The document discusses z-scores and how they are used to standardize scores in a normal distribution. A z-score represents the number of standard deviations a data point is from the mean. It is calculated by subtracting the mean from the data value and dividing by the standard deviation. Examples are given to demonstrate how to calculate z-scores and use them to determine percentiles and the percentage of cases above or below a given score.

1. The z -score represents the number of standard deviations that a data value is from the mean. It is obtained by subtracting the mean from the data value and dividing this result by the standard deviation. The z-score is unitless with a mean of 0 and a standard deviation of 1.

2. Population Z - score Sample Z - score

3. EXAMPLE Using Z-Scores The mean height of males 20 years or older is 69.1 inches with a standard deviation of 2.8 inches. The mean height of females 20 years or older is 63.7 inches with a standard deviation of 2.7 inches. Data based on information obtained from National Health and Examination Survey. Who is relatively taller: Shaquille O’Neal whose height is 78 inches or Lisa Leslie whose height is 70 inches.

4. Shaquille O’Neal standard height z = 78 – 69.9 z = 2.89 2.8 Lisa Leslie standard height z = 70 – 63.7 z = 2.33 2.7

5. The maximum height is at the mean asymptotic – the tail never touches the base line and extends to infinity skewness is zero Kurtosis is mesokurtic Shaquille O’Neal’s height 99.81% of the cases are below Shaquille O’ Neil .19% of the cases are above Shaquille O’ Neil

6. Areas of the Normal Curve 99.81% of the cases are below Shaquille O’ Neil .19% of the cases are above Shaquille O’ Neil z score height of Shaquille is 2.89 Appendix B – Area in Larger portion=.9981 - Area in smaller portion=.0019 From the mean Shaquille is 41.81% of the cases below Appendix B – Area from mean=.4981

7. There are 19.9962 cases below Shaquille There are 0.038 cases above Shaquille 20 (.9981) = 19.9962 20(.0019) = .038 Areas of the Normal Curve

8. Problem #1 There are 45 students who took an achievement test with a mean of 51.4 and a standard deviation of 3.6. A student got a score of 45. What is his standard score? How much percent of cases is below him? Above him? How many cases are below him?

9. Illustration for problem #1 1.78 – position of the student 96.16%=43 cases 3.84%=4 cases

10. A student A got a standard score of .25 and student B got a standard score of 1.5. What is the percentage of students in between them? Problem #2

11. Illustration for problem #2 .25 .0978 1.5 .4332 area away from the mean 33.54%

12. Student C got a standard score of – 0.75 and student D got standard score of +1.75 among 50 students. What is the percentage of cases between them? How many cases? What is the percentage of the rest of the cases? How many cases are there? Problem #3

13. Illustration for problem #3 -0.75 .2734 -1.75 .4599 73.33% 22.56% 4%

14. A student got a navy score (T score) of 65 from an aptitude test. There were 150 students who took the test. What is his standard score? How many cases are below the student? How many cases are above the student? Problem #4

15. A student in a math test is in the 25 th percentile. The mean performance of the entire sample is 80 with a standard deviation of 5.3. What is the score of the student? What is the corresponding t-score of the student? Problem #5