Different Formulas for Calculating Mode

Date: 09/11/2008 at 11:00:41
From: Saptarshi
Subject: about mode formula

i am a M.B.A student. our teacher tells a formula to find out mode.
that is Z=L1+(F1-F0/2F1-F0-F2)*i

where: L1 = lower limit of modal class
       F1 = modal class frequency.
       F2 = just after the modal class frequency.
       F0 = just previous the modal class frequency.
        i = class interval.
        Z = the mode value.

but i saw in most of cases the highest frequency is the mode. they
don't use that formula. [i saw that when searching about mode in
google]. so why we need that formula? can you please explain me.

Date: 09/11/2008 at 11:42:16
From: Doctor Peterson
Subject: Re: about mode formula

Hi, Saptarshi.

This formula gives a linear interpolation to estimate the actual 
value of the mode from grouped data; otherwise, all you really know 
is the modal class (which is sufficient for many purposes).

Your formula can be written differently if we take

  d1 = F1 - F0  (difference between modal class and previous class)
  d2 = F1 - F2  (difference between modal class and next class)

Then d1 + d2 = (F1 - F0) + (F1 - F2) = 2F1 - F0 - F1, so the formula 
is

  Z = L1 + d1/(d1 + d2) * i

I was asked about this formula a year ago, with specific reference to 
the case where the modal class is the first class.  I had not seen the
formula previously, but could see how it arose:

Here are two pages I found explaining the formula, which you may find 
helpful if they say more than your text says:

  A Statistical Manual for Forestry Research
    http://www.fao.org/docrep/003/X6831E/X6831E04.htm 

  Hong Kong Institute of Vocational Education (Tsing Yi)(PDF file)
 
http://ictlab.tyict.vtc.edu.hk/~kenli/ESS_Bank/1_2_IndexNo/SummaryStats.pdf 

The formula these give, with definitions of the variables, is (using
the second site's version):

  When data are already grouped in a frequency distribution, we can
  assume that the mode is located in the class with the most items.
  In order to determine a single value for the mode from this modal
  class, we use

    mode = LBMo + [d1 /(d1+d2)] (Width)

  where

    LBMo = lower boundary of the modal class
    Width = width of the modal class interval
    d1 = frequency of the modal class minus the frequency of the
         class directly below it
    d2 = frequency of the modal class minus the frequency of the
         class directly above it

Note that d1 and d2 relate to the classes on the left and on the right
in the histogram.  If there is no class on the left, then you can
imagine a class with frequency zero.  Then the formula applies easily.

The purpose of this formula is to identify one value within the modal
class that seems likely to be the peak of the curve if you smoothed
out the histogram.  It does this by taking the value within the
interval whose distance from the class on either side is proportional
to how much less the frequency is on either side.   You can see this
by rewriting the formula:

  mode - LBMo     d1
  ----------- = -------
     Width      d1 + d2

There is a simple geometrical way you could find this point.  Just
draw lines from the top corners of the modal bar to the near corners
of the neighboring bars, and the mode estimate lies at the intersection:

            +---------+
            |  \    / |d2
          d1|     X   |
            |   / :   +---------+
            | /   :   |         |
  +---------+     :   |         |
  |         |     :   |         |
  |         |     :   |         |
  |         |     :   |         |
  |         |     :   |         |
  +---------+-----:---+---------+
           LBmo  mode
            |<------->|
               width

For an example, take these classes:

  85<91          10
  91<97           8
  97<103          3
  103<109         8
  109<115         0
  115<121         7

The modal class is 85<91.

  LBmo = 85
  width = 6
  d1 = 10 - 0 = 10 (since the frequency on the left is 0)
  d2 = 10 - 8 = 2  (since the frequency on the right is 8)

  mode = LBMo + [d1 /(d1+d2)] (Width)
       = 85 + (10/12)(6)
       = 85 + 5
       = 90

This is 5 from the left and 1 from the right, a ratio of 5:1, while
the differences in frequency are 10:2.

Does this help?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 09/12/2008 at 00:55:15
From: Saptarshi
Subject: Thank you (about mode formula)

thanks..i am very much grateful to you..now i really understand what
is the need of that formula...i really very thankful to you.