Lesson 5.2 · 20 min

The Modal Class and the Median Class

Finding the class with the highest frequency and the class that contains the median, and seeing how regrouping can change them.

In this lesson you will learn to…

Identify the modal class of a grouped frequency table
Find the position of the median using $\frac{n + 1}{2}$
Use a running total to find the class the median lies in
Explain how changing the groups can change the modal class

Why this matters

The mean is not always the most useful average. A shop ordering stock cares about which size range sells most often (the mode). A government setting a minimum wage may care about the middle earner (the median), because a few very high earners pull the mean up.

These are not the same thing. Fifteen long-jump distances (cm):

$152, 152, 155, 161, 163, 164, 165, 166, 168, 169, 170, 172, 175, 181, 182$

The mode is 152 cm (the only value that appears twice). But grouped into tens:

Length ( $l$ , cm)	Frequency
$150 \le l < 160$	3
$160 \le l < 170$	7
$170 \le l < 180$	3
$180 \le l < 190$	2

the modal class is $160 \le l < 170$ — and 152 is not in it.

Reading a histogram

In a histogram with equal class widths, the modal class is simply the tallest bar. To find the median class, add the heights of the bars from the left until you pass the middle position.

Remember. From grouped data you can only say which class the median is in — not its exact value.

Key rules and formulas

Modal class
The class interval with the highest frequency. It is the easiest "average" to find — just look for the biggest number in the frequency column.
Position of the median
$median position = \frac{n + 1}{2}$
With $n = 100$ values, the median is the $\frac{101}{2} = 50.5$ th value — halfway between the 50th and 51st.
Running totals find the median class
Add up the frequencies from the first class until the running total reaches the median position. The class where that happens contains the median.
Groups change the answer
The modal class depends on how the data is grouped. Regroup the same data with wider classes and a different class can become the modal one.

Try it

Find the median class

Frequency, 30 ≤ x < 354

Frequency, 35 ≤ x < 407

Frequency, 40 ≤ x < 4523

Frequency, 45 ≤ x < 5030

Frequency, 50 ≤ x < 5516

Frequency, 55 ≤ x < 6011

Frequency, 60 ≤ x < 655

Frequency, 65 ≤ x < 704

Total frequency, Σf: 100
Σfx (midpoint × frequency): 4850
Estimated mean = Σfx ÷ Σf: 48.5
Modal class: 45 ≤ x < 50
Median class (value number 50.5): 45 ≤ x < 50
Estimated range: 40

The masses of 100 school children. The median is the 50.5th value. The first three classes hold 34 children, and the next holds 30 more — so the running total passes 50.5 inside $45 \leq m < 50$ .

Worked examples

Worked example 1

Modal class and median class

Masses of 100 children:

30

–

35

: 4,

35

–

40

: 7,

40

–

45

: 23,

45

–

50

: 30,

50

–

55

: 16,

55

–

60

: 11,

60

–

65

: 5,

65

–

70

: 4 (kg). Find the modal class and the class containing the median.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

When regrouping changes the mode

Puzzle times:

15

–

19

: 5,

19

–

23

: 3,

23

–

27

: 15,

27

–

31

: 12,

31

–

35

: 13,

35

–

39

: 2 seconds. Farook took 24 s, which is in the modal class. He regroups into

15 \leq t < 27

and

27 \leq t < 39

. Is he still in the modal class?

Try it yourself first, then reveal the solution one step at a time.

Practice: modal class and median class

9 questions

Nine questions. For median questions, write the running totals down the side of the table.

Question 1 of 9Multiple choiceFoundation
Oranges: $120$ – $125$ g: 5, $125$ – $130$ g: 16, $130$ – $135$ g: 13, $135$ – $140$ g: 4, $140$ – $145$ g: 2. What is the modal class?
Question 2 of 9Complete the table
Complete the running totals for the heights of 30 plants, then use them.
Height (h, cm) Frequency Running total
10 ≤ h < 20 4 Row 1, Running total
20 ≤ h < 30 8 Row 2, Running total
30 ≤ h < 40 11 Row 3, Running total
40 ≤ h < 50 7 Row 4, Running total
Question 3 of 9Fill the gaps
Complete the working for the plant heights.
The median lies in position $\frac{30 + 1}{2} =$ Gap a. There are 12 values smaller than Gap b cm. The 13th to the 23rd values lie in the class $□ \leq h < □$ , so the median class starts at Gap c and ends at Gap d.
Question 4 of 9Step by step
Find the median class one step at a time.
Ages in a village of 500 people: $0$ – $10$ : 92, $10$ – $20$ : 88, $20$ – $30$ : 85, $30$ – $40$ : 68, $40$ – $50$ : 55, $50$ – $60$ : 52, $60$ – $70$ : 42, $70$ – $80$ : 18.
1. In which position is the median?
  Step 1 answer
Question 5 of 9Multiple choice
Angelique regroups the village ages into $0$ – $20$ : 180, $20$ – $40$ : 153, $40$ – $60$ : 107, $60$ – $80$ : 60. What is the new modal class?
Question 6 of 9True or false
Long jumps (cm): 152, 152, 155, 161, 163, 164, 165, 166, 168, 169, 170, 172, 175, 181, 182. Grouped in tens, the mode of the raw data (152) lies in the modal class.
Question 7 of 9Choose all that apply
Puzzle times (s): $15$ – $19$ : 5, $19$ – $23$ : 3, $23$ – $27$ : 15, $27$ – $31$ : 12, $31$ – $35$ : 13, $35$ – $39$ : 2. Select every true statement.
Select every correct answer — there may be more than one.
The median lies in $23 \leq t < 27$ The estimated range is 24 sThe median is the 25.5th valueThe modal class is $23 \leq t < 27$ The median lies in $27 \leq t < 31$
Question 8 of 9Type the answer
A histogram shows the ages of 40 people in a sports club: $0$ – $10$ : 4, $10$ – $20$ : 9, $20$ – $30$ : 16, $30$ – $40$ : 11. Estimate the mean age.
Your answeryears
Question 9 of 9Explore and matchChallenge
This histogram shows 36 exam marks. Increase the frequency of $60 \leq x < 80$ so that it becomes the only modal class, using the smallest possible number.
Move the controls until you match the question, then press Check.
Frequency, 0 ≤ x < 202
Frequency, 20 ≤ x < 406
Frequency, 40 ≤ x < 6012
Frequency, 60 ≤ x < 809
Frequency, 80 ≤ x < 1007
Total frequency, Σf
36
Σfx (midpoint × frequency)
2060
Estimated mean = Σfx ÷ Σf
57.22
Modal class
40 ≤ x < 60
Median class (value number 18.5)
40 ≤ x < 60
Estimated range
100

Height (h, cm)	Frequency	Running total
10 ≤ h < 20	4	Row 1, Running total
20 ≤ h < 30	8	Row 2, Running total
30 ≤ h < 40	11	Row 3, Running total
40 ≤ h < 50	7	Row 4, Running total

In this lesson you will learn to…

Why this matters

Mode of the raw data vs modal class

Reading a histogram

Key rules and formulas

Modal class

Position of the median

Running totals find the median class

Groups change the answer

Find the median class

Worked examples

Practice: modal class and median class