Percentiles Help (page 2)

By — McGraw-Hill Professional
Updated on Aug 26, 2011

Percential Points, Ranks, and Inversion

Percential Points

We get around the foregoing conundrum by defining a scheme for calculating the positions of the percentile points in a set of ranked data elements. A set of ranked data elements is a set arranged in a table from ''worst to best,'' such as that in Table 4-1. Once we have defined the percentile positioning scheme, we accept it as a convention, ending all confusion forever and ever.

So – suppose we are given the task of finding the position of the pth percentile in a set of n ranked data elements. First, multiply p by n, and then divide the product by 100. This gives us a number i called the index:

            i = pn/100

Here are the rules:

  • If i is not a whole number, then the location of the pth percentile point is i + 1.
  • If i is a whole number, then the location of the pth percentile point is i + 0.5.

Percential Ranks

If we want to find the percentile rank p for a given element or position s in a ranked data set, we use a different definition. We divide the number of elements less than s (call this number t) by the total number of elements n, and multiply this quantity by 100, getting a tentative percentile p*:

            p* = 100 t/n

Then we round p* to the nearest whole number between, and including, 1 and 99 to get the percentile rank p for that element or position in the set.

Percentile ranks defined in this way are intervals whose centers are at the percentile boundaries as defined above. The 1st and 99th percentile ranks are often a little bit oversized according to this scheme, especially if the population is large. This is because the 1st and 99th percentile ranks encompass outliers, which are elements at the very extremes of a set or distribution.

Percentile Inversion

Once in a while you'll hear people use the term ''percentile'' in an inverted, or upside-down, sense. They'll talk about the ''first percentile'' when they really mean the 99th, the ''second percentile'' when they really mean the 98th, and so on. Beware of this! If you get done with a test and think you have done well, and then you're told that you're in the ''4th percentile,'' don't panic. Ask the teacher or test administrator, ''What does that mean, exactly? The top 4%? The top 3%? The top 3.5%? Or what?'' Don't be surprised if the teacher or test administrator is not certain.

Percentile Practice Problems

Practice 1

Where is the 56th percentile point in the data set shown by Table 4-1?

Solution 1

There are 1000 students (data elements), so n = 1000. We want to find the 56th percentile point, so p = 56. First, calculate the index:

    i = (56 × 1000)/100
        = 56,000/100
        = 560

This is a whole number, so we must add 0.5 to it, getting i + 0.5 = 560.5. This means the 56th percentile is the boundary between the ''560th worst'' and ''561st worst'' test papers. To find out what score this represents, we must check the cumulative absolute frequencies in Table 4-1. The cumulative frequency corresponding to a score of 25 is 531 (that's less than 560.5); the cumulative frequency corresponding to a score of 26 is 565 (that's more than 560.5). The 56th percentile point thus lies between scores of 25 and 26.

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