How to Find Z-Scores Given Area (With Examples)


There are three ways to find the z-score that corresponds to a given area under a normal distribution curve

1. Use the z-table.

2. Use the Percentile to Z-Score Calculator.

3. Use the invNorm() Function on a TI-84 Calculator.

The following examples show how to use each of these methods to find the z-score that corresponds to a given area under a normal distribution curve.

Example 1: Find Z-Score Given Area to the Left

Find the z-score that has 15.62% of the distribution’s area to the left.

Method 1: Use the z-table.

The z-score that corresponds to a value of .1562 in the z-table is -1.01.

2. Use the Percentile to Z-Score Calculator.

According to the Percentile to Z-Score Calculator, the z-score that corresponds to a percentile of .1562 is -1.01.

3. Use the invNorm() function on a TI-84 calculator.

Using the invNorm() function on a TI-84 calculator, the z-score that corresponds to an area of .1562 to the left is -1.01.

Notice that all three methods lead to the same result.

Example 2: Find Z-Score Given Area to the Right

Find the z-score that has 37.83% of the distribution’s area to the right.

Method 1: Use the z-table.

The z table shows the area to the left of various z-scores. Thus, if we know the area to the right is .3783 then the area to the left is 1 – .3783 = .6217

The z-score that corresponds to a value of .6217 in the z-table is .31

2. Use the Percentile to Z-Score Calculator.

According to the Percentile to Z-Score Calculator, the z-score that corresponds to a percentile of .6217 is .3099.

3. Use the invNorm() function on a TI-84 calculator.

Using the invNorm() function on a TI-84 calculator, the z-score that corresponds to an area of .6217 to the left is .3099.

Example 3: Find Z-Scores Given Area Between Two Values

Find the z-scores that have 95% of the distribution’s area between them.

Method 1: Use the z-table.

If 95% of the distribution is located between two z-scores, it means that 5% of the distribution lies outside of the z-scores.

Thus, 2.5% of the distribution is less than one of the z-scores and 2.5% of the distribution is greater than the other z-score.

Thus, we can look up .025 in the z-table. The z-score that corresponds to .025 in the z-table is -1.96.

Thus, the z-scores that contain 95% of the distribution between them are -1.96 and 1.96.

2. Use the Percentile to Z-Score Calculator.

According to the Percentile to Z-Score Calculator, the z-score that corresponds to a percentile of .025 is -1.96.

Thus, the z-scores that contain 95% of the distribution between them are -1.96 and 1.96.

3. Use the invNorm() function on a TI-84 calculator.

Using the invNorm() function on a TI-84 calculator, the z-score that corresponds to an area of .025 to the left is -1.96.

Thus, the z-scores that contain 95% of the distribution between them are -1.96 and 1.96.

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