How to Perform Inverse Normal Probability Calculations

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

The normal probability density function, the bell-shaped curve, is amazingly useful for explaining random events. In this lesson, we show how to invert this function using a table to calculate intervals corresponding to given probabilities.

Arrival Times

Fred has just joined a large group of bird watchers. Although the general meeting for the group is scheduled to start at 7 PM, the meeting begins much later. This is a non-punctual group! Fred decides use arrival times to play with some of the math he has recently learned. Let's follow Fred as he attempts to understand the behavior of this group.

Gathering Some Information

There are 1,000 members who regularly attend the monthly meetings. Fred gets to the meeting place early and keeps a running record of arrival times relative to the 7 PM start time. The mean of these arrival times, μ, is 15 minutes. In other words, on average, people show up 15 minutes late. The standard deviation, σ, of Fred's data is 5 minutes.

After plotting the data, Fred sees a familiar bell-shaped curve, the Normal Probability Density Function, also known as the Gaussian distribution.

The peak located at the mean of 15

This is an idealized curve. Fred's data roughly resembles this curve. The area under the curve is a probability. Integrating from -∞ to +∞ includes all the arrival times and the probability is 1.

The X on the horizontal axis is the random variable, X. We can ask a question like, ''What is the probability the random variable X (the arrival time) is less than some value?'' In the figure, there is an orange shaded area. This area is the probability the person will arrive less than 5 minutes late. Now, let's ask this type of question in a slightly different way.

Finding Inverse Normal Probability Values

Fred would like to know at what time, a, there will be less than 23 members present. This is a probability of 23/1000 = 0.023. Thus, we are asking for the value of X which will give an area under the curve equal to 0.023. This is the inverse normal probability value. We can write this as P(X < a) = 0.023.

This 0.023 probability is the area under the curve. In principle, we would integrate the normal curve from -∞ to a. The problems are we don't know a, and the integral itself does not have a closed-form solution. We can use numerical methods to approximate the integration very accurately, but we won't have a result as a function of a. If we did, we could set this result equal to 0.023 and use algebra to solve for a.

Also, it would be impossible to tabulate all possible combinations of means and standard deviations. Instead, tables are published for a mean of 0 and a standard deviation of 1. The random variable is called Z. A portion of one of these tables looks like:

Area under the curve from negative infinity to the Z value

Fred looks at the table for the number closest to 0.0223. He finds:

  • P = 0.0233 for Z = -1.99
  • P = 0.0228 for Z = -2.00
  • P = 0.0222 for Z = -2.01

Do you see how the numbers in the first column and the numbers in the first row are combined to locate a probability value? We choose the closest: Z = -2.00. Now to relate this value to our bird watchers group.

The relationship between the random variables X and Z:


In our case, μ = 15 and σ = 5:


Thus, (a - 15)/5 = -2.00. Solving for a we get a = 5.

Fred is deliriously happy! He expects to see at least 23 members present after waiting 5 minutes.

The Very Late Arrivers

Fred wonders at what time everyone will be there except the last 159 members.

Probability all members will have arrived except the last 159

This probability is P(X > a). The table, however, describes integration from -∞ to a. We are looking for a. Fred has an idea. Since the total probability is 1, he writes:


And, 1 - P(X > a) = 1 - 0.159 = 0.841.

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