## Wednesday, December 5, 2012

### Percentiles?!

We take tests throughout our school years and most of us understand how the scoring works.  Take the number of points earned, divide by the number of points possible, move the decimal a couple of spaces right and, voila, we have a percent that represents our score.  (If you want to learn more about percents, there's a song for that.)

But then, once a year, we take a standardized test and when the score is reported, we are given a percentile.  What in the world is a percentile?

Unfortunately, we don’t learn much about percentiles in school.  However, we do learn about the median, which I discussed in my last blog entry, and also about quartiles.  Let’s review briefly:

If all of the values in a data set are arranged in order, then the middle number (or, if there are two middle numbers, their average) is the median.  This number divides the data set into two equal groups:  Half of the values in the set are above the median, and half of the values are below the median.

If all of the values in a data set are arranged in order, then the quartiles (there are three of them) are the values that split the data set into four equal groups:

One-fourth (25%) of the values in the set are less than or equal to the 1st quartile, and three-fourths (75%) of the values are greater than or equal to the 1st quartile.

Half (50%) of the values in the set are less than or equal to the 2nd quartile, and half (50%) of the values are greater than or equal to the 2nd quartile.  (The 2nd quartile is the same as the median.)

Three-fourths (75%) of the values in the set are less than or equal to the 3rd quartile, and one-fourth (25%) of the values are greater than or equal to the 3rd quartile.

If you understand how quartiles work, then it’s not much of a leap to understand percentiles.

If all of the values in a data set are arranged in order, then the percentiles (there are 99 of them) are the values that split the data into 100 equal groups.  (Note: some people call these centiles.)

1% of the values in the set are less than or equal to the 1st percentile, and 99% of the values are greater than or equal to the 1st percentile.

2% of the values in the set are less than or equal to the 2nd percentile, and 98% of the values are greater than or equal to the 2nd percentile.

And so on…

Because they split the data set into so many groups, percentiles are only useful in analyzing data sets that are very large – like the number of students who take standardized tests.

If we understand how percentiles work, then interpreting standardized test scores becomes much easier.

If a score is reported as 20th percentile, then 20% of test-takers scored at or below that level.

If a score is reported as 90th percentile, then 90% of test-takers scored at or below that level.

That wasn’t so hard was it?

So, why are there so many people who struggle with percentiles?

I blame politicians.

In the recent years, politicians arguing over tax rates have consistently discussed “Americans in the top two percent of earners”.  They would do us all a great service if they would instead argue about “Americans at or above the 98th percentile of earnings”.  It wouldn’t make politicians any more likely to work well together, but at least people would understand standardized test scores better!

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