A z-score, also known as a standard score, is a measure of how many standard deviations a particular data point is from the mean of a set of data. It is used to compare how different a particular data point is from the mean of the data set and to determine how likely it is to occur.

For example, consider a set of data representing the heights of a group of people. The mean height of this group is 5 feet 6 inches, and the standard deviation is 2 inches. If one person in this group is 6 feet tall, their z-score would be (72-66)/2 = 3. This means that this person’s height is 3 standard deviations above the mean, and is therefore quite unusual in comparison to the rest of the group.

On the other hand, if another person in this group is 5 feet 2 inches tall, their z-score would be (62-66)/2 = -2. This means that their height is 2 standard deviations below the mean, and is therefore not as unusual in comparison to the rest of the group.

In addition to comparing data points to the mean of a data set, z-scores can also be used to compare data from different sets. For example, consider two groups of students who took the same test. The mean score for the first group was 80, with a standard deviation of 10. The mean score for the second group was 70, with a standard deviation of 5. If a student from the first group scored a 90, their z-score would be (90-80)/10 = 1. If a student from the second group scored a 90, their z-score would be (90-70)/5 = 4. While both students scored the same absolute score, the student from the second group had a higher z-score, indicating that their score was more unusual in comparison to the mean of their group.

Z-scores can also be used to determine the likelihood of a particular data point occurring. For example, consider a data set with a mean of 50 and a standard deviation of 10. If a data point has a z-score of 1, it is 1 standard deviation above the mean, which means it is likely to occur approximately 68% of the time. If a data point has a z-score of 2, it is 2 standard deviations above the mean, which means it is likely to occur approximately 95% of the time. On the other hand, if a data point has a z-score of -1, it is 1 standard deviation below the mean, which means it is likely to occur approximately 32% of the time. If a data point has a z-score of -2, it is 2 standard deviations below the mean, which means it is likely to occur approximately 5% of the time.

Overall, z-scores are a useful tool for comparing data points to the mean of a data set and for determining the likelihood of a particular data point occurring. They allow for the standardization of data and provide a way to compare data from different sets or populations.