Question 178741
I think people shy away from percentiles because the math world cant get their act together on one definition:...there are really 3 ways of looking at percentiles .:
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There is no universally accepted definition of a percentile. Using the 65th percentile as an example, the 65th percentile can be defined as the lowest score that is greater than 65% of the scores. Lets call this "Definition 1". The 65th percentile can also be defined as the smallest score that is greater than or equal to 65% of the scores. This we will call "Definition 2". Unfortunately, these two definitions can lead to dramatically different results, especially when there is relatively little data. Moreover, neither of these definitions is explicit about how to handle rounding. For instance, what score is required to be higher than 65% of the scores when the total number of scores is 50? This is tricky because 65% of 50 is 32.5. How do we find the lowest number that is less than 32.5% of the scores? A third way to compute percentiles is a weighted average of the percentiles computed according to the first two definitions. This third definition handles rounding more gracefully than the other two and has the advantage that it allows the median to be defined conveniently as the 50th percentile.
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I will use the 3rd method but give answers to the other 2
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first line this up in ascending order:
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20,22,22,19,22,50,24,15,34,43,22,20,17,38,18,21,21,23,18,23
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15,17,18,18,19,20,20,21,21,22,22,22,22,23,23,24,34,38,43,50
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The first step is to compute the rank (R) of the 20th and 75th percentiles. This is done using the following formula: R=P/100(N+1) where P is the desired percentile and N is the number of terms.
In our cases P=20 and 75 and N=20
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so in our case for the 20 percentile: R=20/100(20+1)=4.2
.......................75 percentile: R=75/100(20+1)=15.75

If R were an integer, the P the percentile would be the number with rank R. When R is not an integer, we compute the Pth percentile by interpolation as follows:

Define IR as the integer portion of R (the number to the left of the decimal point). For our problems IR = 4 and 15
Define FR as the fractional portion of R. In our problems FR= .2 and .75
Find the scores with Rank IR and with Rank IR+1. For our problems this would be
the 4th and 5th terms for 20th percentile and 15 and 16th for the 75th percentile.  For P=20 this would be 18 and 19. for P=75 it would be 23 and 24
Interpolate by multiplying the difference between the scores by FR and add the result to the lower score. So for P=20 you would take .2(19-18)+18={{{highlight(18.2)}}}. For P=75 it would be .75(24-23)+23={{{highlight(23.75)}}}
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15,17,18,18,19,20,20,21,21,22,22,22,22,23,23,24,34,38,43,50
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Therefore, the 20th percentile is 18.2 and the 75th percentile is 23.75. If we had used the first definition (the smallest score greater than 20% and 75% of the scores) the 20th percentile would have been 19(5th term) and for the 75 percentile would have been 24(16th term). If we had used the second definition ( the smallest score greater than or equal to 25% of the scores) the 20th percentile would have been 18(4th term) and the 75 percentile would have been 23(15th term).
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As you can see these definitions need to be standardized so everyone is on the same page.
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hope that helps