AP®StatisticsCollege BoardStudy GuidesUnit 2: Probability, Random Variables, and Probability DistributionsNormal DistributionInverse Normal Calculations

Inverse Normal Calculations (College Board AP® Statistics): Study Guide

Syllabus Edition

First teaching 2026

First exams 2027

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on 22 May 2026

Inverse normal calculations

Given the value P(X < a) how do I find the value of a using the standard normal table?

The proportion of a distribution of a variable $X$ that lies below a given value is the area under the curve to the left of the given value
- This is also the probability that the variable lies below this value
- $P (X < a) = p$

A normal distribution curve with mean μ and shaded area to the left of a vertical dashed line at a. The mean is labeled μ and point a is labeled a. — **Normal distribution with proportion of distribution less than a shaded**

$P (X < a) = P (Z < z)$
- where $z$ is the $z$ -score of the value $a$
If you are given the proportion (or probability) and want to work out the value of $a$ :
- find the cell in the standard normal table that is
  - equal to the proportion
  - or the highest value that is less than the proportion
- identify the $z$ -score by listing the relevant row and column
- convert the $z$ -score back into an actual value

Given the value of P(X < a) how do I find the value of a using a calculator?

Your calculator will have a function called Inverse Normal Distribution
- Some calculators call this InvN
Given that $P (X < a) = p$ you will need to enter:
- The proportion (or probability), $p$
  - This is the area of the distribution to the left of $a$
- The mean, $μ$
- The standard deviation, $σ$
Some calculators might ask for the tail
- For $P (X < a)$ this is the left tail

Examiner Tips and Tricks

Always check your answer makes sense!

If $P (X < a)$ is less than 0.5 then $a$ should be smaller than the mean,
If $P (X < a)$ is more than 0.5 then $a$ should be bigger than the mean.

A sketch will help you see this.

Given the value of P(X > b) how do I find the value of a using the standard normal table?

The proportion of a a distribution of a variable $X$ that lies above a given value is the area under the curve to the right of the given value
- This is also the probability that the variable lies above this value
- $P (X > b) = p$

A normal distribution curve with mean μ and shaded area to the right of a vertical dashed line at b. The mean is labeled μ and point b is labeled b. — **Normal distribution with proportion of distribution above b shaded**

$P (X > b) = P (Z > z)$
- where $z$ is the $z$ -score of the value $b$
The total area under the curve is 1
- So $P (X > b) = 1 - P (X < b)$
If you are given the proportion (or probability) and want to work out the value of $b$ :
- subtract the proportion from 1
- find the cell in the standard normal table that is
  - equal to this result
  - or the lowest value that is greater than the proportion
- identify the $z$ -score by listing the relevant row and column
- convert the $z$ -score back into an actual value

Given the value of P(X > b) how do I find the value of b using a calculator?

You will need to use the Inverse Normal Distribution function again
Given $P (X > b) = p$
- Use $P (X < b) = 1 - P (X > b)$ to rewrite this as $P (X < b) = 1 - p$
- Then use the method for $P (X < b)$ to find $b$
Your calculator may have the tail option (left, right or centre)
- If so, you can use the Inverse Normal Distribution function straightaway by:
  - selecting right for the tail
  - and entering the area as $p$

Examiner Tips and Tricks

Always check your answer makes sense!

If $P (X > b)$ is less than 0.5 then $b$ should be bigger than the mean,
If $P (X > b)$ is more than 0.5 then $b$ should be smaller than the mean.

A sketch will help you see this.

How can I use the table for t-distribution critical values?

You will later learn that the t-distribution tends to the standard normal distribution when the degree of freedom tends to infinity
In your exam, you have a table of t-distribution critical values
- These give the values of $t$ such that $P (T > t) = p$
You can use the row for infinity (∞) to find z-scores that give probabilities $P (Z > z) = p$
The table only gives certain common values of $p$
You can use symmetries of the normal distribution to find negative z-scores
- For example, the table tells you $P (Z > z) = 0.025 \Rightarrow z = 1.960$
  - This means $P (Z < z) = 0.025 \Rightarrow z = - 1.960$
To use this table, write the probability statement in the form $P (Z > z) = p$
- If the probability is greater than 0.5, subtract it from 1 and flip the inequality
  - For example, if you want $P (Z < z) = 0.9$ then rewrite as $P (Z > z) = 0.1$
- If the sign is $<$ , then flip the inequality and change the sign of the z-score
  - For example, if you want $P (Z < z) = 0.1$ then find $P (Z > z) = 0.1$ and change the z-score to a negative

Worked Example

The distribution of heights of all adult male bison in a particular herd of bison is approximately normal with a mean of 5.9 feet and standard deviation 0.16 feet. The rancher wants to select the tallest twenty percent of the bison to use for a breeding program.

What is the minimum height that an adult male bison must be to qualify for the breeding program?

Answer:

Draw a sketch of the situation

Because $P (X > b) = 0.2$ , is less than 0.5, $b$ should be bigger than the mean, i.e. $b$ lies to the right of the mean

A normal distribution with the mean, μ, at 6.8. The area under the curve to the right of 'b' is highlighted and has probability, P(X > b) = 0.2.

Method 1: Using the tables

Using the standard normal table, look at the section with the positive $z$ -values as $b$ is above the mean

You can use the ∞ row in the table for t-distribution critical values

$P (Z > z) = 0.20 \Rightarrow z = 0.841$

Convert the $z$ -score into an actual value, using $z = \frac{x - μ}{σ}$

$\begin{array}{rcl} 0.841 & = & \frac{b - 5.9}{0.16} \\ 0.841 \cdot 0.16 & = & b - 5.9 \\ 0.841 \cdot 0.16 + 5.9 & = & b \\ b & = & 6.03456 \end{array}$

Explain the value in the context of the question

The minimum height that an adult male bison must be to qualify for the breeding program is 6.04 feet (to 3 significant figures)

Method 2: Using a calculator

Write down the parameters for the situation

$P (X < b) = 0.8$

$μ = 5.9$

$σ = 0.16$

Enter these values into the Inverse Normal Distribution function on your calculator and calculate $b$

$b = 6.036 . . .$

Explain the value in the context of the question

The minimum height that an adult male bison must be to qualify for the breeding program is 6.04 feet (to 3 significant figures)

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Inverse Normal Calculations (College Board AP® Statistics): Study Guide

Inverse normal calculations

Given the value P(X < a) how do I find the value of a using the standard normal table?

Given the value of P(X < a) how do I find the value of a using a calculator?

Examiner Tips and Tricks

Given the value of P(X > b) how do I find the value of a using the standard normal table?

Given the value of P(X > b) how do I find the value of b using a calculator?

Examiner Tips and Tricks

How can I use the table for t-distribution critical values?

Worked Example

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Author: Naomi C

Reviewer: Dan Finlay