Exercise 13.3

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the median, mean and mode of the data and compare them.

$$\begin{array}{|c|c|}

\hline
\text{Monthly consumption (in units)} & \text{Number of consumers} \\
\hline
65\text{ – }85   & 4  \\
85\text{ – }105  & 5  \\
105\text{ – }125 & 13 \\
125\text{ – }145 & 20 \\
145\text{ – }165 & 14 \\
165\text{ – }185 & 8  \\
185\text{ – }205 & 4  \\
\hline
\end{array}$$

If the median of the distribution given below is $28.5$, find the values of $x$ and $y$.

$$\begin{array}{|c|c|}

\hline
\text{Class interval} & \text{Frequency} \\
\hline
0 - 10 & 5 \\
\hline
10 - 20 & x \\
\hline
20 - 30 & 20 \\
\hline
30 - 40 & 15 \\
\hline
40 - 50 & y \\
\hline
50 - 60 & 5 \\
\hline
\text{Total} & 60 \\
\hline
\end{array}$$

A life insurance agent found the following data for distribution of ages of 100 policy holders.
Calculate the median age, if policies are given only to persons having age $18$ years onwards but less than $60$ years.

$$\begin{array}{|c|c|}

\hline
\text{Age (in years)} & \text{Number of policy holders (cumulative)}\\
\hline
\text{Below }20 & 2\\
\hline
\text{Below }25 & 6\\
\hline
\text{Below }30 & 24\\
\hline
\text{Below }35 & 45\\
\hline
\text{Below }40 & 78\\
\hline
\text{Below }45 & 89\\
\hline
\text{Below }50 & 92\\
\hline
\text{Below }55 & 98\\
\hline
\text{Below }60 & 100\\
\hline
\end{array}$$

The lengths of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table:

$$\begin{array}{|c|c|}

\hline
\text{Length (in mm)} & \text{Number of leaves} \\
\hline
118 - 126 & 3 \\
\hline
127 - 135 & 5 \\
\hline
136 - 144 & 9 \\
\hline
145 - 153 & 12 \\
\hline
154 - 162 & 5 \\
\hline
163 - 171 & 4 \\
\hline
172 - 180 & 2 \\
\hline
\end{array}$$

Find the median length of the leaves.

(Hint: The data needs to be converted to continuous classes for finding the median, since the formula assumes continuous classes. The classes then change to $117.5 - 126.5$, $126.5 - 135.5$, $\dots$, $171.5 - 180.5$.)

The following table gives the distribution of the life time of 400 neon lamps:

$$\begin{array}{|c|c|}

\hline
\text{Life time (in hours)} & \text{Number of lamps} \\
\hline
1500 - 2000 & 14 \\
\hline
2000 - 2500 & 56 \\
\hline
2500 - 3000 & 60 \\
\hline
3000 - 3500 & 86 \\
\hline
3500 - 4000 & 74 \\
\hline
4000 - 4500 & 62 \\
\hline
4500 - 5000 & 48 \\
\hline
\end{array}$$

Find the median life time of a lamp.

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabets in the surnames was obtained as follows:

$$\begin{array}{|c|c|c|c|c|c|c|}

\hline
\text{Number of letters} & 1 - 4 & 4 - 7 & 7 - 10 & 10 - 13 & 13 - 16 & 16 - 19 \\
\hline
\text{Number of surnames} & 6 & 30 & 40 & 16 & 4 & 4 \\
\hline
\end{array}$$

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames? Also, find the modal size of the surnames.

The distribution below gives the weights of 30 students of a class. Find the median weight of the students.

$$\begin{array}{|c|c|c|c|c|c|c|c|}

\hline
\text{Weight (in kg)} & 40 - 45 & 45 - 50 & 50 - 55 & 55 - 60 & 60 - 65 & 65 - 70 & 70 - 75 \\
\hline
\text{Number of students} & 2 & 3 & 8 & 6 & 6 & 3 & 2 \\
\hline
\end{array}$$