Different Formulas for Calculating ModeDate: 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. |
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