Question 13
The following is the frequency distribution of duration for 100 calls made on a mobile phone :

$\begin{matrix} D u r a t i o n (i n s) & N u m b e r o f c a l l s 95 - 125 & 14 125 - 155 & 22 155 - 185 & 28 185 - 215 & 21 215 - 245 & 15 \end{matrix}$
Calculate the average duration (in sec) of a call and also find the median from a cumulative frequency curve.

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Solution

First, we calculate class marks as follows.

$\begin{matrix} D u r a t i o n (i n s) & N u m b e r o f c a l l s (f_{i}) & C l a s s m a r k s & u_{i} = \frac{x_{i} - a}{h} & f_{i} u_{i} 95 - 125 & 14 & 110 & - 2 & - 28 125 - 155 & 22 & 140 & - 1 & - 22 155 - 185 & 28 & a = 170 & 0 & 0 185 - 215 & 21 & 200 & 1 & 21 215 - 245 & 15 & 230 & 2 & 30 \sum f_{i} = 100 & \sum f_{i} u_{i} = 1 \end{matrix}$

Here, (assumed mean) a=170,
and (class width) h=30
By step deviation method,
Average $(¯ x) = a + \frac{\sum f_{i} u_{i}}{\sum f_{i}} \times h = 170 + \frac{1}{100} \times 30 = 170 + 0.3 = 170.3$
Hence, average duration is 170.3s.
For calculating median from a cumulative frequency curve
We prepare less than type or more than type ogive
We observe that, number of calls in less than 95 s is 0. Similarly, in less than 125 s include the number of calls in less than 95 s as well as the number of calls from 95-125s. So, the total number of calls less than 125 s is 0+14=14. Continuing in this manner, we will get remaining in less than 155, 185, 215 and 245 s.
Now, we construct a table for less than ogive(cumulative frequency curve).
$L e s s t h a n t y p e$
$\begin{matrix} D u r a t i o n (i n s) & N u m b e r o f c l a s s L e s s t h a n 95 & 0 L e s s t h a n 125 & 0 + 14 = 14 L e s s t h a n 155 & 14 + 22 = 36 L e s s t h a n 185 & 36 + 28 = 64 L e s s t h a n 215 & 64 + 21 = 85 L e s s t h a n 245 & 85 + 15 = 100 \end{matrix}$

To draw less than type ogive we plot them the points (95,0),(125,14),(155,36),(185,64),(215,85),(245,100) on the paper and join them by free hand.

$∵$ Total number of calls(n) =100
$∴ \frac{n}{2} = \frac{100}{2} = 50.$
Now, point 50 taking on Y-axis draw a line parallel to X-axis meet at a point P and draw a perpendicular line from P to the X-axis, the intersection point of X-axis is the median.
Hence, the required median is 170.

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