2. Chapter 4 Translating to and from Z scores, the standard error of the mean and confidence intervals Welcome Back! NEXT

3. Concepts behind Z scores zZ scores represent standard deviations above and below the mean. zIf you know the mean and standard deviation of a population,then you can always convert a raw score to a Z score. zIf you know a Z score, then you can look up in the Z table the proportion of the population between the mean and score.

4. Z scores continued zThe proportion above or below the score. zThe percentile rank equivalent. zThe proportion of scores between two scores. If you know the proportion from the mean to the score, then you can easily calculate:

5. Definition Z = score - mean standard deviation If we know mu and sigma, any score can be translated into a Z score: = X -  

6. Definition Conversely, as long as you know mu and sigma, a Z score can be translated into any other type of score : Score =  + ( Z *  )

7. Raw scores to Z scores Given mu = 100 and sigma = 4.00, convert the following raw scores to Z scores. 106 100 6.00 4.00 1.50 87 100 -13.00 4.00 -3.25 95 100 -5.00 4.00 -1.25 X  (X-  )  (X-  )/ 

8. Calculating z scores Z = score - mean standard deviation What is the Z score for someone 6’ tall, if the mean is 5’8” and the standard deviation is 3 inches? Z = 6’ - 5’8” 3” = 72 - 68 3 = 4 3 = 1.33

9. 2100 2080 22802030 2330 Production FrequencyFrequency units 2180 What is the Z score for a daily production of 2100, given a mean of 2180 units and a standard deviation of 50 units? Z score = ( 2100 - 2180) / 50 3 2 1 0 1 2 3 Standard deviations = -80 / 50 = -1.60 22302130

10. 200 800 470 300 700 Verbal SAT Scores FrequencyFrequency score 500 What percentage of test takers obtain a verbal score of 470 or less, given a mean of 500 and a standard deviation of 100? Z score = ( 470 - 500) / 100 3 2 1 0 1 2 3 Standard deviations = -30 / 100 = -0.30 600400 Proportion mu to Z for.30 =.1179 Proportion below score =.5000 -.1179 =. 3821 = 38.21%

11. Convert to Z scores to find the proportion of scores between two raw scores. Given mu = 100 and sigma = 15, what proportion of the population falls between 85 and 115? Z score = (85 - 100) / 15 = -15 / 15 = -1.00 Z score = (115 - 100) / 15 = 15 / 15 = 1.00 Proportion =.3413 +.3413 =.6826 What proportion of the population falls between 95 and 110? Z score = (95 - 100) / 15 = -5 / 15 = -0.33 Z score = (110 - 100) / 15 = 10 / 15 = 0.67 Proportion =.2486 +.1293 =.3779

12. Equal sized intervals, close to and further from the mean: More scores close to the mean! Given mu = 100 and sigma = 15, what proportion of the population falls between 95 and 105? Z score = (95 - 100) / 15 = -5 / 15 = -.33 Z score = (115 - 100) / 15 = 5 / 15 =.33 Proportion =.1293 +.1293 =.2586 What proportion of the population falls between 105 and 115? Z score = (105 - 100) / 15 = 5 / 15 = 0.33 Z score = (115 - 100) / 15 = 105/ 15 = 1.00 Proportion =..3413 -.1293 =.2120

13. Concepts behind Scale Scores zScale scores are raw scores expressed in a standardized way. zThe most basic scale score is the Z score itself, with mu = 0.00 and sigma = 1.00. zRaw scores can be converted to Z scores, which in turn can be converted to other scale scores. zAnd Scale scores can be converted to Z scores, that in turn can be converted to raw scores.

14. You need to memorize these scale scores Z scores have been standardized so that they always have a mean of 0.00 and a standard deviation of 1.00. Other scales use other means and standard deviations. Examples: IQ -  =100;  = 15 SAT/GRE -  =500;  = 100 Normal scores -  =50;  = 10

15. Convert Z scores to IQ scores Z  (Z*  )  IQ=  + (Z *  ) +2.67 -.060 15 -9.00 100 91 +2.67 15 40.05 +2.67 15 +2.67 15 40.05 100 +2.67 15 40.05 100 140

16. Scale score translation: first always translate to Z scores Convert IQ scores of 120 & 80 to percentiles. 120 100 20.0 15 1.33 80 100 -20.0 15 -1.33 mu-Z =.4082,.5000 +.4082 =.9082 = 91st percentile, Similarly 80 =.5000 -.4082 = 9th percentile X  (X-  )  (X-  )/  Convert an IQ score of 100 to a percentile. An IQ of 100 is right at the mean and that’s the 50th percentile.

17. SAT to percentile – first transform to a Z scores If a person scores 592 on the SATs, what percentile is she at? Proportion mu to Z =.3212 592 500 92 100 0.92 SAT  (X-  )  (X-  )/  Percentile = (.5000 +.3212) * 100 = 82.12 = 82nd

18. Reverse the order: %tile to scale score If someone scores at the 58th percentile on the SAT-verbal, what SAT-verbal score did he receive? Look at Column 2 of the Z table on page 54. Closest Z score for area of.0800 is 0.20 58th Percentile is above the mean. This will be a positive Z score. The mean is the 50 th percentile. So the 58 th percentile is 8% or a proportion of.0800 above mu. So we have to find the Z score that gives us a proportion of.0800 of the scores between mu and Z. 0.20 100 20 500 520 Z  (Z*  )  SAT=  + (Z *  )

19. SAT / GRE scores - Examples How many people out of 400 can be expected to score between 550 and 650 on the SAT? Proportion difference =.4332 -.1915 =.2417 550 500 50 100 0.50 SAT  (X-  )  (X-  )/  650 500 150 100 1.50 Proportion mu to Z 0.50 =.1915 Proportion mu to Z 1.50 =.4332 Expected frequency =.2417 * 400 = 96.68 people

20. Normal scores - examples 1. Convert a normal score of 37 to a Z score. 37 50 -13.0 10 -1.30 X  (X-  )  (X-  )/  2. Convert a Z score of 0.00 to a normal score. 0.00 10 0 50 50 Z  (Z*  )   + (Z *  ) EASY

21. Midterm type problems: Double translations 35 25.00 10.00 6.00 1.67 On the verbal portion of the Wechsler IQ test, John scores 35 correct responses. The mean on this part of the IQ test is 25.00 and the standard deviation is 6.00. What is John’s verbal IQ score? Raw  (X-  ) Scale Scale Scale score (raw) (raw)  Z   score 6.00 1.67 100 15 125 IQ score = 100 + (1.67 * 15) = 125 Z score = 10.00 / 6.00 = 1.67

22. Double translations again 116 125.00 -9.00 12.00 -0.75 On the GRE-Advanced Psychology exam, there are 225 questions. The mean is 125.00 correct with a standard deviation of 12.00. Joan gets 116. What is her GRE score on this test? Raw  (X-  )  Scale Scale Scale score (raw) (raw) (raw) Z   score 500 100 425