The Mode

The mode, the value that occurs most often: from a list, a frequency table, a dot diagram, a pictogram, a histogram and a stem-and-leaf diagram; one mode, two modes or none; the mode of categories.

What You Should Already Know

Lesson 12.3: the value with the highest frequency is the mode, the value under the tallest bar of a histogram.
Lesson 12.4: for grouped data, the class with the highest frequency is the modal class.
The mean (lesson 12.7) and the median (lesson 12.8) are the other two averages.

The Most Common Value

The mode

The mode of a set of data is the value that occurs most often: the value with the highest frequency.

The mode is a value of the data, not its frequency. If the score 5 occurs 8 times, the mode is 5, not 8.

If every value occurs the same number of times, there is no mode (5, 7, 9; or 3, 3, 5, 5). Otherwise, if two values share the highest frequency, both are modes (the data are bimodal); if more than two values share it, there is no mode.

Example 1: The mode of a list

Ten pupils read these numbers of books during the school holidays: 8, 9, 10, 10, 3, 5, 6, 10, 6, 1. Find the mode.

Solution

Count each value: 1, 3, 5, 8 and 9 occur once; 6 occurs twice; 10 occurs 3 times.

The highest frequency is 3, and the value with that frequency is 10.

The mode is 10 books.

Example 2: Two modes, and no mode

(a) The lengths, in cm, of 10 ribbons are 100, 110, 95, 60, 20, 60, 110, 88, 102, 120. Find the mode. (b) The ages of six cousins are 5, 7, 9, 11, 13 and 15 years. Find the mode. (c) A goalkeeper let in 3, 3, 5, 5 goals in four matches. Find the mode.

Solution

(a) 60 and 110 each occur twice; every other length occurs once.

Two values share the highest frequency, so both are modes. Do not pick one of them.

(b) Every age occurs once: no value is more common than another.

(c) 3 occurs twice and 5 occurs twice. Two values are equally common, but so is every value: each occurs the same number of times, so no value is more common than another.

(a) The modes are 60 cm and 110 cm. (b) There is no mode. (c) There is no mode.

The Mode from a Table or a Diagram

In a frequency table, look for the highest frequency and give the value beside it. In a dot diagram, look for the tallest column of dots; in a histogram, the tallest bar; in a pictogram, the longest row.

Example 3: The mode from a frequency table, in words

The table shows the commission earned in one month by the 11 salespeople of a shop. (a) Find the mode. (b) Is it true that most of the salespeople earned this commission?

Commission ($)	Number of salespeople
$1000$	$2$
$1200$	$5$
$1500$	$3$
$2000$	$1$
Total	$11$

Solution

(a) The highest frequency is 5, beside $1200.

(b) "Most" means more than half. Only 5 of the 11 salespeople earned $1200; the other 6 earned something else.

The mode is the most common value, not the value that most of the data have.

(a) The mode is $1200. (b) No: $1200 is the most common commission, but only 5 of the 11 salespeople earned it.

Example 4: The mode from a pictogram

The pictogram shows the favourite chocolate flavours of a group of pupils. Find the mode.

Solution

Count the squares: Sea Salt 1, Caramel 4, Peppermint 3, Hazelnut 4. Each square stands for 20 pupils, so Caramel and Hazelnut have $4 \times 20 = 80$ pupils each.

Caramel and Hazelnut share the longest row, so both are modes.

The data are flavours, not numbers, and the mode is a flavour, not 80.

The modes are Caramel and Hazelnut.

Example 5: The mode from a stem-and-leaf diagram

The stem-and-leaf diagram shows the ages of the 18 members of a line-dance class at a community centre. Find the mode.

Solution

A value is repeated when the same leaf appears more than once on the same stem. Look along each row:

Stem 4: 5, 5 (45 twice). Stem 5: 2, 2, 2 (52 three times). Stem 6: 3, 3 and 7, 7 (63 and 67 twice each).

The highest frequency is 3, for 52.

Do not count a leaf across different stems: the leaf 7 appears six times, but 37, 47, 57, 67, 67 and 77 are different ages.

The mode is 52 years.

Example 6: Two tallest bars

The histogram shows how many pieces of fruit each of 28 pupils ate yesterday. Find the mode.

Solution

The bars over 2 and 5 are equally the tallest, at 7.

Both values are modes, even though they are not next to each other.

The modes are 2 pieces and 5 pieces of fruit.

