5.7 - Finding Probabilities using a Standard Normal Table

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When finding the probability associated with a score on a normal distribution it may be necessary to first convert the observation to a z score in order to use the z table to find a probability. Recall from Lesson 2 the formula for computing the z-score for an individual observation: 

z Score

\[z=\frac{x - \overline{x}}{s}\]

z = z score
x = original individual score
\(\overline{x}\) = mean of the original distribution
s = standard deviation of the original distribution

This formula can also be written using population parameters: \(z=\frac{x-\mu}{\sigma}\)

We will be using Table A in Appendix A of the Agresti, Franklin, and Klingenberg textbook. Table A in the textbook gives normal curve cumulative probabilities for standardized scores. This is also known as a z table. Row labels of Table A give possible z-scores up to one decimal place. The column labels give the second decimal place of the z-score. The cumulative probability for a value equals the cumulative probability for that value's z-score.



The examples on the following pages will walk you through examples of finding the probability less than, greater than, or between two values on the normal distribution. Remember, when finding the probability associated with an observation that is on a scale other than the standard normal distirbution (i.e., \(\mu=0\) and \(\sigma=1\)), you must first translate the score to a z score before using the table.