1. 1 Standard Normal Distribution Curve Standard Score

2. 2 Class Objective After this class, you will be able to 1.Calculate standard scores (Z-scores) 2.Convert standard and raw scores 3.Use standardized scores to compare data

3. Homework and Reading Check Assignment: – Chapter 2 – Exercise 2.98 and 2.103 Reading: – Chapter 2 – p. 47-49 3

4. Suggested Answer – 2.98 4

5. Suggested Answer – 2.103 5

6. Construct a Normal Curve in terms of the mean and standard deviation 6 Review

7. 7 Given: 1. SAT Math scores: Normally Distributed 2. Population Mean: 500 3. Population Standard Deviation: 100 Haydee scored 600. What does the score 600 tell you? Explain You have 3 minutes

8. Interpretation 8 Haydee is at the “84 th percentile That is: Haydee’s score is 84% higher than those who took the test OR Haydee score is ONE standard deviation above the mean (standard score)

9. Standard Score / Z-Score Definition – Observations/data expressed in standard deviations above or below the mean of a distribution 9

10. Now You try it! Given: 1. SAT Math scores: Normally Distributed 2. Mean: 500 3. Standard Deviation: 100 John scored 300. Question: What does the score 300 tell you? Explain You have 3 minutes. 10

11. Interpretation 11

12. Standard Score The standard score for any observation/datum is Standard Score = Observation/datum – mean standard deviation Example: Observation = 120 Mean: 100 Standard Deviation = 40 Standard Score = 120-100 = 0.5 40 Interpretation: The observation is 0.5 standard deviation above the mean. (12

13. Now you try it The standard score for any observation/datum is Standard Score = Observation/datum – mean standard deviation Example: Observation = 80 Mean: 100 Standard Deviation = 40 Standard Score = __________ Interpretation: The observation is _______________ You have 2 minutes (13

14. SAT VS ACT Scores Given: 1. Both SAT and ACT Scores are normally Distributed 2. For SAT, the mean is 500 and the standard deviation is 100 3. For ACT, the mean is 18 and the standard deviation is 6 Jeff scored 600 on the SAT Math Eric scored 21 on the ACT Math Question: Who has the higher score? 14

15. Jeff’s raw score = 600 – Z-score = 600-500 = 1 – 100 Eric’s raw score = 21 – Z-score = 21-18 = 0.5 – 6 Since Jeff’s z-score is 1 standard deviation above the mean and Eric’s z-score is only 0.5 above the mean, Jeff has a higher score. 15

16. More on the standard score Raw Score in Normal Curve 16

17. Now, you try it! Chris and Trent are in 2 different sections of the same course in Three Rivers Community College The scores on the Mid-term exam are normally distributed Chris’ section – Mean = 64 Standard deviation = 8 Trent’s section – Mean = 72 Standard deviation = 16 Question: Chris’ score is 74 and Trent’s score is 82. – 1. Sketch the normal distribution curves of the scores of Chris’ and Trent’s sections – 2. Who performs better (with respect to the other students in their own sections)? And why? 17

18. More on the standard score Raw Score in Normal Curve 18

19. More on the standard score Standard Score in Standard Normal Distribution Curve 19

20. Standard Normal Distribution Curve Properties 1.A normal distribution 2.Mean = 0 3.Standard deviation = 1 20 20

21. Standard Normal Distribution Curve VS Normal Distribution Curve 21

22. Z-score back to raw score Given: 1. Both SAT and ACT Scores are normally Distributed 2. For SAT, the mean is 500 and the standard deviation is 100 3. For ACT, the mean is 18 and the standard deviation is 6 Jeff has a z-score of 1 on the SAT Math Eric has a z-score of 0.5 on the ACT Math Question: What are their raw scores? 22

23. IF Convert raw score to Z-score Standard Score = Observation (Raw Score)– mean standard deviation THEN Convert z-score to raw score Raw Score =Standard Score x Standard Deviation + mean Jeff raw score = 1 X 100 + 500 = 600 Eric raw score = 0.5 X 6 + 18 = 21 23

24. 24 Standardized z-Scores Standardized score or z-score: Example: Mean resting pulse rate for adult men is 70 beats per minute (bpm), standard deviation is 8 bpm. The standardized score for a resting pulse rate of 80: A pulse rate of 80 is 1.25 standard deviations above the mean pulse rate for adult men.

25. 25 The Empirical Rule Restated – Standard Normal Distribution Curve For bell-shaped data, About 68% of values have z-scores between –1 and +1. About 95% of values have z-scores between –2 and +2. About 99.7% of values have z-scores between –3 and +3.

26. Homework Assignment: – Standard Score Homework Assignment (Ex 13.13, 13.14 – p.305) Reading: – Chapter 2 – p. 47-49 26