Mode, median, mean & range
🎯 In this topic you will
- Find the mode and median of sets of data.
- Find the mean and range of sets of data.
- Use the average and range to describe sets of data and answer questions.
🧠 Key Words
- average
- bimodal
- mean
- median
- mode
- range
Show Definitions
- average: A value used to represent a set of data, often calculated using the mean, median, or mode.
- bimodal: A data set that has two values occurring most frequently.
- mean: The total of all values divided by the number of values in a data set.
- median: The middle value in an ordered list of numbers.
- mode: The value that appears most often in a data set.
- range: The difference between the highest and lowest values in a data set.
📊 Understanding Averages
Averages can help you solve problems and make decisions. When people review a book out of 10, all their reviews are put together and the average number is found. This average gives a general idea of what people think about the book.
📚 What Averages Show
The average represents what a typical person might think about a book. It helps you understand overall opinions and can guide you when deciding whether you might enjoy reading a book.
❓ EXERCISES
$1.$ Write the modes and median of each set of measures.
$a.$ $4\ \text{cm},\ 4\ \text{cm},\ 5\ \text{cm},\ 5\ \text{cm},\ 6\ \text{cm},\ 7\ \text{cm}$
$b.$ $51\ \text{mm},\ 47\ \text{mm},\ 51\ \text{mm},\ 53\ \text{mm},\ 59\ \text{mm},\ 59\ \text{mm}$
$c.$ $1.2\ \text{m},\ 1.8\ \text{m},\ 1.1\ \text{m},\ 2.1\ \text{m},\ 1.2\ \text{m},\ 1.8\ \text{m},\ 1.6\ \text{m},\ 1.4\ \text{m}$
$d.$ $101\ \text{cm},\ 106\ \text{cm},\ 95\ \text{cm},\ 105\ \text{cm},\ 102\ \text{cm},\ 102\ \text{cm},\ 97\ \text{cm},\ 101\ \text{cm}$
👀 Show answer
$a.$ Modes: $4\ \text{cm}$ and $5\ \text{cm}$. Median: $5\ \text{cm}$.
$b.$ Modes: $51\ \text{mm}$ and $59\ \text{mm}$. Median: $52\ \text{mm}$.
$c.$ Modes: $1.2\ \text{m}$ and $1.8\ \text{m}$. Median: $1.5\ \text{m}$.
$d.$ Modes: $101\ \text{cm}$ and $102\ \text{cm}$. Median: $101.5\ \text{cm}$.
$2.$ What is the mean average for each set of numbers?
$a.$ $5,\ 6,\ 7$
$b.$ $9,\ 9,\ 2,\ 8$
$c.$ $10,\ 12$
$d.$ $2,\ 3,\ 4,\ 5,\ 6$
👀 Show answer
$a.\ \dfrac{5+6+7}{3}=6$
$b.\ \dfrac{9+9+2+8}{4}=7$
$c.\ \dfrac{10+12}{2}=11$
$d.\ \dfrac{2+3+4+5+6}{5}=4$
$3.$ Which of these bowlers has the highest mean average bowling score after six games?
| Player | Game $1$ | Game $2$ | Game $3$ | Game $4$ | Game $5$ | Game $6$ |
|---|---|---|---|---|---|---|
| Player A | $95$ | $108$ | $99$ | $120$ | $95$ | $101$ |
| Player B | $109$ | $130$ | $124$ | $111$ | $145$ | $131$ |
| Player C | $138$ | $130$ | $151$ | $157$ | $153$ | $165$ |
| Player D | $98$ | $154$ | $160$ | $91$ | $129$ | $118$ |
$b.$ Players B and D both have the same average bowling score. Which do you think is the better player? Write a sentence to convince your partner that B or D is better. Use information from the table in your sentence.
👀 Show answer
Mean scores:
Player A: $\dfrac{95+108+99+120+95+101}{6}=103$
Player B: $\dfrac{109+130+124+111+145+131}{6}=125$
Player C: $\dfrac{138+130+151+157+153+165}{6}=149$
Player D: $\dfrac{98+154+160+91+129+118}{6}=125$
So the bowler with the highest mean average score is Player C.
For part $b$, a good sentence is: Player B is the better player because B and D both average $125$, but B is more consistent, scoring between $109$ and $145$, while D scores between $91$ and $160$.
$4.$ Find the range of each set of numbers.
