Question

Find the mean and the mode of the following frequency distribution:

Class	Frequency
0 − 15	9
15 − 30	15
30 − 45	35
45 − 60	20
60 − 75	11
75 − 90	13
90 − 105	17

Answer 1

Class	Frequency (fi)	Class Mark (xi)	fixi
0 − 15	9	7.5	67.5
15 − 30	15	22.5	337.5
30 − 45	35	37.5	1312.5
45 − 60	20	52.5	1050.0
60 − 75	11	67.5	742.5
75 − 90	13	82.5	1072.5
90 − 105	17	97.5	1657.5
Total	`∑ f_i = 120`		`∑ f_ix_i = 6240`

Mean formula: `barx = (∑ f_ix_i )/(∑ f_i )`

`barx = 6240/120`

= 52

Calculation of Mode:

The class with the highest frequency (35) is 30 - 45. This is our Modal Class.

• Lower limit l = 30
• Frequency of modal class f₁ = 35
• Frequency of preceding class f₀ = 15
• Frequency of succeeding class f₂ = 20
• Class size (\(h\)) = 15

Mode = `l + [(fm − f1) / (2fm − f1 − f2)] · h`   ...[Use grouped‑data mode formula]

Substitute values: (fm − f1)

= 35 − 15 = 20, (2fm − f1 − f2)

= 70 − 15 − 20

= 35, fraction

= `20/35`

= `4/7`

Mode = 30 + `(4/7)`·15

= 30 + `60/7`

= 30 + 8.571...

= 38.571... ≈ 38.57.