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 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.

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