D1.5 Determine the mean and the median and identify the mode(s), if any, for various data sets involving whole numbers and decimal numbers, and explain what each of these measures indicates about the data.
Activity 1: Median and Mean
Pose the following statements to the students and let them explore them in teams. Afterwards, pair up the teams so that they can share their findings.
- Even if a large number is part of a set of data, 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
Have students create two data sets 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; for example: 5; 5.7; 6.3 compared to 1.3; 5.8; 9.9.
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 the following data set to students.
Price of a Flour Bag (2.5 kg)
| Store A | Store B | Store C | Store D | Store E | Store F |
|---|---|---|---|---|---|
|
$5.97 |
$5.75 |
$4.99 |
$5.25 |
$6.50 |
$6.25 |
The mean price of a flour bag, at all six stores, is $5.79. Do a Think-Pair-Share activity with students asking them what we mean by the "mean". (Students can see that if you add all the prices together and divide the total evenly into six, it comes to an average of $5.79. They can also tell that $5.79 does not necessarily mean that the cost of a flour bag is $5.79 at each store. It may be that the price is higher in one store and lower in another)
The median of this set is $5.86. Do another Think-Pair-Share activity with students asking them what the amount $5.86, the median, tells you.
- Does the mean $5.79 appropriately represent the price of a flour bag in a store? (Yes, because there is not much difference between the highest price ($6.50) and the lowest ($4.99).)
- Does the median $5.86 appropriately represent the price of a flour bag in a store? (Yes, because there is not much difference between the highest price ($6.50) and the lowest ($4.99).)
- If we added a price of $9.86 or removed stores A, E and F, what effect would that have on the mean? on the median? on the graphic representation? Why? (Adding a price of $9.86 would have a slight impact on the median, since the median would be $5.97 rather than $5.86. On the other hand, the mean would be $6.37 instead $5.79, a sufficiently significant increase. The mean would then not be representative of the price of a flour bag in a store, since five of the prices would be lower than the mean, and two of the prices would be higher than the mean. However, if stores A, E and F were removed, then the mean would be $5.33 as opposed to $5.79 and the median would be $5.25 as opposed to $5.86, since the highest prices high would be removed.)
Point out that when adding data to a set with extreme values, the median is a particularly good measure of central tendency because these extreme values do not affect the median. However, this is not the case with the mean. Discuss this with students to consolidate what the median and mean represent.
Activity 5: Mode, Median and Mean
Present the following data set to students.
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 measure of central tendency 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.)
- How does the value 30 of student F affect the data set? (It affects the mean. The fact that the 30 value is lower than the other values, 90 minutes, it lowers the mean and gives a false impression that students averaged 60 minutes of exercise in a day.)
- If the data from student F were removed, how would this affect the mode, median and mean?
- If we added another student who exercised for 90 minutes? 30 minutes? 5 minutes? what impact would this have on the mode, median and mean?
Source: adapted from Making Math Meaningful, Marian Small, p. 575.