The Mode of Categories

Categories such as colours, flavours or CCAs are names, not numbers. They cannot be added, so they have no mean, and they have no order from smallest to largest, so they have no middle one and no median. The mode is the only average that categories with no order have. Some categories do have an order, such as the letter sizes S, M, L, XL of T-shirts: they have a median as well as a mode, but still no mean. (Shoe sizes 5, 6, 7 are different: they are numbers that go up in equal steps, so they have a mean too.)

Example 7: Favourite colours

Ten pupils named their favourite colour: Red, Blue, Red, Green, Blue, Red, Yellow, Red, Blue, Red. (a) Find the mode. (b) Can the mean or the median favourite colour be found?

Solution

(a) Count the colours:

Colour	Frequency
Red	$5$
Blue	$3$
Green	$1$
Yellow	$1$
Total	$10$

Red has the highest frequency, 5.

(b) No. Colours cannot be added, so there is no mean, and they have no order, so there is no middle one. Giving the colours numbers (Red 1, Blue 2, …) does not help: the numbers are only labels, and other labels would give another "mean".

(a) The mode is Red. (b) No: only the mode can be found for colours.

Changing the Mode

Example 8: Adding values

Priya slept 6, 7, 7, 8, 9, 9, 9 and 10 hours on eight nights. (a) Find the mode. (b) After one more night, the data have two modes. How long did she sleep that night? (c) After two more nights instead, 7 hours is the only mode. How long did she sleep on those two nights?

Solution

(a) 9 occurs 3 times, 7 twice, the rest once: the mode is 9 hours.

(b) For two modes, another value must also occur 3 times. Only 7 (now twice) can get there with one more night. With 7 added:

(a) 9 hours (b) 7 hours (c) 7 hours on each night

The Modal Class and the Mode

For grouped data, the modal class is the class with the highest frequency (lesson 12.4). It is the most common class, not where most of the data lie. A grouped table does not show the single values, so it cannot give the mode, and the mode of the original data need not even lie in the modal class.

Example 9: The mode outside the modal class

Seven buses were 1, 1, 1, 5, 6, 7 and 8 minutes late. (a) Find the mode. (b) Group the times into the classes $0 \le t < 5$ and $5 \le t < 10$ , and find the modal class. (c) Does the mode lie in the modal class?

Solution

(a) 1 occurs 3 times, every other time once: the mode is 1 minute.

(b) 1, 1, 1 are in $0 \le t < 5$ ; 5, 6, 7, 8 are in $5 \le t < 10$ .

Delay, $t$ min	Tally	Frequency
$0 \le t < 5$		$3$
$5 \le t < 10$		$4$
Total		$7$

The modal class is $5 \le t < 10$ , with 4 buses.

(a) 1 minute (b) $5 \le t < 10$ (c) No: the mode need not lie in the modal class.

Common Mistakes

Giving the highest frequency as the mode. The mode is the value with that frequency.
Giving one mode when two values share the highest frequency (unless every value occurs equally often, when there is no mode).
Saying "the mode is 0" or "the mode is every value" when every value occurs equally often (even 3, 3, 5, 5): there is no mode.
Saying "most of the data have the mode". The mode is only the most common value: in Example 3, 5 of 11.
Counting the same leaf on different stems of a stem-and-leaf diagram as one value.
Giving categories numbers and then finding a mean of the numbers (or, for categories with no order, a median).
Calling a mode a "modal class", or saying the mode must lie in the modal class.

Practice Questions

Try each question first, then open the solution to check your working.

Question 1

State the mode or modes of each set of data, or write "no mode". (a) Test marks: 65, 70, 70, 75, 80, 85, 70, 90. (b) Ages: 12, 13, 14, 15, 16, 17. (c) Shoe sizes: 6, 7, 7, 8, 8, 9, 10. (d) Goals scored: 2, 2, 3, 3, 4, 4, 6.

ShowHide solution

(a) 70 occurs 3 times, every other mark once.

(b) Every age occurs once.

(d) 2, 3 and 4 each occur twice and 6 once: three values share the highest frequency, so there is no mode.

(a) 70 (b) no mode (c) 7 and 8 (d) no mode

Question 2

Kai says the mode of the scores in this table is 7. Explain his mistake and give the mode.

Score	$1$	$2$	$3$	Total
Number of pupils	$3$	$7$	$2$	$12$

ShowHide solution

7 is the highest frequency: 7 pupils had the same score.

The mode is the score those 7 pupils had, which is 2.

He gave the frequency, not the value. The mode is 2.

Question 3

The dot diagram shows how many times each of 20 pupils ate at a hawker centre last week. Find the mode.

ShowHide solution

The tallest column has 6 dots, over 2.

The mode is 2 times.

