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Z-scores, also known as standard scores, are a statistical measure that indicates the number of standard deviations a data point is above or below the mean of a normal distribution. They are useful for comparing data points from different distributions and determining the relative position of a data point within a distribution.
To find the Z-score when given a specific area, one can use a Z-score table or a statistical calculator. These tools provide the Z-score corresponding to a given area under the normal curve. For example, if we want to find the Z-score for an area of 0.95, we can look up the corresponding Z-score of 1.96 in the Z-score table.
Another way to find Z-scores is by using the Z-score formula, which is Z = (x – μ) / σ, where x is the data point, μ is the mean, and σ is the standard deviation. This formula calculates the Z-score for a specific data point in a normal distribution.
For example, if we have a normal distribution with a mean of 50 and a standard deviation of 10, and we want to find the Z-score for a data point of 60, we can use the formula Z = (60-50) / 10 = 1. This means that the data point of 60 is one standard deviation above the mean and has a Z-score of 1.
In summary, Z-scores can be found when given a specific area by using a Z-score table, a statistical calculator, or the Z-score formula. They are a useful tool for understanding the relative position of a data point within a normal distribution.
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 .
2. Use the .
3. Use the .
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 is -1.01.
2. Use the Percentile to Z-Score Calculator.
According to the , 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 on a TI-84 calculator, the z-score that corresponds to an area of .1562 to the left is -1.01.
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 is .31
2. Use the Percentile to Z-Score Calculator.
According to the , the z-score that corresponds to a percentile of .6217 is .3099.
3. Use the invNorm() function on a TI-84 calculator.
Using the 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 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 , 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 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.