Derivation of the formula for mode

JUNAID AKHTAR
DIRECTOR
J.AKHTAR'S CLASSES

What is the derivation of the formula for mode?

Answer :-

Consider a bar graph with a bar for each of the classes of data. Then f1 is the height of the bar of the modal class, f0 is the height of the bar on the left of it, and f2 is the height of the bar on the right of it.
The quantity f1−f0 measures how far the modal class's bar "sticks up" above the bar on its left. The quantity f1−f2 measures how far the modal class's bar "sticks up" above the bar on its right.
Now, observe that
(f1−f0)/(2f1−f0−f2) + (f1−f2)/(2f1−f0−f2) =
(f1−f0)/{(f1−f0)+(f1−f2)} + (f1−f2)/{(f1−f0)+(f1−f2)}=1
So if we want to divide an interval of width h into two pieces, where the ratio of sizes of those two pieces is (f1−f0):(f1−f2), the first piece will have width (f1−f0)*h / (2f1−f0−f2).
This is what the formula for estimating the mode does. It splits the width of the modal bar into two pieces whose ratio of widths is (f1−f0):(f1−f2), and it says the mode is at the line separating those two pieces, that is, at a distance (f1−f0)*h / (2f1−f0−f2) from the left edge of that bar, l.
Hence,  
Mode = l + (f1−f0)*h / (2f1−f0−f2)
If f1−f0=f1−f2, that is, the modal bar is equally far above the bars on both its left and right, then this formula estimates the mode right in the middle of the modal class:
l +  (f1−f0)*h / (2f1−f0−f2) = l + 1*h/2.
But if height of the bar on the left is closer to the modal bar's height, then the estimated mode is to the left of the centerline of the modal class. In the extreme case where the bar on the left is exactly the height of the modal bar, and both are taller than the bar on the right, that is, when
f1−f0=0 but f1−f2>0, the formula estimates the mode at l exactly, that is, at the left edge of the modal bar. In the other extreme case, where the bar on the left is shorter but the bar on the right is the same height as the modal bar (f1−f0>0 but f1−f2=0), the formula estimates the mode at l+h, that is, at the right edge of the modal bar.