Percentiles
A percentile is a measure of relative
standing, meaning we get information
about the position of a value relative
to the rest of the data set.
1
An example: last 10 golf scores for 18 holes, sorted:
82, 83, 84, 85, 85, 88, 90, 90, 93, 95
1
2 3 4 5 6
7 8 9 10
Here I have an example where I wrote down the last 10 golf
scores I had. Note that I have sorted the scores from low to high
score (although that is not the order I shot them – but for what I
want to do next I need to have the data sorted from low to high –
ascending order).
Below each score I put the “location” of the score in the
ascending order.
2
Here is an approximation method to get percentiles.
Let p = the percentile of interest,
n = the number of data points or observations, then
Lp = (n + 1)(p/100)n is a location index number (a fancy name
for a handy little device) we will use to find pth percentile.
Now, if n = 10 and if we want the 25th percentile the location
index is
Lp = (10 + 1)(25/100) = 11(.25) = 2.75.
In my example data points have locations with whole numbers.
The location index of 2.75 is between 2 and 3. The 25th
percentile value will be 75% of the way between the 2nd and 3rd
number.
3
The 2nd number is 83 and the third number is 84. To get the 25th
percentile number take the lower number, 83 and add .75 of the
difference between the 2nd and 3rd numbers:
83 + .75(84 – 83) = 83.75.
Note if the index is a whole number the value in that location is
the percentile of interest.
The 50th percentile here is found by:
Lp = (10 + 1)(50/100) = 5.5 or half the way between the 5th and
6th numbers. So,
85 + .5(88 – 85) = 86.5 is the 50th percentile.
4
Quartiles
Quartiles are just special percentiles. The 25th percentile is
the 1st quartile, the 50th percentile is the 2nd quartile (and also
called median) and the 75th percentile is the 3rd quartile.
What is most important here, I think, is that you understand
the meaning of a percentile and the special percentiles called
quartiles.
5
special percentiles
•The 25th and 75th percentiles are called the 1st and 3rd
quartiles(Q1 and Q3), respectively. They are just the
medians of the lower and upper halves of the arranged
values.
The following is a visual to see the percentiles.
lowest
25%
of observations
first quartile is
a value
next
25%
next
25%
median is a value
highest
25%
third
quartile is
a value
number line
where we
measure
values
of the
variable
Interquartile range
•Variation can be indicated by the interquartile range,
IQR = Q3 - Q1. The smaller the IQR, the closer Q3 and Q1
are in the graph and thus the lower the spread!
•Hey, please check out the box plot, or what is sometimes
called the box and whiskers plot in the text. I know you
will!