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Chapter 4

Chapter 4. Translating to and from Z scores, the standard error of the mean and confidence intervals. Welcome Back!. NEXT. Where we have been: Z scores. The proportion above or below the score. The percentile rank equivalent. The proportion of scores between two Z scores.

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Chapter 4

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

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

  3. Concepts behind Z scores • Z scores represent standard deviations above and below the mean. • Positive Z scores are scores higher than the mean. Negative Z scores are scores lower than the mean. • If you know the mean and standard deviation of a population, then you can always convert a raw score to a Z score. • If you know a Z score, the Z table will show you the proportion of the population between the mean and that Z score.

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

  5. 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 *  )

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

  7. 2100 Standard deviations 3 2 1 0 1 2 3 Production F r e q u e n c y Z score = ( 2100 - 2180) / 50 = -80 / 50 = -1.60 units 2030 2330 2080 2280 2130 2180 2230 What is the Z score for a daily production of 2100, given a mean of 2180 units and a standard deviation of 50 units?

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

  9. Concepts behind Scale Scores • Scale scores are raw scores expressed in a standardized way. • The most basic scale score is the Z score itself, with mu = 0.00 and sigma = 1.00. • Raw scores can be converted to Z scores, which in turn can be converted to other scale scores. • And Scale scores can be converted to Z scores, that in turn can be converted to raw scores.

  10. 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

  11. