SOLUTION: The following are distances (in miles) traveled to the workplace by 18 employees of a certain hospital. 10,14,6,32,9,2,28,27,13,14,37,40,24,15,31,7,22,36 Find 25th and 90th

Question 178140: The following are distances (in miles) traveled to the workplace by 18 employees of a certain hospital.
10,14,6,32,9,2,28,27,13,14,37,40,24,15,31,7,22,36
Find 25th and 90th percentiles for these distinces.

Answer by Mathtut(3670) (Show Source): You can put this solution on YOUR website!
10,14,6,32,9,2,28,27,13,14,37,40,24,15,31,7,22,36
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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 as I understand:
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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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10,14,6,32,9,2,28,27,13,14,37,40,24,15,31,7,22,36
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2,6,7,9,10,13,14,14,15,22,24,27,28,31,32,36,37,40
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The first step is to compute the rank (R) of the 25th and 90th 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=25 and 90 and N=18
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so in our case for the 25 percentile: R=25/100(18+1)=4.75
.......................90 percentile: R=90/100(18+1)=17.1

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 17
Define FR as the fractional portion of R. In our problems FR= .75 and .1
Find the scores with Rank IR and with Rank IR+1 For our problems this would be
the 4th and 5th terms for 25th percentile and 17 and 18th for the 90th percentile. For P=25 this would be 9 and 10. for P=90 it would be 37 and 40
Interpolate by multiplying the difference between the scores by FR and add the result to the lower score. So for P=25 you would take .75(10-9)+9=. For P=90 it would be .1(40-37)+37=
Therefore, the 25th percentile is 9.75 and the 90th percentile is 37.3. If we had used the first definition (the smallest score greater than 25% and 90% of the scores) the 25th percentile would have been 10 and for the 90 percentile would have been 37. If we had used the second definition ( the smallest score greater than or equal to 25% of the scores) the 25th percentile would have been 10 and the 90 percentile would have been 37)
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if the number of terms were divisible by 4 then the first and 2nd definitions would yield different answers.....for instant the second term would fall on the 25% point so for definition 1 the 25 percentile would be the 3rd term but for definition 2 it would be the 2nd term. As you can see these definitions need to be standardized so everyone is on the same page.
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hope that helps

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