Question 7
The table below shows the salaries of 280 persons.
$\begin{matrix} Salary(in Rs.thousand) & N u m b e r o f p e r s o n s 5 - 10 & 49 10 - 15 & 133 15 - 20 & 63 20 - 25 & 15 25 - 30 & 6 30 - 35 & 7 35 - 40 & 4 40 - 45 & 2 45 - 50 & 1 \end{matrix}$
Calculate the median and mode of the data.

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Solution

First, we construct a cumulative frequency table
$\begin{matrix} Salary (in Rs.thousand) & N u m b e r o f p e r s o n s (f_{i}) & Cumulative frequency (c_{f}) 5 - 10 & 49 = f_{1} & 49 = (c_{f}) 10 - 15 & f_{m} = 133 = f & 133 + 49 = 182 15 - 20 & 63 = f_{2} & 182 + 63 = 245 20 - 25 & 15 & 245 + 15 = 260 25 - 30 & 6 & 260 + 6 = 266 30 - 35 & 7 & 266 + 7 = 273 35 - 40 & 4 & 273 + 4 = 277 40 - 45 & 2 & 277 + 2 = 279 45 - 50 & 1 & 279 + 1 = 280 N = 280 \end{matrix}$
$∴ \frac{N}{2} = \frac{280}{2} = 140$
(i) Here, median class is 10-15, because 140 lies in it.
Lower limit(l)=10, Frequency(f)=133,
Cumulative frequency $c_{f}$ =49 and class width(h)=5
$∴ M e d i a n = l + \frac{(\frac{N}{2} - c_{f})}{f} \times h = 10 + \frac{(140 - 49)}{133} \times 5 = 10 + \frac{91 \times 5}{133} = 10 + \frac{455}{133} = 10 + 3.421 = R s . 13.421 (i n t h o u s a n d) = 13.421 \times 1000 = R s .13421$
(ii) Here, the highest frequency is 133, which lies in the interval 10-15, called modal class.
Lower limit(l)=10, class width(h)=5, $f_{m} = 133, f_{1} = 49$ and $f_{2} = 63.$
$∴ M o d e = l + (\frac{f_{m} - f_{1}}{2 f_{m} - f_{1} - f_{2}} \times h) = 10 + {\frac{133 - 49}{(2 \times 133) - 49 - 63}} \times 5 = 10 + \frac{84 \times 5}{266 - 112} = 10 + \frac{84 \times 5}{154} = 10 + 2.727 = R s .12 .727 (i n t h o u s a n d) = 12.727 \times 1000 = R s .12727$
Hence, the median and modal salary are Rs. 13421 and Rs. 12727, respectively.

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The table below shows the salaries of 280 persons.

\begin{matrix} Salary(in Rs.thousand) & N u m b e r o f p e r s o n s 5 - 10 & 49 10 - 15 & 133 15 - 20 & 63 20 - 25 & 15 25 - 30 & 6 30 - 35 & 7 35 - 40 & 4 40 - 45 & 2 45 - 50 & 1 \end{matrix}

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\begin{matrix} Salary(In thousand Rs.) & No. of Persons 5 - 10 & 49 10 - 15 & 133 15 - 20 & 63 20 - 25 & 15 25 - 30 & 6 30 - 35 & 7 35 - 40 & 4 40 - 45 & 2 45 - 50 & 1 \end{matrix}

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$5 - 10$	$49$
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$20 - 25$	$15$
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Calculate the median salary of the following data giving salaries of $280$ persons:

$\begin{matrix} Salary (in thousands) & 5 - 10 & 10 - 15 & 15 - 20 & 20 - 25 & 25 - 30 & 30 - 35 & 35 - 40 & 40 - 45 & 45 - 50 Number of persons & 49 & 133 & 63 & 15 & 6 & 7 & 4 & 2 & 1 \end{matrix}$

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Question 7 The table below shows the salaries of 280 persons. Salary(in Rs.thousand) Number of persons 5−10 49 10−15 133 15−20 63 20−25 15 25−30 6 30−35 7 35−40 4 40−45 2 45−50 1 Calculate the median and mode of the data.