D1.5 determine the mean and the median and identify the mode(s), if any, for various data sets involving whole numbers, and explain what each of these measures indicates about the data.
Activity 1: Median and Mean
Present the following statement to the students and let them explore the idea in pairs. Then, pair up the teams to present their findings to each other. Facilitate a mathematical discussion to pool the results.
- Even if a large number is part of the data set, it will not affect the median. Show an example.
- Would this be the case for the mean? Why?
Source: Making Math Meaningful, Marian Small, p. 575.
Activity 2: Mean
Ask students to invent a mean for a given situation, then imagine what the data in the series might be to arrive at that mean.
Example
At the field hockey tournament, six friends scored an average of four goals. What is the number of goals each friend scored? Explain your set of numbers and compare it with that of a peer.
Facilitate a mathematical discussion to highlight students' strategies and share their learning.
This kind of exercise will help students understand what a mean is, and it reveals a lot about their understanding of the concept.
Source: translated from L'@telier - Ressources pédagogiques en ligne (atelier.on.ca).
Activity 3: Mean
Ask students to create two sets of data such that they both have the same mean, and in one set the values are close to the mean and in the other set the data values are far from the mean. Example: 4, 5, 6 vs. 1, 5, 9.
Which set does the mean best describe and why?
(The set whose values are close to the mean; if you used the mean for the other set, it would not give you any sense as to the range of the data.)
Source: Making Math Meaningful, Marian Small, p. 575.
Activity 4: Median and Mean
Present students with the following data set.
Number of students in groups
|
Group A |
Group B |
Group C |
Group D |
Group E |
Group F |
|
3 |
5 |
4 |
7 |
5 |
6 |
The average number of students in the six groups is 5. Have students participate in a Think-Pair-Share by asking them what is meant by average. (Students can see that if we put all the values from the groups together and divide the total evenly into six, it gives us an average of five students. We can also say that five does not necessarily mean five students in each group. There may be groups with more or fewer students.)
The median of this set is 5.5. Do another “Think-Talk-Share” by asking students what is meant by the median.
- Does the average of 5 appropriately represent the number of students in the groups? (Yes, because there are few groups that have slightly more or slightly less than five students. Two groups have five students)
- Does the median of 5.5 appropriately represent the number of students in the groups? (Yes, because there are few groups that have slightly more or slightly less than 5.5 students. Two groups have five students.)
- If we added a group of 30 students and we removed group A, B, or D, how would that affect the mean? the median? the graph? Why? (If we added a group of 30 students, it wouldn't have much of an effect, since the median would be 5 instead of 5.5. However, the mean would increase to 10 instead of 5. The mean of 10 would no longer be representative of the number of students in the groups)
Point out that by adding an extreme data value (or two extreme data values) to a set, the median will not be siginifcantly impacted compared to the mean, thus the median is a particularly good measure of central tendency for these types of data sets. Discuss this with students to point out what the median and mean represent.
Activity 5: Mode, Median and Mean
Present students with the following data set.
Number of Minutes of Exercise in a Day
|
Student A |
Student B |
Student C |
Student D |
Student E |
Student F |
|
90 |
90 |
90 |
90 |
90 |
30 |
- Which central measure would be more appropriate to describe this data set, mode, median or mean? Why?(The mode 90 since it repeats very often. Also the median since 90 is in the center of the data. The mean of 60 is not the best indicator of how many minutes students exercise)
- What impact does the student F value of 30 have on the mean of the data set? (the lower value of 30 gives a false impression that the group of students only did about 80 minutes of exercise in a day)
Source: adapted from Making Math Meaningful, Marian Small, p. 575.