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/
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