Percentile Rank

Fomula

First, sort all n raw scores in ascending order. Next, find the rank (r), or position in the [Page 1027]ordered listing of scores, of the percentile of interest (p):

To find the score representing that percentile, Y(p),

where k is the integer portion of r, and d is the decimal portion of r, such that k + d = r.

Example

The raw scores of a test administered to seven people are listed below, along with their respective ranks. You are interested in finding the 80th percentile.

The rank, r, of the 80th percentile is

Using the definitions above, k = 6 and d = 0.4, giving k + d = r. The 80th percentile can now be calculated as follows:

That is, an examinee who received a raw score of 90.4 on this exam performed as well as or better than 80% of the other test takers. For a sample size of only seven people, one can see that this is neither very informative nor useful. However, when dealing with larger samples, percentiles can be very helpful in describing where one stands in relation to others.

The Sample Median

The sample median is a useful statistic in that, unlike the mean and mode of a set of observations, it is unaffected by outliers and skewness in the data because it depends only on the values in the very middle of all the observations. The median is often used to report statistics whose distributions are widely varied or skewed, such as annual household income for a community or the ages of students enrolled in a program.

By definition, the median is the halfway point of a data set: Exactly half of the observations are less than it, and therefore, exactly half are greater than it. Thus, the median is also the 50th percentile of a set of data.

Example

One can see that, if r is an integer, no interpolation is necessary, and the percentile is the value with rank r. An example illustrates this point by finding the 50th percentile, or median, of the seven scores above.

Now, k = 4 and d = 0.0. To find the

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