Q1. The following table shows the ages of the patients admitted to a hospital during a year. Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency.

Solution:

Modal class = 35–45

Frequency of the modal class (f1) = 23

Frequency of the class preceding the modal class (f0) = 21

Frequency of the class succeeding the modal class (f2) = 14

Class size (h) = 10

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 35 + \frac{23-21}{46-21-14} \cdot 10 = 36.8
\]

Mode = 36.8 years

Mean formula and calculation:

\[
\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{2830}{80} = 35.375
\]

Mean = 35.375 years

Q2. The following data gives the observed lifetimes (in hours) of 225 electrical components. Determine the modal lifetimes of the components.

Solution:

Modal class = 60–80

Frequency of the modal class (f1) = 61

Frequency of the class preceding the modal class (f0) = 52

Frequency of the class succeeding the modal class (f2) = 38

Class size (h) = 20

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 60 + \frac{61-52}{122-52-38} \cdot 20 = 65.625
\]

Modal lifetime = 65.625 hours

Q3. The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure and the mean monthly expenditure.

Solution:

Modal class = 1500–2000

Frequency of the modal class (f1) = 40

Frequency of the class preceding the modal class (f0) = 24

Frequency of the class succeeding the modal class (f2) = 33

Class size (h) = 500

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 1500 + \frac{40-24}{80-24-33} \cdot 500 = 1847.83
\]

Mode = Rs. 1847.83

Mean formula and calculation:

\[
\bar{x} = a + \frac{\sum f_i u_i}{\sum f_i} \cdot h = 2750 + \frac{-35}{200} \cdot 500 = 2662.5
\]

Mean = Rs. 2662.50

Q4. The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data.

Solution:

Modal class = 30–35

Frequency of the modal class (f1) = 10

Frequency of the class preceding the modal class (f0) = 9

Frequency of the class succeeding the modal class (f2) = 3

Class size (h) = 5

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 30 + \frac{10-9}{20-9-3} \cdot 5 = 30.625
\]

Mode = 30.625

Mean formula and calculation:

\[
\bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{1022.5}{35} \approx 29.2
\]

Mean = 29.2

Q5. The given distribution shows the number of runs scored by some top batsmen of the world in one-day international cricket matches. Find the mode of the data.

Solution:

Modal class = 4000–5000

Frequency of the modal class (f1) = 18

Frequency of the class preceding the modal class (f0) = 4

Frequency of the class succeeding the modal class (f2) = 9

Class size (h) = 1000

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 4000 + \frac{18-4}{36-4-9} \cdot 1000 \approx 4608.7
\]

Mode = 4608.7 runs

Q6. A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarized it in the table given below. Find the mode of the data.

Solution:

Modal class = 40–50

Frequency of the modal class (f1) = 20

Frequency of the class preceding the modal class (f0) = 12

Frequency of the class succeeding the modal class (f2) = 11

Class size (h) = 10

Mode formula:

\[
\text{Mode} = l + \frac{f_1 – f_0}{2f_1 – f_0 – f_2} \cdot h
\]

\[
\text{Mode} = 40 + \frac{20-12}{40-12-11} \cdot 10 \approx 44.7
\]

Mode = 44.7 cars