Question 4

The stem-and-leaf diagram shows the marks of 15 pupils in a quiz marked out of 50. (a) Find the mode. (b) One more pupil, who scored 23, takes the quiz. What are the modes now?

ShowHide solution

(a) On stem 2 the leaf 3 appears twice (23, 23). On stem 3 the leaf 3 appears three times (33, 33, 33). The leaf 3 on stem 4 is 43, another value.

The mode is 33.

(b) Now 23 also occurs 3 times, as often as 33.

(a) 33 marks (b) 23 marks and 33 marks

Question 5

The histogram shows the number of people sitting at each table of a hawker centre at noon. Find the mode.

ShowHide solution

The bars over 2 and 4 are equally the tallest, at 9.

There are two modes: 2 people and 4 people.

Question 6

The table shows the monthly salaries of the 150 employees of a company. (a) State the mode. (b) A report says, "Most employees earn $3000 a month." Explain why this is wrong, and write a correct sentence.

Monthly salary ($)	Number of employees
$2000$	$10$
$3000$	$56$
$3500$	$42$
$4500$	$25$
$5000$	$17$
Total	$150$

ShowHide solution

(a) The highest frequency is 56, beside $3000.

(b) 56 of 150 employees is less than half ( $\frac{56}{150} \times 100\% = 37.33\ldots\% \approx 37.3\%\text{),}$ so it is not "most". The other 94 earn other salaries.

(a) $3000 (b) Only 56 of 150 employees earn $3000. Correct: "$3000 is the most common monthly salary."

Question 7

A shop sold T-shirts in these sizes in one week: S, M, M, M, L, L, XL, XL, XL, XL, XXL. (a) Find the mode. (b) A pupil numbers the sizes S = 1, M = 2, L = 3, XL = 4, XXL = 5 and finds a "mean size" of 3.09. Why does this mean nothing? (c) Which average helps the shop decide what to order?

ShowHide solution

(a) XL was sold 4 times, more than any other size.

(b) The numbers are only labels for sizes. An XL is not "4" of anything, and the gap from L to XL is not the same as from S to M. Other labels would give another "mean". Letter sizes are categories, so the mean cannot be found. (Letter sizes do have an order, S < M < L < XL < XXL, so a median size can be found: the 6th of the 11, L. But there is no mean.)

(a) XL (b) The numbers are labels, not amounts. (c) The mode.

Question 8

The bar graph shows the drinks sold at a school canteen stall one morning. (a) Find the mode. (b) Jia Hui says the mode is 36. What has she done wrong?

ShowHide solution

(a) The tallest bar is Milo, with 36 cups.

(b) 36 is the frequency, the number of cups of Milo. The mode is the drink itself.

(a) Milo (b) She gave the frequency; the mode is Milo.

Question 9

The six numbers 2, 4, 4, 7, 9 and $x$ have two modes. Find every possible value of $x$ .

ShowHide solution

Without $x$ , 4 occurs twice and every other number once.

For two modes, $x$ must make another number occur twice: $x = 2$ , $7$ or $9$ .

If $x = 4$ , 4 occurs 3 times and is the only mode. If $x$ is any other number, 4 is still the only mode.

$x = 2$ , $x = 7$ or $x = 9$

Question 10

The table shows the masses of 30 pupils. (a) State the modal class. (b) Can you find the mode of the masses? Explain.

Mass, $m$ kg	Frequency
$40 \le m < 45$	$6$
$45 \le m < 50$	$11$
$50 \le m < 55$	$8$
$55 \le m < 60$	$5$
Total	$30$

ShowHide solution

(a) The highest frequency is 11, in $45 \le m < 50$ .

(b) No. The table shows only how many masses are in each class, not the masses themselves, so we cannot tell which mass occurs most often. It need not even be in $45 \le m < 50$ .

(a) $45 \le m < 50$ (b) No: grouped data do not show the single values.

Key Points

The mode is the value with the highest frequency: a value of the data, not the frequency.
Data can have one mode, two modes, or no mode. If every value occurs the same number of times, there is no mode; otherwise two values sharing the highest frequency are both modes, and more than two sharing it give no mode.
Read the mode from the highest frequency in a table, the tallest column of a dot diagram, the tallest bar of a histogram or the longest row of a pictogram.
In a stem-and-leaf diagram, the mode is the leaf repeated most often on one stem, read with that stem.
The mode is the most common value, not the value that most of the data have.
The mode is the only average for categories with no order (colours, flavours, CCAs); numbers given to categories are labels, not amounts.
The modal class of grouped data is not the mode, and the mode need not lie in it.