$a.$ $1,\ 2,\ 3,\ 4$
$b.$ $7,\ 7,\ 12,\ 2$
$c.$ $34,\ 33,\ 70,\ 5,\ 6,\ 8$
$d.$ $26,\ 21,\ 35,\ 63,\ 30$
$e.$ $11,\ 10,\ 15,\ 13,\ 11$
$f.$ $25,\ 34,\ 28,\ 29$
$g.$ $91,\ 105,\ 116$
👀 Show answer
$a.\ 4-1=3$
$b.\ 12-2=10$
$c.\ 70-5=65$
$d.\ 63-21=42$
$e.\ 15-10=5$
$f.\ 34-25=9$
$g.\ 116-91=25$
$5.$ Find the range in heights of these two groups of children.
| Group $1$ | Group $2$ |
|---|---|
| $127\ \text{cm},\ 130\ \text{cm}$ | $137\ \text{cm},\ 131\ \text{cm}$ |
| $152\ \text{cm},\ 138\ \text{cm}$ | $129\ \text{cm},\ 143\ \text{cm}$ |
| $135\ \text{cm},\ 138\ \text{cm}$ | $136\ \text{cm},\ 143\ \text{cm}$ |
| $141\ \text{cm}$ | $132\ \text{cm}$ |
Which group has the largest range?
Describe how the two groups would look different because of their different ranges.
👀 Show answer
Group $1$: highest $=152\ \text{cm}$, lowest $=127\ \text{cm}$, so range $=25\ \text{cm}$.
Group $2$: highest $=143\ \text{cm}$, lowest $=129\ \text{cm}$, so range $=14\ \text{cm}$.
So Group $1$ has the larger range.
The children in Group $1$ would look more spread out in height, with bigger differences between the shortest and tallest children. Group $2$ would look more similar in height.
$6.$ Kali, Summer, Benji and Kyle are learning to skip.
While they were practising they recorded how many skips they did in a row.
Here are their attempts:
| Name | $1$st try | $2$nd try | $3$rd try | $4$th try | $5$th try | $6$th try | $7$th try |
|---|---|---|---|---|---|---|---|
| Kali | $6$ | $5$ | $6$ | $6$ | $8$ | $11$ | $7$ |
| Summer | $3$ | $0$ | $3$ | $8$ | $0$ | $7$ | $0$ |
| Benji | $0$ | $0$ | $1$ | $0$ | $4$ | $0$ | $2$ |
| Kyle | $4$ | $7$ | $7$ | $6$ | $2$ | $5$ | $4$ |
$a.$ Copy and complete this table:
| Name | Range | Mode | Median | Mean |
|---|---|---|---|---|
| Kali | ||||
| Summer | ||||
| Benji | ||||
| Kyle |
$b.$ Who do you think has been most successful at skipping? Explain your answer using the information in your table.
👀 Show answer
| Name | Range | Mode | Median | Mean |
|---|---|---|---|---|
| Kali | $6$ | $6$ | $6$ | $7$ |
| Summer | $8$ | $0$ | $3$ | $3$ |
| Benji | $4$ | $0$ | $0$ | $1$ |
| Kyle | $5$ | $4$ and $7$ | $5$ | $5$ |
Kali has been the most successful because Kali has the highest mean, and also strong values for the mode and median.
$7.$ Gabriella and Demi have recorded the temperature in the shade at midday on every day of their six week school holiday.
| Week | Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |
|---|---|---|---|---|---|---|---|
| Week $1$ | $25^\circ\text{C}$ | $20^\circ\text{C}$ | $20^\circ\text{C}$ | $19^\circ\text{C}$ | $20^\circ\text{C}$ | $24^\circ\text{C}$ | $26^\circ\text{C}$ |
| Week $2$ | $25^\circ\text{C}$ | $25^\circ\text{C}$ | $28^\circ\text{C}$ | $27^\circ\text{C}$ | $28^\circ\text{C}$ | $31^\circ\text{C}$ | $25^\circ\text{C}$ |
| Week $3$ | $28^\circ\text{C}$ | $27^\circ\text{C}$ | $27^\circ\text{C}$ | $27^\circ\text{C}$ | $24^\circ\text{C}$ | $22^\circ\text{C}$ | $20^\circ\text{C}$ |
| Week $4$ | $15^\circ\text{C}$ | $19^\circ\text{C}$ | $23^\circ\text{C}$ | $19^\circ\text{C}$ | $16^\circ\text{C}$ | $15^\circ\text{C}$ | $19^\circ\text{C}$ |
| Week $5$ | $18^\circ\text{C}$ | $19^\circ\text{C}$ | $20^\circ\text{C}$ | $23^\circ\text{C}$ | $25^\circ\text{C}$ | $28^\circ\text{C}$ | $28^\circ\text{C}$ |
| Week $6$ | $32^\circ\text{C}$ | $32^\circ\text{C}$ | $32^\circ\text{C}$ | $27^\circ\text{C}$ | $29^\circ\text{C}$ | $28^\circ\text{C}$ | $30^\circ\text{C}$ |
$a.$ Design a table to record the three types of average and the range for each week of the holiday. Complete your table with the averages and range.
