The number of telephone calls received at an exchange in 245 successive one-minute intervals are shown in the following frequency distribution :

$\begin{matrix} Number of calls & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 Frequency & 14 & 21 & 25 & 43 & 51 & 40 & 39 & 12 \end{matrix}$

Compute the mean deviation about median.

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Solution

We will first calculate the median

$\begin{matrix} x_{i} & f_{i} & Cumulative frequency & | d_{i} | = | x_{i} - 61 | & f_{i} | d_{i} | 0 & 14 & 14 & 4 & 56 1 & 21 & 35 & 3 & 63 2 & 25 & 60 & 2 & 50 3 & 43 & 103 & 1 & 43 4 & 51 & 154 & 0 & 0 5 & 40 & 194 & 1 & 40 6 & 39 & 233 & 2 & 78 7 & 12 & 245 & 3 & 36 N = \sum f_{i} = 245 & \sum_{i = 1}^{n} f_{i} | d_{i} | = 366 \end{matrix}$

Here,

$\frac{N}{2} = \frac{245}{2} = 122.5$

The cumulative frequency just greater than 122.5 is 154 and the corresponding value of x is 4.

$∴$ Median, M = 4

$M . D . = \frac{1}{N} \sum_{i = 1}^{n} f_{i} | d_{i} | = \frac{1}{245} \times 366 = 1.493$

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The number of telephone calls received at an exchange in 245 successive one-minute intervals are shown in the following frequency distribution : Number of calls01234567Frequency1421254351403912 Compute the mean deviation about median.