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Q.

The following distribution gives the state-wise teacher student ratio in higher secondary schools of India. What is the mode and mean of this data?

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a

Mode = 31.7, Mean = 28.2

b

Mode = 33.6, Mean = 25.3

c

Mode = 35.7, Mean = 26.3

d

Mode = 30.6, Mean = 29.2

answer is D.

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Detailed Solution

It is given the state-wise teacher student ratio in higher secondary schools of India.
Question Image

Question Image

We convert the data in terms of the following expression.

x_{i} = \frac{lower limit + upper limit}{2}

Number of students	$x_{i}$	Number of state $(f_{i})$	${(x}_{i}) \times (f_{i}) = (x_{i} f_{i})$
15 - 20	17.5	3	52.5
20 - 25	22.5	8	180
25 - 30	27.5	9	247.5
30 - 35	32.5	10	325
35 - 40	37.5	3	112.5
40 - 45	42.5	0	0
45 - 50	47.5	0	0
50 - 55	52.5	2	105
		$Σ f_{i} = 35$	$Σ f_{i} x_{i} = 1022.5$

From the table, we get

Σ f_{i} = 35

Σ f_{i} x_{i} = 1022.5

We know that,

Mean = \frac{Σ x_{i} f_{i}}{Σ f_{i}}

\Rightarrow Mean = \frac{1022.5}{35}

\Rightarrow Mean = 29.2

We also know that,
Mode

= l + \frac{f - f_{1}}{2 f - f_{1} - f_{2}} \times h

Where:

l \to

lower limit of the modal class

f \to

frequency of the modal class

f_{1} \to

frequency of the class preceding the modal class

f_{2} \to

frequency of the class succeeding the modal class

h \to

class width
From the given table, we get that the maximum frequency is

10

and the corresponding class is

30 - 35

. Therefore,

30 - 35

is the modal class.
So, we have

l = 30, h = 5, f = 10, f_{1} = 9

and

f_{2} = 3

Putting the values in the formula, we get

Mode = 30 + \frac{10 - 9}{20 - 9 - 3} \times 5

\Rightarrow Mode = 30 + \frac{1}{8} \times 5

\Rightarrow Mode = 30 + \frac{5}{8}

\Rightarrow Mode = 30 + 0.625

\Rightarrow Mode = 32.625

Therefore, the mode of the given frequency distribution is 32.625 and the mean is 29.2.
Hence, the correct option is 4.

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