Derivation of Linear Interpolation Median Formula

Date: 09/29/2007 at 05:16:31
From: Daya
Subject: Median,  m  =  L + [ (N/2 – F) / f ]C    

Median,  m  =  L + [ (N/2 – F) / f ]C.  

How does this median formula come?  My teacher did not show and proof
how does this formula come.  Therefore, I just substitute and blindly
use the formula.  Can you help me?

This formula is used to find the median in a group data with class 
interval.  The median is the value of the data in the middle position
of the set when the data is arranged in numerical order.  The class 
where the middle position is located is called the median class and
this is also the class where the median is located.  This formula is
used to find the median in a group data which is located in the median
class.

Median,  m  =  L + [ (N/2 – F) / f ]C  

  L means lower boundary of the median class

  N means sum of frequencies

  F means cumulative frequency before the median class.  Meaning that
    the class before the median class what is the frequency

  f means frequency of the median class
 
  C means the size of the median class

I have tried to use an ogive graph to understand, but I still did not
get how did this formula come.

Date: 09/30/2007 at 23:19:53
From: Doctor Peterson
Subject: Re: Median,  m  =  L + [ (N/2 – F) / f ]C

Hi, Daya.

This is a linear interpolation (on the ogive graph, as you suggested),
which finds where the actual median WOULD be if you assume that the
data are uniformly distributed within the median class.

One way to derive the formula is just to note that N/2 is the number
of data values BELOW the median, so N/2 - F is the number of data
values in median class that are below the median.  Therefore, (N/2 -
F)/f is the fraction of values in the median class that are below the
median.  This times C is that fraction of the class width; adding L
gives the value at that position in the class.

In terms of the ogive (cumulative distribution), let's first just plot
the actual cumulative frequency before each class, something like

  N+                             *
   |                        *
   |                   *
   |
   |              * ---
   + . . . . . .     ^
   |                 |f
   |                 v
  F|         *      ---
   |    *
   *----+----+----+----+----+----+
             L
             |<-->|
                C

We don't know where the actual data points are, but if they are
uniformly distributed within each class, we could connect the points
above with straight lines.  Your formula gives the x coordinate
corresponding to y=N/2.  See if you can derive it this way.

If you have any further questions, feel free to write back.


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