$b.$ Which week had the greatest range of temperatures?
$c.$ Use the information in your table to argue which was the warmest week of the holiday.
$d.$ Gabriella and Demi were taking part in a conservation project on every Tuesday of the holiday. What was the range of temperatures on Tuesdays? What was the average (mode, median and mean) temperature on Tuesdays?
👀 Show answer
| Week | Mode | Median | Mean | Range |
|---|---|---|---|---|
| Week $1$ | $20^\circ\text{C}$ | $20^\circ\text{C}$ | $22^\circ\text{C}$ | $7^\circ\text{C}$ |
| Week $2$ | $25^\circ\text{C}$ | $27^\circ\text{C}$ | $27^\circ\text{C}$ | $6^\circ\text{C}$ |
| Week $3$ | $27^\circ\text{C}$ | $27^\circ\text{C}$ | $25^\circ\text{C}$ | $8^\circ\text{C}$ |
| Week $4$ | $19^\circ\text{C}$ | $19^\circ\text{C}$ | $18^\circ\text{C}$ | $8^\circ\text{C}$ |
| Week $5$ | $28^\circ\text{C}$ | $23^\circ\text{C}$ | $23^\circ\text{C}$ | $10^\circ\text{C}$ |
| Week $6$ | $32^\circ\text{C}$ | $30^\circ\text{C}$ | $30^\circ\text{C}$ | $5^\circ\text{C}$ |
Week $5$ had the greatest range of temperatures, with a range of $10^\circ\text{C}$.
The warmest week was Week $6$ because it has the highest mean, the highest median, and the highest mode.
On Tuesdays the temperatures were $20^\circ\text{C},\ 28^\circ\text{C},\ 27^\circ\text{C},\ 23^\circ\text{C},\ 20^\circ\text{C},\ 32^\circ\text{C}$. The range is $32-20=12^\circ\text{C}$. The mode is $20^\circ\text{C}$, the median is $25^\circ\text{C}$, and the mean is $25^\circ\text{C}$.
$8.$ The four fictional countries of Fratania, Spanila, Brimland and Gretilli celebrate a dry weather festival during the months of January to May. They each try to encourage tourists to visit their own countries. Here are graphs of each country's rainfall last year for the five months of the festival.
Which average (mode, median, or mean) would be best for each country to advertise the lowest possible average rainfall for the season?
👀 Show answer
Using the rainfall values shown on the graphs:
Fratania: the mean is the lowest average.
Spanila: the mode is the lowest average.
Brimland: the median is the lowest average.
Gretilli: the median is the lowest average.
🧠 Think like a Mathematician
Daphne the dog had four litters of puppies. The mean average number of puppies in a litter was $5$. Investigate how many puppies could be in each litter. Find different ways that make the mean average $5$.
What do you notice about the total number of puppies in each solution where the mean is $5$?
Check that it is true for another solution. Explain what you find out.
Show Answers
Possible solutions:
- $4,\ 5,\ 5,\ 6$
- $3,\ 4,\ 6,\ 7$
- $2,\ 5,\ 5,\ 8$
- $1,\ 4,\ 7,\ 8$
- $5,\ 5,\ 5,\ 5$
What do you notice?
In every case, the total number of puppies is $20$, because there are $4$ litters and the mean is $5$, so $4 \times 5 = 20$.
Check with another solution:
For example, $2 + 6 + 4 + 8 = 20$, and $20 \div 4 = 5$, so the mean is still $5$.
Conclusion:
If the mean number of puppies is $5$ for $4$ litters, then the total number of puppies must always be $20$.