A z-score (also called a standard score) tells you how many standard deviations a value is above or below the mean of a dataset.
Example 1: If the Z-score is 1.5. Find the probability that a randomly selected data point falls below this Z-score.
Solution:
To determine the probability that a randomly selected data point falls below the Z score, we can do the following.
Using a Z-score table or calculator, look for a Z-score of 1.5 and get a corresponding probability of about 0.9332. This means there is a 93.32% probability that the data point falls below a Z-score of 1.5 in the standard normal distribution.
Example 2: Find the probability that the Z score is greater than -1.2
Solution:
To determine the probability that the Z-score is greater than -1.2.
Using the Z-score table, find the cumulative probability associated with -1.2, which would be 0.1151. Subtract this value from 1 to find the probability of greater than -1.2:
1 − 0.1151 = 0.8849
Thus, the probability that the Z-score is greater than -1.2 is approximately 0.8849 or 88.49%.
Practice Questions
- A class of 100 students took a math test. The mean score is 75, with a standard deviation of 10. What is the Z-score of a student who scored 85 on the test?
- In applied physics, students measure the time it takes for a ball to fall from a certain height. It is 3 seconds with a standard deviation of 0.5 seconds. If a student measures a fall time of 2.2 seconds, what is the Z-score for this measurement?
- A company is conducting an employee compensation audit. The average salary is 50,000 with a standard deviation of 8,000. What is the Z-score of an employee with a salary of 56,000?
- A doctor is measuring the height of a child to compare it with a group of children of the same age. The height of this group is 120 cm and the standard deviation is 5 cm. If the child is 130 cm tall, what is the Z-score for this measurement?
- In a study of test anxiety among students, the average test anxiety score was 60, with a standard deviation of 10. If a student scores a test anxiety score of 75, what is the Z-score of this score?
