Z-ScoresHonors Advanced Algebra

Presentation 1-6

VocabularyZ-Score – the number of standard deviations from the mean. Used to tabulate the area under a normal curve.

- Sample meanμ – Population meanS – Sample standard deviationσ – Population standard deviation

Z-ScoresA z-score is used to determine how many

standard deviations away a data point is from the mean.

Based on a z-score, we can determine probabilities of achieving certain goals.

Formula for calculating a z-score:

Example 1Find the z-score of a data point of 21 when the mean is 24 and the standard deviation is 2

Example 2Find the z-score of a data point of 58 when the mean is 50 and the standard deviation is 3.5

Analyzing Z-ScoresZ-scores are interpreted from a z-score table.The table tells the area of the curve to the

left of a z-score.

Example 3If a data point has a z-score of .5, what is the probability of getting lower than that score?

Example 4If a data point has a z-score of -1, what is the probability of getting higher than that score?

Example 5What is the probability of getting a score

between the z-score of -1 and the z-score of .5?

Example 6Scores on a test are normally distributed with a mean of 75 and a standard deviation of 8.1. Estimate the probability that a randomly

selected student scored less than an 87.

2. Estimate the probability that a randomly selected student scored more than a 79.

Example 6 (cont’d)Scores on a test are normally distributed with a mean of 75 and a standard deviation of 8.3. Estimate the probability that a randomly selected student scored between 71 and 75.

Example 7Heights of females are normally distributed with a mean of 5’4” with a standard deviation of 2.7 inches.1. What is the probability that a randomly

selected female will be shorter than 5’?

2. What is the probability that a randomly selected female will be taller than 6’?

Example 7 (cont’d)Heights of females are normally distributed with a mean of 5’4” with a standard deviation of 2.7 inches.3. What is the probability that a randomly

selected female will be between 5’2” and 5’8”?

4. What is the probability that a randomly selected female will be either shorter than 5’2” or taller than 5’8”?

Example 7 (cont’d)Heights of females are normally distributed with a mean of 5’4” inches with a standard deviation of 2.7 inches.5. If a female’s height gave her a z-score of 2.3, how tall was she?

6. If a female wants to be taller than 99.3% of other females, how tall does she have to be?