57, 55, 58, 55, 57, 57, 56, 58, 54, 58, 54, 54, 53, 56, 58, 57, 53, 55, 57\end{align*} What is the mode of his times?
- The number of students attending class was recorded for thirty consecutive days. The recorded attendance was: \begin{align*}& 30, 32, 28, 28, 29, 30, 31, 28, 27, 27, 31, 28, 32, 28, 27 \\ & 28, 30, 30, 29, 32, 32, 28, 29, 30, 31, 30, 32, 31, 29, 29 \\\end{align*} What is the modal attendance?
- The Vince Ryan Hockey Tournament attracts teams from Canada and the United States. The host team has recorded their results over the past fifteen years of the tournament and has published the results in the local newspaper.
Year | Wins (2 Points) | Ties (1 Point) | Loses (0 Point) |
---|---|---|---|
1995 | 3 | 4 | 3 |
1996 | 4 | 0 | 6 |
1997 | 7 | 0 | 3 |
1998 | 3 | 2 | 5 |
1999 | 8 | 0 | 2 |
2000 | 5 | 0 | 5 |
2001 | 6 | 2 | 2 |
2002 | 7 | 2 | 1 |
2003 | 4 | 2 | 4 |
2004 | 5 | 1 | 4 |
2005 | 6 | 2 | 2 |
2006 | 5 | 4 | 1 |
2007 | 6 | 2 | 4 |
2008 | 6 | 0 | 4 |
2009 | 2 | 4 | 4 |
What is the mode for the host team’s points?
- Two-color counters are often used when teaching students how to add and subtract integers. These counters are red on one side and yellow on the other. Three counters are tossed simultaneously 20 times. Each counter either landed Red (R) or Yellow(Y). The results of the tosses are shown below:
Counter 1 | Counter 2 | Counter 3 | Counter 1 | Counter 2 | Counter 3 |
---|---|---|---|---|---|
Y | Y | Y | |||
Y | Y | Y | Y | ||
Y | Y | Y | |||
Y | Y | ||||
Y | Y | Y | Y | ||
Y | Y | Y | Y | Y | Y |
Y | Y | Y | |||
Y | Y | Y | |||
Y | Y | Y | |||
Y | Y |
Which set of results is the mode
3 Reds
3 Yellows
2 Reds and 1 Yellow
or 1 Red and 2 Yellows?
- The temperature in \begin{align*}^\circ F\end{align*} on 20 days during the winter was: \begin{align*}& 40^\circ F, 36^\circ F, 36^\circ F, 34^\circ F, 30^\circ F, 30^\circ F, 32^\circ F, 34^\circ F, 38^\circ F, 40^\circ F \\ & 34^\circ F, 34^\circ F, 38^\circ F, 36^\circ F, 38^\circ F, 36^\circ F, 34^\circ F, 38^\circ F, 40^\circ F, 36^\circ F\end{align*} What was the modal temperature?
Review Answers
- There are two modes (6 and 7).
- 6 goals
- (8)
- 57 minutes
- (28)
- 14 points
Counter 1 | Counter 2 | Counter 3 | Counter 1 | Counter 2 | Counter 3 |
---|---|---|---|---|---|
R | R | R | |||
R | R | ||||
R | R | R | |||
R | R | R | R | ||
R | R | ||||
R | R | R | |||
R | R | R | |||
R | R | R | |||
R | R | R | R |
(1 Red and 2 Yellows)
- \begin{align*}34^\circ F\end{align*}
Answer Key for Review Questions (even numbers)
2. 6 goals
4. 57 minutes
6. 14 points
8. \begin{align*}340^\circ F\end{align*}
Vocabulary
- Frequency Table
- A table that shows how often each data value, or group of data values, occurs.
- Mean
- A number that is typical of a set of data. The mean is calculated by adding all the data values and dividing the sum by the number of values.
- Median
- The value of a data set that occupies the middle position. For an odd- number set of data, it is the value such that there is an equal number of data before and after this middle value. For an even-number of data, the median is the average of the two values in the middle position.
- Mode
- The value or values that occur the most often in a set of data.