# Find the median from following data?

## There are $12$ data-points between $10$ and $19$; $19$ data-points between $20$ and $29$; $31$ data-points between $30$ and $39$; $27$ data-points between $40$ and $49$; $16$ data-points between $50$ and $59$ and $8$ data-points between $60$ and $69$.

Jan 16, 2017

Median is $37.73$

#### Explanation:

As there are $12$ data-points between $10$ and $19$ and $19$ data-points between $20$ and $29$. As such we have $12 + 19 = 31$ data-points say up to $29$ and hence cumulative frequency up to $29$. And similarly $31 + 31 = 62$ data points up to $39$. This way we can derive cumulative frequency for other class intervals (C.I.) , as given below and final cumulative frequency is $113$, which is equal to sum of all the frequencies.

Note that classes are $10 - 19$, $20 - 29$,... means that while $19$ comes in first C.I., $20$ comes in second C.I. There is nothing common between the two as is usually when classes are defined as $10 - 20$, $20 - 30$ and so on. This is as data points take only discreet points. To rectify this let us term C.I.'s as $9.50 - 19.50$, $19.50 - 29.50$, $29.50 - 39.50$ and so on.

As the total data points are $113$, the middle one representing $\frac{113 + 1}{2} = {57}^{t h}$ item lies in C.I. $29.50 - 39.50$ (marked with red colored arrow) and lower bound for this C.I. i.e. $L$ is $29.50$. This is the median class. Other variable used in formula are ${f}_{1}$, the frequency of median class, which is $31$, $n = \sum f$, which is $113$ and $c {f}_{1}$, cumulative frequency of the class just before median class, which too is $31$.

C.I. $\textcolor{w h i t e}{X X X X X X X}$Frequency$\textcolor{w h i t e}{X X X}$Cum. Frequency

$09.50 - 19.50 \textcolor{w h i t e}{X X X X X X X x} 12 \textcolor{w h i t e}{X X X X X X X X X x} 12$
$19.50 - 29.50 \textcolor{w h i t e}{X X X X X X \times} 19 \textcolor{w h i t e}{X X X X X X X X X X} 31$
$29.50 - 39.50 \textcolor{w h i t e}{X X X X X X \times} 31 \textcolor{red}{\to} \textcolor{w h i t e}{X X X X X X X X} 62$
$39.50 - 49.50 \textcolor{w h i t e}{X X X X X X \times} 27 \textcolor{w h i t e}{X X X X X X X X X X} 89$
$49.50 - 59.50 \textcolor{w h i t e}{X X X X X X \times} 16 \textcolor{w h i t e}{X X X X X X X X X x} 105$
$59.50 - 69.50 \textcolor{w h i t e}{X X X X X X X X} 8 \textcolor{w h i t e}{X X X X X X X X \times} 113$

Formula for Median is $M e \mathrm{di} a n = L + \frac{\frac{n}{2} - c {f}_{1}}{{f}_{m}} \times i$, hence

$M e \mathrm{di} a n = 29.50 + \frac{\frac{113}{2} - 31}{31} \times 10$

$= 29.50 + \frac{113 - 62}{2 \times 31} \times 10 = 29.50 + \frac{510}{62}$

$= 29.50 + 8.23 = 37.73$