1. The mode

2. The most popular destination
This graph shows the destinations of U.S. vacations in 2011.
Holiday destinations abroad of U.S. citizens 30 25 20 15
Percentage of holidays
abroad 2011 10 5
Mexico
Asia
Europe
Other
Canada
Oceania
Caribbean
S. America
Middle East
Central America
Teacher notes
The most popular destination is Europe.
Use this slide to review what students remember of the mode from middle school. Ensure students understand the meaning of the word “frequency”. Draw attention to the fact that the bars are separated because the data is discrete.
Photo credit: © Darren Baker, Shutterstock.com 2012
Destination
Which is the most popular destination?

3. The mode The most common item is called the mode.
The mode is the item that occurs the most in a data set.
In the graph the mode is “Europe” because it is represented by the highest bar.
We could also say “The modal holiday destination abroad is Europe.”
Teacher notes
It is possible to have more than one modal value; there could be two or more events that are equally the most popular. It is also possible to have no modal value; all events could be equally popular.
Is it possible to have more than one modal value?
Is it possible to have no modal value?

4. Percentage of adult population
The mode
Top ten activities undertaken on tourism trips (types A and B)
Eating/drinking out
Shopping
Visiting friends or relatives
Entertainment
Leisure attraction/place of interest
TRIP TYPE A
TRIP TYPE B
Hobby or special interest
Walking/hiking
Watching live sport event
Sports/active pursuits
Driving/sightseeing/picnicing
Teacher notes
Trip type A has a clear mode; eating/drinking out. Activities for trip type B are more evenly distributed and there is more than one mode; leisure attractions/places of interest and walking/hiking.
Due to the popularity of the activities, the left graph could represent city trips and the right countryside trips. 30 25 20 15 10 5 5 10 15 20 25 30
Percentage of adult population
Compare the two graphs. What is the mode for each trip type? Suggest two types of holiday that A and B could represent in the graph.

5. The mode
These figures show the number of juniors that attended a baseball practice each week.
Discuss the following questions:
Over how many weeks were the results collected?
What is the modal number of students attending?
What is the outlier in the data set? Can you think of any possible reasons for the outlier in this data set?
Teacher notes
Ask students to discuss these questions in pairs.
The answers are: 18 weeks, 14 and 15 are the joint mode [4 weeks each], the outlier is 0. A possible reason for the outlier is that the juniors were on a trip, or the coach/ teacher was absent. It would be sensible to put the results into a tally chart or frequency table if the data set were very large.
Mathematical Practices
2) Reason abstractly and quantitatively.
When answering the questions on this slide, students should be able to step back from the data and think about the meanings of the values; the total number of values represents the number of weeks, the modal number of students attending is the most common number of students going to practice each week, and the outlier may represent a week of vacation or an absent coach.
If the data set were very large, what would be the best way to find the mode?

6. The mode from bar charts
A group of students were asked how many pets they had.
This graph shows the results. 14 12 10
Frequency 8 6 4 2
Teacher notes
14 students have more than two pets. The modal number of pets is students took part in the survey.
This graph represents numerical data. Some students may confuse the frequency with the number of pets, i.e. give 13 as the mode rather than 1. In this situation, you could begin writing out the whole list of results to illustrate the meanings of the number on the two axes.
Photo credit: © Eric Isselee, Shutterstock.com 2012 1 2 3 4 5 6
Numbers of pets
How many students have more than two pets?
What is the modal number of pets?
How many students took part in the survey?

7. The mode from bar charts
A survey on participation in sport asked people the number of days in the past week when they had taken part in a moderate intensity sport.
Number of days in the week that adults exercised 70 60 50 40
Percentage of adults 30 20
Teacher notes
The graph shows that the modal number of days was zero. This statistic may cause students to make comments such as “There were more people who did no exercise at all than those who did.” However, the percentages of adults who did zero days of exercise is just under 50%, so in fact more people exercised. Encourage students to discuss the implications of the groups chosen for the bar graph; for example, if the graph were to show simply “Did exercise” and “Did no exercise,” the mode would be “Did exercise.” 10 1 2 3+
Number of days of exercise in the week
Discuss what the mode of this graph shows.

8. Unfair mode Another survey is carried out among college students.
The results are represented in this table:
No. of sports played
Frequency 20 1 17 2 15 3 10 4 9 5+ 5
A newspaper reporter writes: “You may be surprised that the average number of sports played by college students is 0.”
Teacher notes
Students may note that the mode is the lowest number of sports, and the majority of students actually do play at least one sport. Students may suggest that the mean or the median is a fairer way of representing the data in this case. It would be less misleading if the newspaper reported changed the wording of his statement, to something like “In a recent survey it was found that more students played no sports at all than any other number of sports.”
Mathematical Practices
3) Construct viable arguments and critique the reasoning of others.
Students should explain their opinion of the newspaper quote and be able to justify their claim mathematically. For example, they may say that it is misleading as the mean and median number of sports are very different from the mode, and “average” usually implies the mean.
4) Model with mathematics.
The mode is an example of a very simple mathematical model for a dataset. Students should see that the mode provides only partial information, and can be misleading.
Photo credit: © rangizz, Shutterstock.com 2012
Discuss the following questions:
Do you think this is a fair comment?
Why is the mode misleading in this example?
Should the reporter say which measure of central tendency has been used?

9. Modal groups
This graph shows grade 8 times for a 50 m wheelbarrow race.
Grade 8 wheelbarrow race results 20 15
Frequency 10 5
Teacher notes
The modal time interval is 20 to 21 seconds. There are 20 students in this interval.
Photo credit: © Mandy Godbehear, Shutterstock.com 2012 16 17 18 19 20 21 22 23 24
Times in seconds
What is the modal time interval?
How many students are in this interval?