Page 101 - Statistics for Dummies

Chapter 5: Means, Medians, and More
Counting from left to right (from the smallest to the largest number in the
data set), you go until you find the 23rd number in the data set. That number
is 98, and it’s the 90th percentile for this data set.
Now say you want to find the 20th percentile. Start by taking 0.20 ∗ 25 = 5;
this is a whole number, so proceed from Step 3a to Step 4, which tells us the
20th percentile is the average of the 5th and 6th numbers in the ordered data
set (62 and 66). The 20th percentile then comes to (62 + 66) ÷ 2 = 64. The
median (the 50th percentile) for the test scores is the 13th score: 77.
There is no single definitive formula for calculating percentiles. The formula
here is designed to make finding the percentile easier and more intuitive,
especially if you’re doing the work by hand; however, other formulas are
used when you’re working with technology. The results you get using various
methods may differ but not by much.
Interpreting percentiles 85
Percentiles report the relative standing of a particular value within a data set.
If that’s what you’re most interested in, the actual mean and standard devia-
tion of the data set are not important, and neither is the actual data value.
What’s important is where you stand — not in relation to the mean, but in
relation to everyone else: That’s what a percentile gives you.
For example, in the case of exam scores, who cares what the mean is, as long
as you scored better than most of the class? Who knows, it may have been
an impossible exam and 40 points out of 100 was a great score (that hap-
pened to me in an advanced math class once; heaven forbid this should ever
happen to you!). In this case, your score itself is meaningless, but your per-
centile tells you everything.
Suppose your exam score is better than 90% of the rest of the class. That
means your exam score is at the 90th percentile (so k = 90), which hope-
fully gets you an A. Conversely, if your score is at the 10th percentile (which
would never happen to you, because you’re such an excellent student), then
k = 10; that means only 10% of the other scores are below yours, and 90% of
them are above yours; in this case an A is not in your future.
A nice property of percentiles is they have a universal interpretation: Being
at the 95th percentile means the same thing no matter if you are looking at
exam scores or weights of packages sent through the postal service; the 95th
percentile always means 95% of the other values lie below yours, and 5% lie
above it. This also allows you to fairly compare two data sets that have differ-
ent means and standard deviations (like ACT scores in reading versus math).
It evens the playing field and gives you a way to compare apples to oranges,
so to speak.

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