Question

How to find mode of the following frequency distribution?

Class `color(white)(XXXXXXX)`Frequency`color(white)(XXX)`
`0-10color(white)(XXXXXXXX)22`
`10-20color(white)(XXXXXXxx)24`
`20-30color(white)(XXXXXXxx)36`
`30-40color(white)(XXXXXXxx)14`
`40-50color(white)(XXXXXXxx)12`

Answer 1

Mode is `23.53`

Explanation:

Let us first work out the cumulative frequency, which is as given below:

Class `color(white)(XXXXXXX)`Frequency`color(white)(XXX)`Cum. Frequency

`0-10color(white)(XXXXXXXX)22color(white)(XXXXXXXXXX)22`
`10-20color(white)(XXXXXXxx)24color(white)(XXXXXXXXXX)46`
`20-30color(white)(XXXXXXxx)36color(white)(XXXXXXXXXX)82`
`30-40color(white)(XXXXXXxx)14color(white)(XXXXXXXXXX)96`
`40-50color(white)(XXXXXXxx)12color(white)(XXXXXXXXXX)108`

As the highest frequency is `f_m=36`, whose modal class is `20-30` and class interval is `10`; the frequency just before is `f_(m-1)=24` and frequency just after is `f_(m+1)=14`.

Formula for Mode is `Mode=L_1+(f_m-f_(m-1))/(2f_m-f_(m-1)-f_(m-2))xxi`

`.:Mode=20+(36-24)/(2xx36-24-14)xx10`

= `20+12/34xx10=20+3.53=23.53`