Percentiles and Box Plots

Percentiles  

We saw that the median splits the data so that half lies below the median.  Often we are interested in the percent of the data that lies below an observed value.   

We call the rth percentile the value such that r percent of the data fall at or below that value.  
Example

If you score in the 75th percentile, then 75% of the population scored lower than you. 

 

Example

Suppose the test scores were 

        22,   34,   68,   75,   79,   79,   81,   83,   84,   87,   90,   92,   96,  and  99

If your score was the 75, in what percentile did you score?

Solution

There were 14 scores reported and there were 4 scores at or below yours.  We divide

           4
                 100%  =  29
          14

So you scored in the 29th percentile.

 

There are special percentile that deserve recognition.

  1. The second quartile (Q2) is the median or the 50th percentile
  2. The first quartile (Q1) is the median of the data that falls below the median.  This is the 25th percentile
  3. The third quartile (Q3) is the median of the data falling above the median.  This is the 75th percentile

We define the interquartile range as the difference between the first and the third quartile

        IQR  =  Q3 - Q1 

An example will be given when we talk about Box Plots.

 

Box Plots

Another way of representing data is with a box plot.  To construct a box plot we do the following:

 

  1. Draw a rectangular box whose bottom is the lower quartile (25th percentile) and whose top is the upper quartile (75th percentile). 

  2. Draw a horizontal line segment inside the box to represent the median.

  3. Extend horizontal line segments ("whiskers") from each end of the box out to the most extreme observations.

Box plots can either be shown vertically or horizontally.  The steps describe how to create a vertical box plot, while the graph below shows an example of a horizontal box plot the shows how student's commuting miles are distributed.

 

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