Inverse Normal Distribution Calculator

How to Use the Inverse Normal Distribution Calculator

Unlike the normal distribution calculator, which finds probability (area under the standard normal curve) from a z-score, the inverse normal distribution calculator works backwards by finding the z-score corresponding to a certain probability. In other words, it calculates the z value corresponding to a certain probability.

To use the calculator, follow these steps:

1. Enter the probability value.
2. Choose the probability type that matches your question.
3. Click Calculate.

The calculator will instantly return the z-score corresponding to your probability and show you exactly how to find the z-score for your probability using the standard normal tables.

Struggling to choose the right probability type? You should choose:

• Left cumulative: P(Z < z) when the probability is to the left of the z score
• Right tail: P(Z > z) when the probability is to the right of the z score
• Central probability: P(-z < Z < z) when the probability is in the middle of the curve
• Outside probability when the probability is split between the two tails

What Is an Inverse Normal Distribution Calculator?

An inverse normal distribution calculator finds the z score that corresponds to a given probability under the standard normal distribution.

A regular z-score probability calculator starts with a z-score and finds the area. On the other hand, an inverse normal calculator does the opposite. It starts with an area or probability and finds the z-score.

For example, if the left-tail probability is 0.975, the calculator finds the z score where 97.5% of the standard normal curve is to the left. The result is approximately: z = 1.96. This value is commonly used in 95% confidence intervals because the middle 95% of the standard normal curve leaves 2.5% in each tail.

Example 1: Find a Z Score from a Left-Tail Probability

A student wants to find the z-score that cuts off the lowest 90% of the standard normal distribution. Find the z score.

Solution

The phrase lowest 90% means the area is to the left of the z score. Thus, we need to find the z value such that P(Z< z) = 0.90

Using the calculator:

1. Choose Left cumulative: P(Z < z)
2. Enter probability as 0.90
3. Click Calculate

The calculator gives z = 1.2816. This means that a z score of about 1.28 cuts off the lowest 90% of the standard normal distribution. In other words, about 90% of values are below this z score, and 10% are above it.

Example 2: Find a Z Score from a Right-Tail Probability

A researcher wants to find the z-score where only 5% of values are above it. Find the z score.

Solution

The phrase 5% of values are above it means the area is to the right of the z score. Thus, we need to find the z value such that P(Z > z) = 0.05.

Using the calculator:

1. Choose Right tail: P(Z > z)
2. Enter probability as 0.05
3. Click Calculate

The calculator gives z = 1.6449. This means that a z score of about 1.64 leaves 5% of the standard normal distribution to the right. In other words, about 95% of values are below this z score, and 5% are above it.

Example 3: Find Z Values from a Central Probability

Find the z values that contain the middle 95% of the standard normal distribution.

Solution

The phrase middle 95% means the area is centered around the mean. Thus, we need to find the z values such that P(-z < Z < z) = 0.95.

Using the calculator:

1. Choose Central probability: P(-z < Z < z)
2. Enter probability as 0.95
3. Click Calculate

The calculator gives z = ±1.9600. This means that the middle 95% of the standard normal distribution lies between z = -1.96 and z = 1.96. In other words, 2.5% of values are below -1.96, and 2.5% are above 1.96.

Example 4: Find Z Values from an Outside Probability

A statistics question asks for the z values that leave 10% of the total area outside the interval. Find the z values.

Solution

The phrase outside the interval means the probability is split between the two tails. Thus, we need to find the z values such that P(Z < -z or Z > z) = 0.10.

Using the calculator:

• Choose the Outside probability
• Enter probability as 0.10
• Click Calculate

The calculator gives z = ±1.6449. This means that about 10% of the standard normal distribution lies outside the interval from z = -1.64 to z = 1.64. In other words, about 5% is in the left tail, 5% is in the right tail, and 90% is between the two z values in the middle.

Common Inverse Normal Values

Here are a few common inverse normal values for the standard normal distribution.

Note. The values in the table are rounded off to two decimal places. Therefore, you should use the calculator when you need a more precise z-score.

When Should You Use This Calculator?

Use the inverse normal calculator when you know the probability or area under the standard normal curve and need to find the matching z score.

It is useful for:

• Finding z-scores from percentiles
• Finding z critical values from tail probabilities
• Finding z values for confidence intervals
• Finding cutoffs for the standard normal distribution
• Checking values from a Z table

However, if you need to calculate a z score from a raw score, mean, and standard deviation, use the z score calculator.

Frequently Asked Questions