LemonmathSelf-study sheet

Grade 12 · Lesson 3.1

Range and interquartile range of grouped data

Name: Date: Time spent: min

Goals

Find the range $R$ of grouped data.
Find the quartiles $Q_1$ , $Q_3$ and the interquartile range $\Delta_Q$ of grouped data.
Compare the spread of two sets of grouped data and explain what it means.
Understand why these measures are only approximations of the measures of the raw data.

Remember

Grouped data $[u_1, u_2)$ , …, $[u_k, u_{k+1})$ , frequencies $n_1$ , …, $n_k$ , sample size $n$ .
$R=u_{k+1}-u_1$ (right end of the last class minus left end of the first class).
$Q_1=u_m+\dfrac{\frac n4-C}{n_m}(u_{m+1}-u_m)$ , where class $m$ is the first class with cumulative frequency $\ge\frac n4$ and $C$ is the cumulative frequency of the classes before it; $Q_3$ : replace $\frac n4$ by $\frac{3n}{4}$ .
$\Delta_Q=Q_3-Q_1$ : the width of the middle half of the data; smaller means more even.
These are approximate values; $\Delta_Q$ is less affected by the two end classes than $R$ .

Exercises

Exercise 1

The masses (g) of $40$ apples are given in the table. Find $R$ , $Q_1$ , $Q_3$ and $\Delta_Q$ .

Mass (g)	$[150, 160)$	$[160, 170)$	$[170, 180)$	$[180, 190)$	$[190, 200)$
Apples	$4$	$10$	$14$	$8$	$4$

Exercise 2

The daily reading times (minutes) of $50$ students are given in the table. Find $Q_1$ , $Q_3$ and $\Delta_Q$ (to two decimal places).

Time (minutes)	$[0, 20)$	$[20, 40)$	$[40, 60)$	$[60, 80)$	$[80, 100)$
Students	$5$	$9$	$16$	$12$	$8$

Exercise 3

A shop tests the battery life (hours) of $50$ phones of each of two models X and Y.

Battery life (hours)	$[8, 10)$	$[10, 12)$	$[12, 14)$	$[14, 16)$	$[16, 18)$
Model X	$3$	$10$	$24$	$10$	$3$
Model Y	$8$	$12$	$10$	$12$	$8$

Find the interquartile range of each model. If a customer wants a steady battery life, which model should they choose?

Exercise 4

The weekly hours of exercise of $30$ students are given in the table.

Hours	$[0, 2)$	$[2, 4)$	$[4, 6)$	$[6, 8)$	$[8, 10)$
Students	$3$	$7$	$10$	$6$	$4$

Is each claim true or false?

a) The range of the data is $10$ .

b) The first quartile lies in the class $[2, 4)$ .

c) $Q_3\approx6.83$ .

d) The interquartile range is greater than $4$ .

Exercise 5

The masses (kg) of $40$ students are given in the table. Find the interquartile range (in kg, to two decimal places).

Mass (kg)	$[40, 45)$	$[45, 50)$	$[50, 55)$	$[55, 60)$	$[60, 65)$
Students	$4$	$9$	$15$	$8$	$4$

Exercise 6

(Revisiting the median of grouped data, grade 11.) For the table of apple masses in Exercise 1, find the median $M_e$ and check that $Q_1<M_e<Q_3$ .

Exercise 7

(Revisiting Measures of dispersion.) Consider the raw data $3, 7, 8, 11, 12, 14, 15, 16, 18, 21, 24, 29$ .

a) Find $R$ and $\Delta_Q$ of the raw data.

b) Group them into the three classes $[0, 10)$ , $[10, 20)$ , $[20, 30)$ and find $R$ and $\Delta_Q$ of the grouped data.

Exercise 8

For class 12A (Example 1) we have $Q_1=15$ and $Q_3=32.5$ minutes. A new student who joins the class takes $62$ minutes to get to school. By the $1.5\,\Delta_Q$ rule, is this time an outlier?

Exercise 9 Challenge

Grouped data have the following frequency table, where $x$ is an unknown positive integer.

Class	$[0, 10)$	$[10, 20)$	$[20, 30)$	$[30, 40)$
Frequency	$4$	$x$	$10$	$6$

We know that $Q_1=13.75$ and that $Q_1$ lies in the class $[10, 20)$ . Find $x$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1.

The times (minutes) that $40$ passengers waited for a bus are given in the table. The range of these grouped data is

Time (minutes)	$[0, 5)$	$[5, 10)$	$[10, 15)$	$[15, 20)$	$[20, 25)$
Passengers	$6$	$14$	$12$	$4$	$4$

5

minutes.B.

20

minutes.C.

25

minutes.D.

10

minutes.

