Statistics Percentiles

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Introduction : Percentiles

The Objectives:

• What are percentiles?


• Computation of percentiles by using formulas.

By using percentile,a set of data is partitioned or divided into 100 equal parts . The position from below is measured through percentiles. In order to determine the position of a person in a population or the position of his rank, we use percentiles. Test score is one of the most important examples ,where we consider the use of percentile. An example can be considered in this regard. To know whether the GRE test score at the 90 percentile will allow a student to get admitted to top schools or not.

Definitions of Percentile

There is no fixed definition of a percentile. Three definitions are there. We will discuss them as follows:

Definition 1:

Percentile indicates a measure to determine the percentage of the total frequency that is scored at or below the measure. The percentage of scores falling at or below a particular score is known as a percentile rank.

• Formula 1 :



In order to compute the percentile rank using definition 1, we will use the following formula:




Percentile rank=(b+0.5e)/n *100




Where n is the number of scores.




b is the number of scores below the given score (say x)




e is the number of scores which is equal to x.

Definition 2:

• Formula 2 :



Secondly, percentile is defined as the measure which determines the percentage of the total frequency that is scored below the measure. The percentage of scores falling below a particular score is known as a percentile rank.




In order to compute the percentile rank using definition 1, we will use the following formula:




Percentile rank=b/n *100




Where n is the number of scores.




b is the number of scores below the given score (say x)

Definition 3

However, the two definitions tend to produce different results. In case of little data , the results will differ a lot. Apart, from this none of these two definitions ,can handle rounding. The third definition is thus, taken into consideration where, a weighted average of the percentiles computed by using the above two definitions is taken into account. Here, in this definition rounding is handled with more care than the first two. In addition to that, median is defined as the 50 th percentile in definition 3.

• Examples :


1. In a class consisting of 150 students if Mary is placed as the 25th one, then suppose 125 students are ranked below. What will be Mary’s percentile rank? Using definition 1:



Using definition 1:




(125+0.5*1)/150=125.5/150=84th percentile.



2. In a class consisting of 150 students if Mary is placed as the 25th one, then suppose 125 students are ranked below. What will be Mary’s percentile rank?



Using definition 2,




125/150=83rd percentile.



3. Now, we will use the 3rd definition of percentiles.


• Examples :The test scores in mechanics were :50, 65, 70, 72, 72, 78, 80, 82, 84, 84, 85, 86, 88, 88, 90, 94, 96, 98, 98, 99. What will be the percentile rank for a score of 84 of this test.

The following formula is used for computation:

R = P/100 x (n + 1)

where, P is the percentile

Here P=84

and n is the number of scores.Here n=20

So,

R = 84/100 x (20 + 1) = 17.64

Two cases :

Case 1:

If R is an integer, the Pth percentile is the number having rank R.

Also R may not be an integer. In such a case ,compute the Pth percentile is computed by means of interpolation as follows:

1. First of all consider the integer portion of R , the integer part is the number before the decimal point.In the above example it is 17.


2. Secondly, consider the fraction part of R. Here, it is =0.64


3. The scores having Rank 17 and Rank (17+1) i.e, Rank 18 are taken into account. Here we see that are the scores corresponding to rank 17 and rank 18 are respectively 96 and 98. The scores are 5 and 7.


4. Then interpolation is performed by taking the product of the difference between the two scores and the fraction part, and then the result is added to the lower score. If we consider the example it will be (0.64)(98 – 96) + 96 = 97.28


5. Hence, the 84th percentile is 97.28. Now, if the first definition and second definition were used then we would have obtained different results. That is, if we considered the first one, then the answer would have been 98 and for the second one, we would have obtain 96 as our answer. Hence, to avoid such confusion The 3rd definition of percentile is the best one.

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