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Z-Scores for Algebra I

Z-Scores for Algebra I. Descriptive Statistics. Mean. Mean = the average of the numbers. To find the mean of a set of numbers, you must add up the numbers then divide by the number of numbers. Example: 18 23 10 39 22 17 16 15 18+23+10+39+22+17+16+15 = 160 160. = 20. 8.

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Z-Scores for Algebra I

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  1. Z-Scores for Algebra I Descriptive Statistics

  2. Mean • Mean = the average of the numbers. To find the mean of a set of numbers, you must add up the numbers then divide by the number of numbers. • Example: 18 23 10 39 22 17 16 15 18+23+10+39+22+17+16+15 = 160 160 = 20 8

  3. Mean Practice • Find the mean of the following sets of numbers. • 30 15 40 20 80 60 50 34 20 56 82 78 30

  4. Mean Practice 20 + 30 + 15 + 40 + 20 + 80 + 60 + 50 = 320 320 34 + 20 + 56 +82 + 78 + 30 = 300 300 = 40 8 = 50 6

  5. How Close is Close? Z-Scores

  6. Z-score • Position of a data value relative to the mean. • Tells you how many standard deviations above or below the mean a particular data point is. z-score = describes the location of a data value within a distribution referred to as a standardized value Population Sample

  7. Z-score In order to calculate a z-score you must know: • a data value • the mean • the standard deviation

  8. Z-scores Here are 23 test scores from Mr. Barnes stat class. • 81 80 77 73 83 74 93 78 80 75 67 73 • 77 83 86 90 79 85 83 89 84 82 What is the mean score? What is the standard deviation?

  9. Z-scores Here are 23 test scores from Mr. Barnes stat class. • 81 80 77 73 83 74 93 78 80 75 67 73 • 77 83 86 90 79 85 83 89 84 82 What is the mean score? 80.5 What is the standard deviation? 6.1

  10. Z-scores Here are 23 test scores from Mr. Barnes stat class. • 81 80 77 73 83 74 93 78 80 75 67 73 • 77 83 86 90 79 85 83 89 84 82 The bold score is Michele’s. How did she perform relative to her classmates?

  11. Z-scores Here are 23 test scores from Mr. Barnes stat class. • 81 80 77 73 83 74 93 78 80 75 67 73 • 77 83 86 90 79 85 83 89 84 82 The bold score is Michele’s. How did she perform relative to her classmates? Michele’s score is “above average”, but how much above average is it?

  12. Z-scores If we convert Michele’s score to a standardized value, then we can determine how many standard deviations her score is away from the mean. • What we need: • Michele’s score • mean of test scores • standard deviation 86 80.5 86 – 80.5 6.1 = 0.90 Z= 6.1 Therefore, Michele’s standardized test score is 0.90. Nearly one standard deviation above the class mean.

  13. 62.2 68.3 74.4 80.5 86.6 92.7 98.8 Z-score Number Line Michele’s Score = 86 Michele’s z-score = .90 -3 -2 -1 0 1 2 3

  14. Calculate a z-score Consider this problem: The mean salary for math teachers in Big State is $45,000 per year with a standard deviation of $5,000. The mean salary of a Food Lion bagger is $21,000 with a standard deviation of $2,000.

  15. Calculate a z-score Who has the better salary relative to the mean? A Big State teacher making 63,000 or a grocery bagger making 30,000? or Grocery Bagger: 30,000 Teacher : 63,000 What is the interpretation of the two z-scores? Who has a better salary relative to the mean?

  16. Calculate a z-score Who has the better salary relative to the mean? A Big State teacher making 63,000 or a grocery bagger making 30,000? Grocery Bagger: 30,000 or Teacher : 63,000 What is the interpretation of the two z-scores? Both scores are above the mean. Who has a better salary relative to the mean? The grocery bagger.

  17. Sample Question for A.9 Which students’ heights have a z-score greater than 1? A) All of them B) Bill, Carrie, Ed and Gus C) Ed and Gus D) None of them

  18. Sample Question for A.9 Which students’ heights have a z-score greater than 1? Mean = 50 Standard Deviation = 5.3 A) All of them B) Bill, Carrie, Ed and Gus C) Ed and Gus D) None of them

  19. Sample Question for A.9 Which students have a z-score less than -2? A) All of them B) Dan and Andy C) Only Dan D) None of them

  20. Sample Question for A.9 Which students have a z-score less than -2? Mean = 50 Standard Deviation = 5.3 A) All of them B) Dan and Andy C) Only Dan D) None of them

  21. Sample Question for A.9 Which student’s height has a z-score of zero? A) Bill B) Carrie C) Frank D) None of them

  22. Sample Question for A.9 Which student’s height has a z-score of zero? A) Bill B) Carrie C) Frank D) None of them

  23. Sample Question for A.9 Given a data set with a mean of 125 and a standard deviation of 20, describe the z-score of a data value of 120? A) Less than -5 B) Between -5 and -1 C) Between -1 and 0 D) Greater than 0

  24. Sample Question for A.9 Given a data set with a mean of 125 and a standard deviation of 20, describe the z-score of a data value of 120? Mean = 125 Standard Deviation = 20 A) Less than -5 B) Between -5 and -1 C) Between -1 and 0 D) Greater than 0

  25. Sample Question for A.9 Given a data set with a mean of 30 and a standard deviation of 2.5, find the data value associated with a z-score of 2? A) 36 B) 35 C) 34.5 D) 32.5

  26. Sample Question for A.9 Given a data set with a mean of 30 and a standard deviation of 2.5, find the data value associated with a z-score of 2? Mean = 30 Standard Deviation = 2.5 A) 36 B) 35 C) 34.5 D) 32.5

  27. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 1) List the z-scores of students that were above the mean.

  28. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 1) List the z-scores of students that were above the mean. 1.5, 1.95, and 0.5

  29. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 2) If the mean of the exam is 80, did any of the students selected have an exam score of 80? Explain.

  30. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 2) If the mean of the exam is 80, did any of the students selected have an exam score of 80? Explain. One student with a z-score of 0.

  31. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 3) If the standard deviation of the exam was 5 and the mean is 80, what was the actual test score for the student having a z-score of 1.95?

  32. Sample Question for A.9 Suppose the test scores on the last exam in Algebra I are normally distributed. The z-scores for some of the students in the course were: 1.5, 0, -1.2, -2, 1.95, 0.5 3) If the standard deviation of the exam was 5 and the mean is 80, what was the actual test score for the student having a z-score of 1.95? 90

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