Question

The following information is available on the talk time (in min.) noted for$70{\rm{ calls}}$ of a certain mobile phone user. Find the mean talk time.

Talk time (min)	Less than $4$	Less than $8$	Less than $12$	Less than $16$	Less than $20$
No. of calls	20	42	57	65	70

Answer 1

Hint: Here, we determine the frequency from the cumulative frequency then find the mean of frequency then the range at which the frequency lies is mean of talk time. Use the relation between frequency and no of observations to determine the mean talk time.


Complete Step-by-step Solution

Talk time (min)	No of calls
Less than $4$	20
Less than $8$	42
Less than $12$	57
Less than 16	65
Less than 20	70

To calculate the mean, we may need frequency so first we need to find frequency from cumulative frequency. The formula to find the frequency from the cumulative frequency is $c{f_{i + 1}} - c{f_i}$ where, $cf$ is the cumulative frequency.

The frequency distribution table is as follows:

Talk time	Mid point(M)	Frequency(f)
0-4	2	20
4-8	6	42-20=22
8-12	10	57-42=15
12-16	14	65-57=8
16-20	18	70-65=5
Total	$$	$\sum {{f_i}} = 70$

The midpoint is taken because the talk time is in intervals so it cannot be multiplied with the frequency.
We know the formula to find mean is given as,
${\rm{mean}} = \dfrac{{\sum\limits_{i = 1}^5 {{x_i}{f_i}} }}{{\sum {{f_i}} }}$
Now, we will substitute the values in the above relation.
$\begin{array}{c}
 = \dfrac{{2 \times 20 + 6 \times 22 + 10 \times 15 + 14 \times 8 + 18 \times 5}}{5}\\
 = \dfrac{{524}}{{70}}\\
 = 7.5
\end{array}$

Hence, the mean talk time is ${\rm{7}}{\rm{.5 min}}$.


Note: The cumulative frequency is not frequency. The mean is the sum of frequency by number of times not cumulative frequency by number of times. In such types of problems cumulative frequency can be used to find out the quantity of observations which lies in between the values given in the data. The mean talk time in this question is calculated by dividing the total number of frequencies with the number of observations.