Question

For the following frequency distribution, find:

1. lower quartile 
2. upper quartile 
3. interquartile range

Weight (in kg)	40	41	42	43	44	45	46	47	48
No. of students	3	7	11	15	18	13	9	6	5

Answer 1

Step 1: Create a cumulative frequency table.

Summing the number of students (f):

n = 3 + 7 + 11 + 15 + 18 + 13 + 9 + 6 + 5

n = 87

Weight (x) in kg	No. of students (f)	Cumulative Frequency (cf)
40	3	3
41	7	10
42	11	21
43	15	36
44	18	54
45	13	67
46	9	76
47	6	82
48	5	87

i. Lower Quartile (Q₁)

The lower quartile is the value at the `(n + 1)/4`th position.

Position of Q₁ = `(87 + 1)/4`

= `88/4`

= 22nd position

Looking at the cumulative frequency table, the 22nd student falls into the weight category of 43 kg. Since the cf for 42 kg ends at 21 and the next 15 students are all 43 kg.

Lower Quartile (Q₁) = 43 kg

ii. Upper Quartile (Q₃)

The upper quartile is the value at the `(3(n + 1))/4`th position.

Position of Q₃ = 3 × 22

= 66th position

Looking at the cumulative frequency table, the 66th student falls into the weight category of 45 kg. Since the cf for 44 kg ends at 54 and the next 13 students, up to the 67th, are all 45 kg.

Upper Quartile (Q₃) = 45 kg

iii. Interquartile Range (IQR)

The interquartile range is the difference between the upper and lower quartiles.

IQR = Q₃ – Q₁

IQR = 45 – 43

Interquartile Range = 2 kg.