TITLE: Derivation of Mode of grouped data
QUESTION [12 upvotes]: A formula to calculate the mode for grouped data's is given in my text book:
Mode = $l + \dfrac{(f_1 - f_0)h}{2f_1 - f_0 - f_2} $
Where, $l = $ lower limit of the modal class,
$h = $ size of the class interval,
$f_1 = $  frequency of the modal class,
$f_0 = $ frequency of the class preceding the modal class,
$f_2 =$ frequency of the class succeeding the modal class.
Can you please explain the derivation of this formula, as it is not given in my T.B. Thanks .

REPLY [5 votes]: we partition the continuous frequency distrbution into intervals. The maximum value is within the modal class. It is assumed that the rate of change of the frequency on both sides of the mode(max. frequency) are equal.

$$
\text{slope, }m_{AB}=-m_{BC}\\
\tan(90-b)=-\tan(90+b)\implies \tan a=\tan b\\
\frac{x}{f_1-f_0}=\frac{h-x}{f_1-f_2}\implies x(f_1-f_2)=h(f_1-f_0)-x(f_1-f_0)\\
x(2f_1-f_0-f_2)=h(f_1-f_0)\implies x=\frac{f_1-f_0}{2f_1-f_0-f_2}.h\\
\text{Mode}=l+x=l+\frac{f_1-f_0}{2f_1-f_0-f_2}.h
$$