# What are Percentiles?

A percentile is best described as a comparison score. The concept is widely used in statistical analysis, but many will be most familiar with the term as it relates to standardized testing in schools. Unlike a percentage, where a student is given a percent number that is related only to her performance in a test or examination, a percentile is a number between 1 and 100 that relates the student’s performance to those of other students who have taken the test. In a set of numbers, the percentile for a given value indicates the percentage of numbers that are less than or equal to that value. For example, if a student scores 85% in a test and is in the 90^{th} percentile, this means that 90% of students had scores that were less than or equal to 85%.

The concept may be easier to understand by comparison with percentage scores. A simple example would be a test with 100 problems, each worth 1% of the test. If a student answers 80 problems correctly, he scores 80%. In some cases, performance may be graded over ranges, for example, an “A” grade may indicate 90% or more, a “B” grade, 80-89% and so on. These grades may be based on the numbers of students who would typically be expected to fall within each category, but in themselves do not constitute a comparison with other students. If this student’s performance in the test is in the 95^{th} percentile, 95% of those who took the test scored 80% or less, or to put it another way, he is in the top 5%.

#### Calculating Percentiles

To calculate percentiles for a set of results, the values are first arranged in ascending order. The percentile for a given value can then be found by subtracting 0.5 from its numerical position in the sequence, dividing by the number of results, then multiplying by 100. For example, if there are 25 results with values ranging from 31 to 93 and the result 47 ranks as the tenth in numerical order, the percentile for this result is 10 – 0.5, divided by 25, then multiplied by 100, giving 38. This means that 38% of results are less than or equal to 47. It is also possible to do the calculation for hypothetical results that did not actually occur, that is, to find what the percentile would be for that result.

#### Percentiles in Education

Although this statistical method has many applications, one of the best known is in education. Especially when testing is standardized, it is meant to serve a diverse group of people and accurately gauge not only individual performance, but also comparative performance. When looking at a data set, percentiles can help better gauge the middle or median performance of students. Many students will cluster into the median area, earning percentiles anywhere from 25 to 75, while a few will far surpass this, reaching into the 90s range. Average and median scores are computed into expected results and can show how most people are performing, as well as how each individual student is performing.

Percentiles can further show if performance in certain areas is poor. If every student taking a test answers the same question incorrectly, or if most of the average students do so, this may indicate a problem. It may be that the question is badly worded or it may be that this area of the subject has not been adequately covered during the course. With increasingly standardized tests in the academic setting, this method can weed out bad questions and identify areas for improvement in courses or teaching methods.

Another educational use for percentiles is to check testing in population groups. For example, a whole high school in an urban neighborhood may score well below average in academic tests. Even if a couple of pupils score well — these are known as outliers — it is clear that there is a problem. Possibly, the children are not prepared properly to take the test, or the test is not understood by the students because of cultural or language barriers. By considering percentiles as well as percentage scores, schools can better address all the needs of their students.

#### Other Applications

Percentiles have many other applications and may be found in many studies that employ statistics. In the scientific analysis of samples, this method may highlight important trends or problems. Scientists may, for example, test a series of air samples taken from an urban area for their sulfur dioxide content in parts per million (ppm). If the safe level is defined as 3 ppm and the percentile for 3ppm is 35%, then it can be said that 65% of samples exceed the safe limit and that action is required to reduce pollution levels. Other uses include analysis of the results of social surveys and opinion polls.

## Discussion Comments

This doesn't make sense: "For example, if a student scores 85% in a test and is in the 90th percentile, this means that 90% of students had scores that were less than or equal to 85%.

It's only true if an 85% score is the lowest score of the top 10th percentile.

@Buer – Just to be sure that everyone is clear on this: percentiles are more a way to compare performances to understand a population and an individual's place in it than to evaluate an individual's abilities or characteristics.

If you have ever taken a standardized test – and chances are you have – then you've had your results placed according to percentile. While it's certainly useful to know your raw score on a test, it is often much more informative to see how you compare to your peers. For example, if a professor wants to be sure that all of his students will be challenged by an exam, he may choose to make it so difficult that no one scores above a sixty; having done that, looking at how all of his students performed is a much more reasonable way to evaluate them than judging them on the one hundred point scale.

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