To use a standard normal distribution table (also known as a Z-table) to find a z-score, you need to understand what the table represents and how to work backward from a probability to find the corresponding z-score.
Understanding the Standard Normal Distribution Table
A standard normal distribution table typically provides the area under the standard normal curve to the left of a given z-score. The standard normal distribution has a mean of 0 and a standard deviation of 1. The area under the curve represents the probability of observing a value less than or equal to that z-score.
Steps to Find a Z-Score Using the Standard Normal Table
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Determine the Probability (Area): Identify the probability associated with the value you're interested in. This probability represents the area under the standard normal curve to the left of the z-score you're trying to find.
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Locate the Probability in the Table: Look for the probability (or the closest probability) within the body of the Z-table. Z-tables usually have z-scores listed on the left column and top row.
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Find the Corresponding Z-Score: Once you've found the probability, trace back to the left column to find the integer part and the first decimal place of the z-score. Then, trace upwards to the top row to find the second decimal place of the z-score.
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Combine the Values: Combine the values from the left column and the top row to get the z-score that corresponds to the probability you identified.
Example
Let's say you want to find the z-score associated with a probability of 0.95. This means you want to find the z-score that has 95% of the data falling below it.
- Probability: P = 0.95
- Locate in the Table: Look for 0.95 in the body of the standard normal distribution table. You might not find exactly 0.95; in that case, find the closest value. Often, you'll find values like 0.9495 and 0.9505 near each other. Select the closest one. In this example, 0.9505 is a common value.
- Find the Z-Score: The value 0.9505 is found at the intersection of the row 1.6 and the column 0.05.
- Combine: Therefore, the z-score is 1.6 + 0.05 = 1.65.
So, the z-score corresponding to a probability of 0.95 is approximately 1.65.
Important Considerations
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Left-Tailed vs. Right-Tailed Probabilities: The standard normal table usually gives the area to the left of the z-score. If you have a right-tailed probability (the area to the right of the z-score), subtract it from 1 to get the left-tailed probability before using the table. For example, if the probability to the right is 0.10, the probability to the left is 1 - 0.10 = 0.90.
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Negative Z-Scores: Some tables include negative z-scores. If you're looking for a z-score associated with a probability less than 0.5, you'll likely find it in the negative z-score section.
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Interpolation: If the probability you're looking for falls exactly between two values in the table, you can interpolate to get a more accurate z-score. This involves calculating a weighted average of the two z-scores corresponding to the probabilities bracketing your target probability. However, in many practical situations, simply choosing the closer probability is sufficient.
By understanding the structure of the standard normal distribution table and practicing these steps, you can effectively use it to find z-scores corresponding to given probabilities.
