# If #X# is #"Normal"(μ = 81.2, σ = 12.4),# what is the 16th percentile of this distribution?

##### 1 Answer

See explanation.

#### Explanation:

A **percentile** is a location in a distribution that has a specified amount (or percentage) of the distribution "below it" (to its left). In other words, if the

#n^"th" " percentile" = x" "# means#" " P(X < x)=n%.#

For example, in a standard normal curve (with

The standard normal distribution

How do we use it? Let's say we want the 25th percentile for the standard normal distribution. We find the value closest to 0.25 in the table (which happens to be 0.2514) and see that it's in row

But wait—how does that help when we want a percentile for any normal distribution *any* curve and the standard normal curve. That connection is found by *shifting* the *stretching/squishing* it so that its standard deviation is

#Z=(X-mu)/sigma#

where

If we know the percentile we want from the

#X=sigma Z + mu# .

As an example, let's use the first question you asked, where

From the table above, the 16th percentile from the

#X=(12.4)("-"0.99)+81.2#

#color(white)X="-"12.276+81.2#

#color(white)X=68.924#

What this says is: if

I'll leave the rest for you as an exercise; with the formulas above, it shouldn't be that hard.

Hope this helps!