Exercises for Question Type 1: Constructing histograms given the frequency distribution table

Question 1

Skill question

A histogram with unequal class widths is used to represent $50$ observations. For the class intervals $0 - 5$ , $5 - 20$ and $20 - 50$ , the bars are drawn with heights (frequency densities) $1.2$ , $0.8$ and $h$ respectively. Given that the total frequency is $50$ , find the value of $h$ .

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Question 2

Skill question

The following raw data show the daily number of pages read by $12$ students: $2, 7, 9, 5, 12, 8, 15, 3, 6, 11, 10, 4$

Group the data into class intervals of width $5$ starting at $0$ (i.e., $0 - 5$ , $5 - 10$ , $10 - 15$ ), then construct the corresponding frequency distribution table.

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Question 3

Skill question

A histogram representing the ages of $120$ participants has the following frequency distribution:

Age (years), $x$	$10 \le x < 20$	$20 \le x < 30$	$30 \le x < 40$	$40 \le x < 50$
Frequency	$24$	$36$	$30$	$30$

Estimate the proportion of participants aged under $30$ .

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Question 4

Skill question

A frequency table for a set of test scores is given below.

Class interval (score)	Frequency
$0 \le x < 20$	$2$
$20 \le x < 40$	$8$
$40 \le x < 60$	$15$
$60 \le x < 80$	$10$
$80 \le x \le 100$	$5$

Calculate the relative frequency for each class and complete the table.

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Question 5

Skill question

A histogram has bars for three class intervals. The vertical axis of the histogram measures the frequency density. The heights of the bars for the respective intervals are as follows:

$0 \leq x < 2$ : height $5$
$2 \leq x < 6$ : height $3$
$6 \leq x < 10$ : height $2$

Complete the following frequency distribution table by finding the class widths, frequency densities, and frequencies.

Class interval	Width	Frequency density	Frequency
$0 \le x < 2$
$2 \le x < 6$
$6 \le x < 10$

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Question 6

Skill question

The following frequency distribution has unequal class widths.

Class interval	Frequency
$0 \le x < 5$	$4$
$5 \le x < 15$	$9$
$15 \le x < 30$	$7$

Construct a histogram for this data by first computing the frequency density for each class.

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Question 7

Skill question

A histogram of exam scores shows that the bar for $70 - 80$ has a greater height than the bar for $80 - 100$ , yet the frequency for $70 - 80$ ( $8$ students) is lower than that for $80 - 100$ ( $12$ students). Explain how this is possible.

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Question 8

Skill question

The frequency distribution below gives heights (in cm) of a group of children:

Height $h$ (cm)	Frequency
$100 \le h < 110$	$5$
$110 \le h < 130$	$12$
$130 \le h < 160$	$8$

1.

Draw a histogram to represent the data.

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2.

From your histogram, state the modal class interval.

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Question 9

Skill question

The frequency distribution table below shows the number of students scoring marks in intervals of width $10$ :

Class interval (marks)	Frequency
$0-10$	$3$
$10-20$	$7$
$20-30$	$12$
$30-40$	$8$

Draw a histogram to represent this data.

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Question 10

Skill question

A histogram has three bars with the following class intervals and frequency densities:

Class intervals: $2 - 4$ , $4 - 9$ , $9 - 15$
Frequency densities: $3$ , $2$ , $1$

Find the frequency for each class and complete the following frequency distribution table.

Class interval	Frequency
$2-4$
$4-9$
$9-15$

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Question 11

Skill question

The daily commute times, $t$ minutes, of $40$ employees are summarised in the table below:

Class interval (minutes)	Frequency
$0 \le t < 10$	$4$
$10 \le t < 20$	$12$
$20 \le t < 30$	$15$
$30 \le t \le 60$	$9$

Construct an augmented table by adding two new columns to the data above: (i) cumulative frequency, (ii) class midpoint.

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IB Mathematics Applications & Interpretation Question Type 1: Constructing Histograms Given the Frequency Distribution Table Exercises