Tails on a Normal Distribution (College Board AP® Statistics)
Study Guide
Written by: Mark Curtis
Reviewed by: Dan Finlay
Tails on a normal distribution
What are tails on a normal distribution?
Tails are the regions under the normal curve on the extreme left- or right-hand sides of the distribution
How do I find the most extreme P% of values from a normal distribution?
The most extreme P% of values in a normal distribution lie in the left-hand tail and the right-hand tail
Both tails are required
As the normal distribution is symmetric, this means
of P% of values lie in the left-hand tail
and of P% of values lie in the right-hand tail
How do I find the middle Q% of values from a normal distribution?
To find the middle Q% of values from a normal distribution
it is easier to work out the percentage of values you do not want
100% - Q%
then halve this value to give the percentage on each tail
%
For example, the middle 90% of values has
10% across both tails
so 5% on each tail
If the boundary values on the standard normal distribution are then
there is a total of 5% + 90% = 95% to the left of
so
Examiner Tips and Tricks
It is a common mistake to say the middle 90% of values has a standard normal boundary value, , that satisfies . The correct statement is .
How do I use the t-distribution tables to find boundary values?
The usual way to find boundary values for the middle 90% is to
let the positive boundary value be , then solve
This is an inverse normal calculation
i.e. find the probability closest to 0.95 in the normal tables
then read off the z-score
However, for certain common percentages that come up a lot (e.g the middle 50%, 90%, 95%, 99% etc of values) it is easier and more accurate to use
the very last row of the t-distribution table ('t-table')
called row infinity,
as row infinity is a row of z-scores from the standard normal distribution
The percentage shown underneath row infinity is the middle percentage of values
not the percentage to the left
e.g. the middle 90% of values has a positive boundary z-score of 1.645 (from the t-tables)
This is more accurate than '1.64 or 1.65' from the normal tables
Examiner Tips and Tricks
You can, of course, use your calculator to do inverse normal calculations, but it is still worth knowing about the useful last row (row infinity) in the t-distribution tables!
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