Find the first and third quartile and determine the interquartile range - High school Statistics & Probabilities Exercises

Find the interquartile range of the following sets of data.

Value	Frequency
1	3
2	4
3	3
4	7
5	2
6	3
7	3
8	4

4.5

The number of values in this set is:

3 + 4 + 3 + 7 + 2 + 3 + 3 + 4 = 29

In the case of the set being ordered in ascending order, the middle point is the 15th element. Therefore:

Elements 1 through 14 comprise the lower half.
Elements 16 through 29 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
1	3
2	4
3	3
4	4

There are 14 elements in the lower half, therefore the lower quartile is the average of the 7th element (2) and 8th element (3):

Q_1 = \dfrac{2+3}{2}=2.5

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
8	4
7	3
6	3
5	2
4	2

There are 14 elements in the upper half, so the upper quartile is the average of the 7th element (6) and 8th element (7):

Q_3 = \dfrac{6+7}{2}=6.5

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 6.5 - 2.5 = 4

The interquartile range for this set of data is 4.

8, 9, 1, 1, 54, 7, 3, 2, 5, 9, 2, 4, 1, 5, 3

5.5

First, arrange the set in order:

Value	Frequency
1	3
2	2
3	2
4	1
5	2
7	1
8	1
9	2
54	1

The number of values in this set is:

3 + 2 + 2 + 1 + 2 + 1 + 1 + 2 + 1 = 15

In the case of the set being ordered in ascending order, the middle point is between the 7th and 8th elements. Therefore:

Elements 1 through 7 comprise the lower half.
Elements 8 through 15 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
1	3
2	2
3	2

There are 7 elements in the lower half, so the lower quartile is equal to the 4th element, (2):

Q_1 = 2

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
5	2
7	1
8	1
9	2
54	1

There are 7 elements in the upper half, so the upper quartile is equal to the 4th element (7):

Q_3 = 8

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 8 - 2 = 6

The interquartile range for this set of data is 6.

{5, 2, 6, 8, 9, 1, 4, 4, 2, 6, 6, 7, 1, 9}

4.5

5.25

First, arrange the set in order:

Value	Frequency
1	2
2	2
4	2
5	1
6	3
7	1
8	1
9	2

The number of values in this set is:

2 + 2 + 2 + 1 + 3 + 1 + 1 + 2 = 14

In the case of the set being ordered in ascending order, the middle point is between the 7th and 8th elements. Therefore:

Elements 1 through 7 comprise the lower half.
Elements 8 through 14 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
1	2
2	2
4	2
5	1

There are 7 elements in the lower half, so the lower quartile is equal to the 4th element (2):

Q_1 = 2

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
6	3
7	1
8	1
9	2

There are 7 elements in the upper half, so the upper quartile is equal to the 4th element (7):

Q_3 = 7

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 7 - 2 = 5

The interquartile range for this set of data is 5.

6, 4, 1, 1, 7, 3, 9, 2, 4, 5, 6, 2, 1

5.5

4.5

3.5

First, arrange the set in order:

Value	Frequency
1	3
2	2
3	1
4	2
5	1
6	2
7	1
9	1

The number of values in this set is:

3 + 2 + 1 + 2 + 1 + 2 + 1 + 1 = 13.

In the case of the set being ordered in ascending order, the middle point is the 7th element. Therefore:

Elements 1 through 6 comprise the lower half.
Elements 8 through 13 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
1	3
2	2
3	1

There are 6 elements in the lower half, so the lower quartile is equal to the average of the 3rd and 4th elements (1 and 2, respectively):

Q_1 = \dfrac{1 + 2}{2} = 1.5

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
4	1
5	1
6	2
7	1
9	1

There are 6 elements in the upper half, so the upper quartile is equal to the average of the 3rd and 4th elements, which are both 6:

Q_3 = 6

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 6 - 1.5 = 4.5

The interquartile range for this set of data is 4.5.

Value	Frequency
1	5
2	2
3	6
4	2
5	7
7	3
11	3
15	2

7.5

The number of values in this set is:

5 + 2 + 6 + 2 + 7 + 3 + 3 + 2 = 30

In the case of the set being ordered in ascending order, the middle point is the 15th element. Therefore:

Elements 1 through 15 comprise the lower half.
Elements 16 through 30 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
1	5
2	2
3	6
4	2

There are 15 elements in the lower half, so the lower quartile is the 8th element (3):

Q_1 = 3

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
5	7
7	3
11	3
15	2

There are 15 elements in the upper half, so the upper quartile is the 8th element (7) :

Q_3 =7

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 7-3=4

The interquartile range for this set of data is 4.

Value	Frequency
2	13
3	2
4	6
6	7
7	2
8	8
10	4
12	8
16	3
18	2
20	4

The number of values in this set is:

13 + 2 + 6 + 7 + 2 + 8 + 4 + 8 + 3 + 2 + 4 = 59

In the case of the set being ordered in ascending order, the middle point is the 15th element. Therefore:

Elements 1 through 29 comprise the lower half.
Elements 31 through 59 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
2	13
3	2
4	6
6	7
7	1

There are 29 elements in the lower half, so the lower quartile is the average of the 15th element (3) :

Q_1 =3

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
8	8
10	4
12	8
16	3
18	2
20	4

There are 29 elements in the upper half, so the upper quartile is the average of the 15th element (12)

Q_3 =12

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 12-3=9

The interquartile range for this set of data is 9.

Value	Frequency
5	1
6	4
8	5
11	1
14	1
15	1
16	2
17	1

8.5

12.5

The number of values in this set is:

1 + 4 + 5 + 1 + 1 + 1 + 2 + 1 = 16.

In the case of the set being ordered in ascending order, the middle point is between the 8th and 9th elements. Therefore:

Elements 1 through 8 comprise the lower half.
Elements 9 through 16 comprise the upper half.

Step 1

Compute Q_1

The set of elements comprising the lower half is presented in the following frequency table:

Value	Frequency
5	1
6	4
8	3

There are 8 elements in the lower half, so the lower quartile is equal to the average of the 4th and 5th elements, which are both 6:

Q_1 = \dfrac{6+6}{2}=6

Step 2

Compute Q_3

The set of elements comprising the upper half is presented in the following frequency table:

Value	Frequency
8	2
11	1
14	1
15	1
16	2
17	1

There are 8 elements in the upper half, so the upper quartile is equal to the average of the 4th and 5th elements (14 and 15, respectively):

Q_3 = \dfrac{14 + 15}{2} = 14.5

Step 3

Compute interquartile range

Knowing Q_1 and Q_3, we can calculate the interquartile range:

IQR = Q_3 - Q_1 = 14.5 - 6 = 8.5

The interquartile range for this set of data is 8.5.