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- 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. <ul><li>Shaquille O’Neal standard height </li></ul><ul><li>z = 78 – 69.9 z = 2.89 </li></ul><ul><li>2.8 </li></ul><ul><li>Lisa Leslie standard height </li></ul><ul><li>z = 70 – 63.7 z = 2.33 </li></ul><ul><li>2.7 </li></ul>
- 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 <ul><li>99.81% of the cases are below Shaquille O’ Neil </li></ul><ul><li>.19% of the cases are above Shaquille O’ Neil </li></ul><ul><li>z score height of Shaquille is 2.89 </li></ul><ul><li>Appendix B – Area in Larger portion=.9981 </li></ul><ul><li>- Area in smaller portion=.0019 </li></ul><ul><li>From the mean Shaquille is 41.81% of the cases below </li></ul><ul><li>Appendix B – Area from mean=.4981 </li></ul>
- 7. <ul><li>There are 19.9962 cases below Shaquille </li></ul><ul><li>There are 0.038 cases above Shaquille </li></ul><ul><li>20 (.9981) = 19.9962 </li></ul><ul><li>20(.0019) = .038 </li></ul>Areas of the Normal Curve
- 8. Problem #1 <ul><li>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. </li></ul><ul><li>What is his standard score? </li></ul><ul><li>How much percent of cases is below him? Above him? </li></ul><ul><li>How many cases are below him? </li></ul>
- 9. Illustration for problem #1 1.78 – position of the student 96.16%=43 cases 3.84%=4 cases
- 10. <ul><li>A student A got a standard score of .25 and student B got a standard score of 1.5. </li></ul><ul><li>What is the percentage of students in between them? </li></ul>Problem #2
- 11. Illustration for problem #2 .25 .0978 1.5 .4332 area away from the mean 33.54%
- 12. <ul><li>Student C got a standard score of – 0.75 and student D got standard score of +1.75 among 50 students. </li></ul><ul><li>What is the percentage of cases between them? How many cases? </li></ul><ul><li>What is the percentage of the rest of the cases? How many cases are there? </li></ul>Problem #3
- 13. Illustration for problem #3 -0.75 .2734 -1.75 .4599 73.33% 22.56% 4%
- 14. <ul><li>A student got a navy score (T score) of 65 from an aptitude test. There were 150 students who took the test. </li></ul><ul><li>What is his standard score? </li></ul><ul><li>How many cases are below the student? </li></ul><ul><li>How many cases are above the student? </li></ul>Problem #4
- 15. <ul><li>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. </li></ul><ul><li>What is the score of the student? </li></ul><ul><li>What is the corresponding t-score of the student? </li></ul>Problem #5

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