Find the mode from the following data?

Age $\textcolor{w h i t e}{X X X X X X X}$Cum. Frequency $0 - 10 \textcolor{w h i t e}{X X X X X X X X} 16$ $10 - 20 \textcolor{w h i t e}{X X X X X X \times} 33$ $20 - 30 \textcolor{w h i t e}{X X X X X X \times} 53$ $30 - 40 \textcolor{w h i t e}{X X X X X X \times} 78$ $40 - 50 \textcolor{w h i t e}{X X X X X X \times} 96$ $50 - 60 \textcolor{w h i t e}{X X X X X X X} 110$ $60 - 70 \textcolor{w h i t e}{X X X X X X X} 125$

Nov 5, 2016

Mode is $34.17$

Explanation:

As there are $16$ members with age less than $10$ and $33$ members with age less than $20$, we have $33 - 16 = 17$ members between age of $10$ and $20$. This way we can derive actual frequency from cumulative frequency, which is as given below:

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

$0 - 10 \textcolor{w h i t e}{X X X X X X X X} 16 \textcolor{w h i t e}{X X X X X X X X X X} 16$
$10 - 20 \textcolor{w h i t e}{X X X X X X \times} 33 \textcolor{w h i t e}{X X X X X X X X X X} 17$
$20 - 30 \textcolor{w h i t e}{X X X X X X \times} 53 \textcolor{w h i t e}{X X X X X X X X X X} 20$
$30 - 40 \textcolor{w h i t e}{X X X X X X \times} 78 \textcolor{w h i t e}{X X X X X X X X X X} 25$
$40 - 50 \textcolor{w h i t e}{X X X X X X \times} 96 \textcolor{w h i t e}{X X X X X X X X X X} 18$
$50 - 60 \textcolor{w h i t e}{X X X X X X X} 110 \textcolor{w h i t e}{X X X X X X X X X X} 14$
$60 - 70 \textcolor{w h i t e}{X X X X X X X} 125 \textcolor{w h i t e}{X X X X X X X X X X} 15$

As the highest frequency is ${f}_{m} = 25$, whose modal class is $30 - 40$ and class interval is $i$; the frequency just before is ${f}_{m - 1} = 20$ and frequency just after is ${f}_{m + 1} = 18$.

Formula for Mode is $M o \mathrm{de} = {L}_{1} + \frac{{f}_{m} - {f}_{m - 1}}{2 {f}_{m} - {f}_{m - 1} - {f}_{m - 2}} \times i$

$. : M o \mathrm{de} = 30 + \frac{25 - 20}{2 \times 25 - 20 - 18} \times 10$

= $30 + \frac{5}{12} \times 10 = 30 + 4.166 \cong 34.17$