COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman

1. COURSE: JUST 3900 TIPS FOR APLIA Developed By: Ethan Cooper (Lead Tutor) John Lohman Michael Mattocks Aubrey Urwick Chapter 5: z-Scores

2. Key Terms and Formulas: Don’t Forget Notecards

3. Describing z-Scores • Question 1: Identify the z-score value corresponding to each of the following locations in a distribution. • Below the mean by 3 standard deviations. • Above the mean by 1/4 standard deviations. • Below the mean by 2.50 standard deviations. • Question 2: Describe the location in the distribution for each of the following z-scores. • z = - 1.50 • z = 0.25 • z = - 3.50 • z = 1.75

4. Describing z-Scores • Question 1 Answer: • z = -3.00 • z = 0.25 • z = -2.50 • Question 2 Answer: • Below the mean by 1.50 standard deviations. • Above the mean by ¼ standard deviations. • Below the mean by 3.50 standard deviations. • Above the mean by 1.75 standard deviations.

5. Describing z-Scores • The numerator in our z-score formula () describes the difference between X and µ. Therefore, if a question asks you to calculate the z-score for a score that is above the mean by 4 pointsand has a standard deviation of σ = 2, you cannot calculate (X - µ) - - i.e., you cannot find the original values for X or µ and calculate the difference between the two. In this case, it has already been provided for you because the question tells you the distance between X and µ (X - µ) = 4. Thus, your formula should read z = 4/2, which comes out to be z = 2.00.

6. Transforming X-Values into z-Scores • Question 3: For a distribution of µ = 40 and σ = 12, find the z-score for each of the following scores. • X = 36 • X = 46 • X = 56 • Question 4: For a population with µ = 30 and σ = 8, find the z-score for each of the following scores. • X = 32 • X = 26 • X = 42

7. Transforming X-Values into z-Scores

8. Using z-Scores to Compare Different Populations • Question 5: A distribution of English exam scores has µ = 70 and σ = 4. A distribution of history exam scores has µ = 60 and σ = 20. For which exam would a score of X = 78 have a higher standing? Explain your answer.

9. Using z-Scores to Compare Different Populations • Question 5 Answer: • For the English exam, X = 78 corresponds to z = 2.00, which is a higher standing than z = 0.90 for the history exam. Remember that 95% of all scores fall between ± 2.00. Thus, a score of +2.00 means that over 95% of the class scored below 78 on the English exam.

10. Using z-Scores to Compare Different Populations • Question 6: A distribution of English exam scores has µ = 50 and σ = 12. A distribution of history exam scores has µ = 58 and σ = 4. For which exam would a score of X = 62 have a higher standing? Explain your answer.

11. Using z-Scores to Compare Different Populations • Question 6 Answer: • The score X = 62 corresponds to z = 1.00 in both distributions. The score has exactly the same standing for both exams.

12. z-Scores and Standardized Scores • Question 7: A population of scores has µ = 73 and σ = 8. If the distribution is standardized to create a new distribution with µ = 100 and σ = 20, what are the new values for each of the following scores from the original distribution? • X = 65 • X = 71 • X = 81 • X = 83

13. z-Scores and Standardized Scores

14. z-Scores and Standardized Scores • Question 8: A population with a mean of µ = 44 and a standard deviation of σ = 6 is standardized to create a new distribution with µ = 50 and σ = 10. • What is the new standardized value for a score of X = 47 from the original distribution? • One individual has a new standardized score of X = 65. What was this person’s score in the original distribution?

15. z-Scores and Standardized Scores X = 47 z = 0.50 X = 55 Old Distribution 32 38 50 56 44 z-Score Distribution New Standardized Distribution

16. z-Scores and Standardized Scores X = 53 z = 1.50 X = 65 Old Distribution 44 32 38 56 50 z-Score Distribution New Standardized Distribution

17. Measure of Relative Location and Detecting Outliers • Question 9: A sample has a mean of M = 30 and a standard deviation of s = 8. • Would a score of X = 36 be considered a central score or an extreme score in the sample? • If the standard deviation were s = 2, would X = 36 be central or extreme?

18. Measure of Relative Location and Detecting Outliers • Question 9 Answer: • X = 36 is not an extreme score because it is within two standard deviations of the mean. • In this case, X = 36 is an extreme score because it is more than two standard deviations above the mean.

19. WARNING!!! • The book defines an extreme score as being more than TWOstandard deviations away from the mean. However, Apliadefines extreme scores as being more than THREE standard deviations from the mean. • When using Aplia, use the THREEdefinition of standard deviation for extreme scores. • On in class exercises and on the test, use the TWOdefinition of standard deviation for extreme scores.

20. Frequently Asked Questions FAQs • How do I find the z-scores from a raw set of scores? • X = 11, 0, 2, 9, 9, 5 • Find the mean: • Find SS: = 96

21. Frequently Asked Questions FAQs • Find σ2: • Find σ: • Find z-score for each X: