Mode and median
🎯 In this topic you will
- Understand the mode of a data set
- Understand the median of a data set
- Compare the mode and the median to determine a suitable average in context
🧠 Key Words
- average
- median
- mode
Show Definitions
- average: A measure that represents the typical or central value of a data set.
- median: The middle value of a data set when the numbers are arranged in order.
- mode: The value that appears most often in a data set.
Understanding Averages
A verages can help you solve problems and make decisions. By combining people’s scores or ratings, you can work out a single value that represents what most people think.
Looking at a Film Score
W hen people give a film a score out of 10, all their scores can be put together to find an average score. This helps show what viewers thought overall.
Different Opinions
N ot everybody gave the film the same score. Some people might have given a very low score, such as 1 out of 10, while others might have given a very high score, such as 10 out of 10.
What the Average Tells Us
T he average represents what many people think about the film when all the reviews are combined. You can use this number to help you decide whether or not you might want to watch the film.
Learning About Averages
I n this section you will learn two different ways to work out the average of a set of data.
❓ EXERCISES
🧠 Reasoning Tip
The value which occurs the most often is the mode. For example, here is a set of animals: cat, cat, mouse, rabbit. ‘Cat’ occurs the most so ‘cat’ is the mode.
1. What is the mode of these sets of numbers?
a. $1,\,1,\,1,\,2,\,3$
b. $30,\,29,\,31,\,29,\,32$
c. $8,\,9,\,9,\,8,\,9,\,8,\,9$
d. $1\tfrac{1}{2},\,1\tfrac{1}{4},\,1\tfrac{1}{2},\,1\tfrac{3}{4},\,1\tfrac{3}{4},\,1\tfrac{3}{4}$
👀 Show answer
b. Mode = $29$
c. Mode = $9$
d. Mode = $1\tfrac{3}{4}$
2. These are the number of bananas in each bunch in a shop.
$6\;\;4\;\;5\;\;6\;\;7\;\;4\;\;4\;\;5\;\;6\;\;5\;\;7\;\;5$
What is the mode of the number of bananas in a bunch?
👀 Show answer
❓ EXERCISES
🧠 Reasoning Tip
The median is the middle value in a sorted list of numbers.
3. These sets of numbers are written in order. Find the median of each set.
a. $1,\,2,\,3,\,4,\,5$
b. $5,\,5,\,6,\,7,\,7,\,8,\,9,\,9,\,9,\,10,\,10$
c. $253,\,257,\,270,\,299,\,308,\,310,\,324,\,740,\,751$
👀 Show answer
b. Median = $8$
c. Median = $308$
4. These sets of numbers are not written in order. Find the median of each set.
a. $3,\,7,\,1,\,5,\,9$
b. $13,\,13,\,11,\,12,\,14,\,12,\,14,\,14,\,11$
c. $535,\,422,\,278,\,567,\,453,\,772,\,329$
👀 Show answer
b. Sorted: $11,\,11,\,12,\,12,\,13,\,14,\,14,\,14,\,13$ → Median = $13$
c. Sorted: $278,\,329,\,422,\,453,\,535,\,567,\,772$ → Median = $453$
5. What is the median mass of these bags?
👀 Show answer
Sorted: $595,\,670,\,722,\,789,\,855,\,998,\,1000$
Median = $789\text{ g}$
6. Find the mode and the median of each set of numbers. Copy and complete this sentence for each set: ‘The mode is _____ and the median is _____’.
a. $5,\,6,\,1,\,2,\,6$
b. $11,\,12,\,9,\,11,\,11,\,12,\,13$
c. $3,\,5,\,6,\,2,\,9,\,3,\,4,\,3,\,5$
d. $5,\,2,\,5,\,6,\,7,\,2,\,3,\,2,\,1$
👀 Show answer
b. Mode = $11$, Median = $11$
c. Mode = $3$, Median = $5$
d. Mode = $2$, Median = $3$
🧠 Think like a Mathematician
Main Task: Find sets of five numbers that have a mode of $3$ and a median of $3$.
Activity Instructions (solo):
- Create one set of five numbers where the mode is $3$ and the median is $3$.
- Write two additional sets of five numbers that also have a mode of $3$ and a median of $3$.
- Check your own sets to confirm they satisfy both conditions.
- Reflect on how you are specialising by constructing different sets that meet the same criteria.
Follow-up Questions:
Show Answers
- Example valid sets:
• $2,\,3,\,3,\,4,\,5$
• $3,\,3,\,3,\,1,\,5$
• $3,\,1,\,3,\,2,\,6$ - 1: For the mode to be $3$, the number $3$ must occur more frequently than any other number. This requires at least two occurrences.
- 2: In a list of five sorted numbers, the median is the third value. Therefore, the third value must be $3$.
- 3: Another valid set: $0,\,3,\,3,\,3,\,7$
❓ EXERCISES
🧠 Reasoning Tip
Sometimes one type of average is more useful than another to solve a problem. Read the question carefully. Work out the mode and the median. Think about which average best solves the problem in the question.
7. A shop sells gloves in sizes $1$, $2$, $3$ and $4$.
In one day the shop sells these glove sizes:
$1,\,2,\,3,\,4,\,1,\,4,\,4,\,2,\,4$
a. What is the mode of the glove sizes sold?
b. What is the median of the glove sizes sold?
c. The shopkeeper wants to buy more gloves to put in the shop, but the shopkeeper can only buy one size of gloves. Should the shopkeeper use the mode or the median to decide what size gloves to buy? Why?
👀 Show answer
a. Mode = $4$
b. Median = $3$
c. The shopkeeper should use the mode because it shows the size sold most often, which is the best indicator of demand.
8. This table shows the rainfall for the first 11 months of the year. Use the table to work out the average amount of rain in a month.
| Month | Jan | Feb | Mar | Apr | May | June | July | Aug | Sept | Oct | Nov |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Rainfall (mm) | $7$ | $6$ | $14$ | $13$ | $5$ | $0$ | $4$ | $4$ | $0$ | $0$ | $3$ |
a. What is the mode amount of rain?
b. What is the median amount of rain?
c. Do you think that the mode or the median best describes the average monthly rainfall? Why?
👀 Show answer
a. Mode = $0$ (appears three times) b. Sorted values: $0,\,0,\,0,\,3,\,4,\,4,\,5,\,6,\,7,\,13,\,14$ → Median = $4$ c. The median best describes the average monthly rainfall because the mode ($0$) is heavily affected by very dry months and does not represent the typical rainfall level.