What is the proof for formula for mode-

L + (fm - f(m-1))\(2fm - f(m-1) - f(m+1))) * c

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The mode M of a frequency distribution with class interval c ,the modal frequency fm and frequencices fm-1 and fm+1 of the preceding and succeeding the modal class interval is given by:

M = L + {(fm -fm-1)/2fm- fm-1 - fm+1)}*c, where L is the lower limit of the class interval.

The proof is in the assumption that the mode corresponds to the variate value M for which the class interval c is divided in the ratio of the weights of the frequencies (fm+1 - fm):(fm - fm-1).

Therefore M divides the class interval (L , L+c) in the ratio (fm+1-fm):(fm-m-1).

Therefore the mode M is somewhere after by (fm- fm-1) (L+c - L)/{(fm-fm-1))+(fm+1-fm).

Therefore M = L + (fm- fm-1) (L+c - L)/{(fm-fm-1))+(fm+1-fm)

M = L +(fm-fm-1)*c/{2fm = fm-1 - fm+1}.

Hope this helps.

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