Question

Find the mean, median and mode of the given data:

Class	65 – 85	85 – 105	105 – 125	125 – 145	145 – 165	165 – 185	185 –205
Frequency	8	7	22	17	13	5	3

Answer 1

Class ${\displaystyle \left(f_i\right)}$	Frequency ${\displaystyle \left(x_i\right)}$	Class mark	${\displaystyle d_i=x_i-A}$ ${\displaystyle h=20}$	${\displaystyle u_i=\frac{d_i}{h}}$	${\displaystyle f_i{u}_i}$
65 – 85	8	75	– 60	– 3	– 24
85 – 105	7	95	– 40	– 2	– 14
105 –125	22	115	– 20	– 1	– 22
125 – 145	17	135 = A	0	0	0
145 –165	13	155	20	1	13
165 – 185	5	175	40	2	10
185 –205	3	195	60	3	9
	${\displaystyle \sum f_i=75}$				${\displaystyle \sum f_i{u}_i=-28}$

We know, Mean ${\displaystyle \overline{\text{X}}=\text{A}+\left(\frac{\sum f_i{u}_i}{\sum f_i}\right)\times h}$

= ${\displaystyle 135+\left(\frac{-28}{75}\right)\times 20}$

= ${\displaystyle 135+\left(\frac{-28}{15}\right)\times 4}$

= 135 – 7.47

= 127.53

Thus, the mean of the data is 127.53.

Now, the maximum frequency is 22 belonging to class interval 105 – 125.

∴ Modal class = 105 – 125

So, L = 105, f₁ = 22, f₀ = 7, f₂ = 17, h = 20.

Mode = ${\displaystyle \text{L}+\left(\frac{f_1-f_0}{2f_1-f_0-f_2}\right)\times h}$

= ${\displaystyle 105+\left(\frac{22-7}{2\left(22\right)-7-17}\right)\times 20}$

= ${\displaystyle 105+\left(\frac{15}{44-24}\right)\times 20}$

= ${\displaystyle 105+\left(\frac{15}{20}\right)\times 20}$

= 105 + 15

= 120

Thus, the mode is 120.

Class	Frequency ${\displaystyle \left(f_i\right)}$	Cumulative frequency
65 – 85	8	8
85 – 105	7	8 + 7 = 15
105 –125	22	15 + 22 = 37
125 – 145	17	37 + 17 = 54
145 –165	13	54 + 13 = 67
165 – 185	5	67 + 5 = 72
185 –205	3	72 + 3 = 75

Here, N = 75

∴ ${\displaystyle {\left(\frac{N}{2}\right)}^{\text{th}}\text{term}=\frac{75}{3}={\left(37.5\right)}^{\text{th}}\text{term}}$

Since, 37.5^th term lies in the class interval 125 – 145.

∴ Median class =125 – 145

${\displaystyle \text{So},L=125,f=17,\frac{N}{2}=37.5,c.f.=37,h=20.}$

Median = ${\displaystyle L+\left(\frac{\frac{\text{N}}{2}-c.f.}{f}\right)\times h}$

= ${\displaystyle 125+\left(\frac{37.5-37}{17}\right)\times 20}$

= ${\displaystyle 125+\left(\frac{0.5}{17}\right)\times 20}$

= ${\displaystyle 125+\left(\frac{5}{17}\right)\times 2}$

= 125 + 0.588

= 125.588

As a result, the median is 125.588.

As a result, the data's mean, median, and mode are 127.53, 125.588, and 120, respectively.