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

3. Where we have been: Z scores zThe proportion above or below the score. zThe percentile rank equivalent. zThe proportion of scores between two Z scores. zThe expected frequency of scores between two Z scores If you know the proportion from the mean to the score, then you can easily calculate:

4. Concepts behind Z scores zZ scores represent standard deviations above and below the mean. zPositive Z scores are scores higher than the mean. Negative Z scores are scores lower than 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, the Z table will show you the proportion of the population between the mean and that Z score.

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

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

7. 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

8. 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

9. If you know a Z score, you can determine theoretical relative frequencies and expected frequencies using the Z table. zYou often start with raw or scale scores and have to convert them to Z scores. zScale scores are public relations versions of Z scores, with preset means and standard deviations.

10. 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.

11. 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

12. 200 800 470 300 700 For example: To solve the problem below, convert an SAT Score to a Z score, then use the Z table as usual. 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 Z score of -.30=.1179 Proportion below score =.5000 -.1179 =. 3821 = 38.21%

13. 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

14. Convert to IQ scores to Z scores to find the proportion of scores between two IQ scores. IQ scores have mu = 100 and sigma = 15. What proportion of the scores 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 scores 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

15. NOTICE: 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

16. Percentile equivalents of scale scores: first 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. Going the other way – Z scores to scale scores Remember: Score =  + ( Z *  )

18. Convert Z scores to IQ scores: Individual scale scores get rounded to nearest integer. 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

19. Tougher problems – like online quiz or midterm

20. If someone scores at the 58 th percentile on the verbal part of the SAT, what is your best estimate of her SAT score?

21. Percentile 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 *  )

22. Slightly tougher –below the mean

23. Percentile to scale score If someone scores at the 38th 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.1200 is 0.31. Z is negative 38 th percentile is below the mean. This will be a negative Z score. The mean is the 50 th percentile. So the 38 th percentile is 12% or a proportion of.1200 below mu. So we have to find the Z score that gives us a proportion of.1200 of the scores between mu and Z. - 0.31 100 -31 500 469 Z  (Z*  )  SAT=  + (Z *  )

24. 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