Calculate Probabilities with A Standard Normal Distribution Table

Introduction to Finding Areas With a Table

A table of z-scores can be used to calculate the areas under the bell curve. This is important in statistics because the areas represent probabilities. These probabilities have numerous applications throughout statistics.

The probabilities are found by applying calculus to the mathematical formula of the bell curve. The probabilities are collected into a table.

Different types of areas require different strategies. The following pages examine how to use a z-score table for all possible scenarios.

Area to the Left of a Positive z Score

To find the area to the left of a positive z-score, simply read this directly from the standard normal distribution table.​

For example, the area to the left of z = 1.02 is given in the table as .846.

Area to the Right of a Positive z Score

To find the area to the right of a positive​ z-score, begin by reading off the area in the standard normal distribution table. Since the total area under the bell curve is 1, we subtract the area from the table from 1.

For example, the area to the left of z = 1.02 is given in the table as .846. Thus the area to the right of z = 1.02 is 1 - .846 = .154.

Area to the Right of a Negative z Score

By the symmetry of the bell curve, finding the area to the right of a negative z-score is equivalent to the area to the left of the corresponding positive z-score.

For example, the area to the right of z = -1.02 is the same as the area to the left of z = 1.02. By use of the appropriate table we find that this area is .846.

Area to the Left of a Negative z Score

By the symmetry of the bell curve, finding the area to the left of a negative z-score is equivalent to the area to the right of the corresponding positive z-score.

For example, the area to the left of z = -1.02 is the same as the area to the right of z = 1.02. By use of the appropriate table we find that this area is 1 - .846 = .154.

Area Between Two Positive z Scores

To find the area between two positive z scores takes a couple of steps. First use the standard normal distribution table to look up the areas that go with the two z scores. Next subtract the smaller area from the larger area.

For example, to find the area between z₁ = .45 and z₂ = 2.13, start with the standard normal table. The area associated with z₁ = .45 is .674. The area associated with z₂ = 2.13 is .983. The desired area is the difference of these two areas from the table: .983 - .674 = .309.

Area Between Two Negative z Scores

To find the area between two negative z scores is, by symmetry of the bell curve, equivalent to finding the area between the corresponding positive z scores. Use the standard normal distribution table to look up the areas that go with the two corresponding positive z scores. Next, subtract the smaller area from the larger area.

For example, finding the area between z₁ = -2.13 and z₂ = -.45, is the same as finding the area between z₁^* = .45 and z₂^* = 2.13. From the standard normal table we know that the area associated with z₁^* = .45 is .674. The area associated with z₂^* = 2.13 is .983. The desired area is the difference of these two areas from the table: .983 - .674 = .309.

Area Between a Negative z Score and a Positive z Score

To find the area between a negative z-score and a positive z-score is perhaps the most difficult scenario to deal with due to how our z-score table is arranged. What we should think about is that this area is the same as subtracting the area to the left of the negative z score from the area to the left of the positive z-score.

For example, the area between z₁ = -2.13 and ​z₂ = .45 is found by first calculating the area to the left of z₁ = -2.13. This area is 1-.983 = .017. The area to the left of z₂ = .45 is .674. So the desired area is .674 - .017 = .657.