Question 2. For the data in Question 1, the class containing the third quartile

Q_3

[5, 10)

.B.

[10, 15)

.C.

[15, 20)

.D.

[20, 25)

Question 3. For the data in Question 1, the best approximation to the first quartile

Q_1

8.33

.B.

5.29

.C.

1.43

.D.

6.43

Question 4. By the interquartile range rule, a value

x

of a data set is regarded as an outlier if

x<Q_1-1.5\,\Delta_Q

x>Q_3+1.5\,\Delta_Q

.B.

x<Q_1

x>Q_3

.C.

x<Q_1-1.5\,R

x>Q_3+1.5\,R

.D.

Q_1-1.5\,\Delta_Q<x<Q_3+1.5\,\Delta_Q

Part II. True or false

Question 1.

The $100$ m running times (seconds) of $32$ students are given in the table.

Time (seconds)	$[12, 14)$	$[14, 16)$	$[16, 18)$	$[18, 20)$	$[20, 22)$
Students	$4$	$8$	$10$	$8$	$2$

a) The range of the data is $10$ seconds.	T / F
b) $Q_1=15$ .	T / F
c) The third quartile $Q_3$ lies in the class $[16, 18)$ .	T / F
d) By the $1.5\,\Delta_Q$ rule, a time of $24$ seconds is an outlier.	T / F

Part III. Short answer

Question 1.

The monthly electricity use (kWh) of $80$ households is given in the table. Find the interquartile range of the data, giving your answer to 1 decimal place.

Electricity use (kWh)	$[50, 100)$	$[100, 150)$	$[150, 200)$	$[200, 250)$	$[250, 300)$
Households	$8$	$20$	$28$	$16$	$8$

Question 2. For the data in the previous question, by the

1.5\,\Delta_Q

rule, above how many kWh a month is a household's electricity use regarded as an outlier? Give your answer to the nearest whole number.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $R=u_{k+1}-u_1$ ; the formulas for $Q_1$ , $Q_3$ of grouped data.

Steps:

Write the cumulative frequencies.
Find the classes containing position $10$ and position $30$ .
Substitute into the formula.

Exercise 2.

What you need: the position of $Q_1$ is $\frac n4=12.5$ and that of $Q_3$ is $\frac{3n}{4}=37.5$ ; the class width is $20$ .

Steps:

Cumulative frequencies: $5$ , $14$ , $30$ , $42$ , $50$ .
Choose the class for each position and substitute into the formula.

Exercise 3.

What you need: a smaller $\Delta_Q$ means the middle of the data is more concentrated.

Steps:

Positions $12.5$ and $37.5$ ; write the cumulative frequencies for each model.
Find $Q_1$ , $Q_3$ and $\Delta_Q$ for each model and compare.

Exercise 4.

What you need: $R=u_{k+1}-u_1$ ; the class containing $Q_1$ is the first class with cumulative frequency $\ge\frac n4$ .

Steps:

Cumulative frequencies: $3$ , $10$ , $20$ , $26$ , $30$ ; $\frac n4=7.5$ , $\frac{3n}{4}=22.5$ .
Find $Q_1$ , $Q_3$ and $\Delta_Q$ .

Exercise 5.

What you need: the formulas for $Q_1$ , $Q_3$ ; the class width is $5$ .

Steps:

Cumulative frequencies: $4$ , $13$ , $28$ , $36$ , $40$ .
Position $10$ lies in $[45, 50)$ , position $30$ in $[55, 60)$ .

Exercise 6.

What you need: $M_e$ is found like $Q_1$ but at position $\frac n2$ .

Steps:

$\frac n2=20$ ; cumulative frequencies $4$ , $14$ , $28$ , …
The class containing $M_e$ is $[170, 180)$ , with $C=14$ , $n_m=14$ .

Exercise 7.

What you need: with $n=12$ even, each half has $6$ values (grade 10); the formulas for grouped data.

Steps:

a) $Q_1$ is the median of the first $6$ values, $Q_3$ the median of the last $6$ .
b) Count the frequency of each class; positions $3$ and $9$ .

Exercise 8.

What you need: a value is an outlier if it is greater than $Q_3+1.5\,\Delta_Q$ or less than $Q_1-1.5\,\Delta_Q$ .

Steps:

Compute $\Delta_Q$ and $1.5\,\Delta_Q$ .
Compute the upper limit and compare it with $62$ .

Exercise 9.

What you need: $n=20+x$ ; $Q_1=u_m+\dfrac{\frac n4-C}{n_m}\cdot h$ .

Steps:

Write $Q_1$ in terms of $x$ with $u_m=10$ , $C=4$ , $n_m=x$ , $h=10$ .
Solve the equation, then check the condition for choosing the class.

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