The class's duplicate stickers
How many duplicate stickers does a typical student in this class have? What if someone shows up with 600?
Number line and tasks
Each dot is a student, placed at the number of duplicate stickers they have in their album. Stacked dots have the same number; a dot with a line down to the axis moved up only so it wouldn't cover its neighbor. Drag the dots, or use Tab and the arrow keys.
What each value means
- The mean is everyone's duplicates added up, divided by the number of students.
- The median is the number of the student in the middle when everyone lines up in order.
- The mode is the number that shows up most often.
- The range is the largest number minus the smallest.
Show the sticker table
| Position | Student | Duplicates |
|---|---|---|
| Sum |
Duda gets more stickers
Duda (D) has 0 duplicates. Write down the four values below the number line, drag Duda to 60, and compare.
A tie for the mode
Move two students onto the same number as a third student. What happens to the mode?
Separate the numbers with "and".
A collector is coming
Marina is joining the class. She has been collecting for years and has 600 duplicate stickers. Before she arrives, place your bet: what will the mean and the median be?
The current values are below the number line. If you have no idea, tap I don't know.
Make your guess first: type the numbers and tap Lock in my guess, or tap I don't know.
| Measure | Before | After |
|---|
Before the challenge, read What's going on, right below. Then tap Next.
Mean 25, median 15
Change the stickers on the number line until the mean is 25 and the median is 15.
The school newspaper
Salaries at a company
At a company, most employees earn little and a few executives earn a lot. The employees say salaries are low; management says they're high. Both sides are using a real number.
What's going on
The four values below the number line are different ways to sum up the class's duplicate stickers.
- The mean is everyone's duplicate stickers added up, divided by the number of students. In the starting class, the sum is 192 and there are 12 students: .
- The median is the middle value when the numbers are in order. With an even number of students, two of them end up in the middle, and the median is the mean of those two.
- The mode is the number that shows up most often. There can be more than one, and if no number repeats, the class has no mode.
- The range is the largest number minus the smallest. It measures how spread out the numbers are.
The rest of this explanation unlocks when Marina arrives, in task 3.
When Marina arrived, her 600 stickers went into the sum. That part alone, divided by 13, already gives stickers, and that's why the mean shoots up. For the median, what counts is Marina's place in the lineup, and she's last. In the starting class, the middle fell between the 6th and 7th students; with 13 students, the middle is the 7th, and the median goes from 12 to 14.
The same thing happens with salaries in the adult world. That's why news stories about income often talk about the median salary: half of the people earn less than it, and half earn more. A few very high salaries pull the mean up, and it stops describing a typical person.
For teachers
Objective
By the end, students can choose between the mean and the median to describe the typical value of a data set, knowing that an extreme value pulls the mean and barely moves the median. This is skill EF08MA25 in Brazil's national curriculum (BNCC), with the range as the measure of spread.
Prerequisites
Adding and dividing (including when the result is a decimal), putting numbers in order, reading a number line. Most students already know how to compute a mean; the median and the mode may be new, and the activity shows how they behave before defining them.
How to use in class
Plan for 20 to 25 minutes. Project the activity and read the opening question. Ask for guesses: how many duplicate stickers does a typical student have? Let the class do the first two tasks in the panel, Duda's and the tie for the mode, for 5 minutes (in pairs, if there are devices). Go back to the projector, tap Start over, go to task 3, and ask the class for one guess for the new mean and another for the new median. Type the guesses the class picks into the fields, tap Lock in my guess, and only then tap The collector arrives. Discuss the before-and-after table: why did the mean go up almost 45 stickers and the median only 2? Read the "What's going on" section together; it's below the simulation. Finish with task 4 (mean 25, median 15) in pairs, which takes about 5 minutes, and task 5, the newspaper one, as a class vote. "Going further," about salaries at a company, works for high school classes or as homework.
Common mistakes
Thinking the mean is always the typical value: it shows up as a guess for the new mean that stays close to the old mean, or as choosing the mean in the newspaper task. Thinking the median is the midpoint of the number line, halfway between the smallest and largest values: students are surprised when the line stretches to 600 and the median stays almost in the same place. Taking only one of the two middle values when the number of students is even: ask them to find the starting median in the table (12, the mean of 10 and 14). Forgetting to sort: ask what the "median" of the list would be in roll-call order. Thinking every list has a mode: the book mentions two conventions for when nothing repeats (every value is a mode, or there is no mode); the activity shows "none."
Discussion questions
- When a website says "on average, people in this country earn this much," what would you want to know before believing it describes a typical person?
- Is there a situation where the mean is better than the median for deciding something? (Think about the total number of stickers the class would bring to a sticker swap.)
- In the task with mean 25 and median 15, the mean ended up larger than the median. What would the stickers look like in a class where the mean is smaller than the median?
- If the mode is 10 stickers, can you say that most of the class has 10 duplicates?
- The range went from 42 to 600 because of one person. Is it a good measure of how different the class's collections are?
Based on Contemporary Mathematics, OpenStax, section 8.3 (and the range from section 8.4), under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.