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Stats: Z Scores

Stats: Z Scores. Think About It:. Two competitors in a decathlon tie in each of the first eight events. In the ninth event, the high jump, one clears the bar 1 inch higher than the other. Then in the 1500 meter run, the other one runs 5 seconds faster. So who wins?.

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Stats: Z Scores

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  1. Stats: Z Scores

  2. Think About It: Two competitors in a decathlon tie in each of the first eight events. In the ninth event, the high jump, one clears the bar 1 inch higher than the other. Then in the 1500 meter run, the other one runs 5 seconds faster. So who wins?

  3. It boils down to…. Which is harder? To jump an inch higher or run 5 seconds faster? But how do we compare two fundamentally different activities? It’s like comparing apples and oranges!

  4. Standard Deviation This is where standard deviation comes into play. IF we know the mean performance (by world class athletes) in each event, and the standard deviation, we could compute how far each performances was from the mean in “standard deviation units”

  5. “Standard Deviation Units” Z Scores:

  6. So….. • The goal is to determine how far away something is from the mean– from there we can compare “apples” and “oranges.”

  7. Example One: You need to consider positive/negative. In the case of Dash, smaller number is better! With Jump/Put a larger number is better

  8. Let’s Get the Z Score for each

  9. Let’s Get the Z Score for each

  10. Overall Z Score So who won??!?!!? Who had the most impressive accomplishment and what activity was it in?

  11. Example 2: • Your Stats teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 80 on test 2. Which score will be dropped?

  12. Example 2 Continued • However, your teacher announces that she is going to grade “on a curve.” She standardized the scores in order to decide which is the lower one. If the mean on the first test is 88 with a standard deviation of 4 and the mean of the second was a 75 with a standard deviation of 5…… • A) Which score will be dropped?

  13. Example Three: • Many colleges require applicants to submit scores on standardized tests. However, there are two tests- SAT and ACT. • Suppose you want to go to a college that has no minimum score requirement, but claims the middle 50% of their students have SAT scores between 1530 and 1850. But you want to take the ACT…. So how do you know how well you have to do to be in the top 25% of the applicants?

  14. We standardize the variable so that we can compare! • We know: • SAT Mean: 1500 ACT Mean: 20.8 • SAT SD: 250 ACT SD: 4.8

  15. To be in the top quarter of applicants, you would need a 1850 or higher on the SAT. What is that z score?

  16. To be in the top quarter of applicants, you would need a 1850 or higher on the SAT. What is that z score? Z= (1850-1500)/250 = 1.40 So an SAT score of 180 is 1.40 standard deviations away from the mean of all the test takers.

  17. So…. • What score would 1.4 standard deviations away from the mean of the ACT be?

  18. So…. • What score would 1.4 standard deviations away from the mean of the ACT be? • 1.4 = (y- 20.8)/ 4.8 = 27.52. • So you need to score a 27.52 on the ACT in order to be in the top 25% of applicants!

  19. Practice • Try a few problems on your own :D • As always call me over if you have a question!

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