# Percentiles and z-scores

## Percentiles

Percentiles are like quartiles, except that they divide the data set into 100 equal parts instead of four equal parts (similarly, there are quintiles and deciles and ...). Percentiles are useful for giving the relative standing of an individual in a population, they are essentially the rank position of an idividual. As with quartiles, there are definitions which vary slightly specifying how to calculate percentiles. All definitions provide that the percentile of a value is approximately the fraction of the population which is less than the specified value. (i.e., percentiles specify how far a datum is from the bottom, not the top. Percentiles are always integers (e.g., 85th, not 85.7th percentile).)

Sullivan specifies that the kth percentile is found by first finding the index i=(k/100)(n+1) [n is the number of data]. Then if i is an integer, the kth percentile is the ith datum from the bottom. If i is not an integer, the kth percentile is the mean of the data in the positions obtained by rounding i up and rounding i down.

For the 30 weights, the 42nd percentile is obtained by first calculationg i=(42/100)(30+1)=13.02, then taking the average of the 13th and 14th data (145+155)/2=150 which is the 42nd percentile. The 25th percentile (which is the first quartile) is obtained as i=(25/100)(31)=7.75, (125+130)/2=127.5. The 75th percentile (which is the third quartile) is obtained as i=(75/100)(31)=23.25, (175+175)/2=175.

The percentile rank of a specified value is found by calculating (y/n)(100) and rounding to the nearest integer where y is the number of data less than the specified value and n is the total number of data. For the 30 weights, 150 is the 43rd percentile ((13/30)(100)=43.33). 127.5 is the 23rd percentile ((7/30)(100)=23.33). 175 is the 73rd percentile ((22/30)(100)=73.33). Note that finding the percentile of a value and finding the value of a percentile are not exact inverse operations.

If one wants to compare someone who graduated 37th out of a class of 250 with someone who graduated 12th in a class of 60, one can calculate (213/250)(100) = 85.2 which is rounded off to the 85th percentile (percentiles measure position from the bottom, 37 from the top means that 213 are below it in a population of 250); similarly (48/60)(100) = 80 or the 80th percentile. Therefore, being 37th out of 250 puts one at the 85th percentile, which is better than 12th out of 60 which is only at the 80th percentile.

## z-scores

Another way to compare individuals in different populations is with z-scores. If mu is the mean of a population and s is the standard deviation, the z-score of a value x is (x-mu)/s (note that z-scores may be positive or negative). A standard example for demonstrating the utility of z-scores is comparing a score on the ACT tests with a score on the SAT tests. Originally, SAT tests had a mean score of 500 with a standard deviation of 100, while ACT tests had a mean score of 18 with a standard deviation of 6 (these are no longer the means and standard deviations for thosae tests). Hence one could compare 680 on the SAT with 25 on the ACT. The respective z-scores are (680-500)/100 = 1.8 and (25-18)/6 = 1.17. Therefore 680 on the SAT is a better score than 25 on the ACT (assuming equal quality among the students who took the two tests). Rearranging z=(x-mu)/s to x=mu+sz, we can calculate that a z-score of 1.8 on the ACT would be obtained with a score of 18+(6)(1.8)=28.8.

For our class weights, a z-score of -.5 corresponds to the weight 153.43+(29.69)(-.5)=138.59. A 175 pound individual would have a z-score (175-153.43)/29.69=.73.

Z-scores measure how outstanding an individual is relative to the mean of a population using the standard deviation for that population to define the scale. Note that percentiles use the median as the average (50th percentile), while z-scores use the mean as average (z-score of 0). Competencies: For the data set {2 5 9 4 6 7 6 8 8}, calculate the quartiles and 5-number summary.

For the class weightsfind the percentile and z-score of the 168 pound individual.

Reflection: When are z-scores versus percentiles a better measure of relative standing?

Challenge:

May 2002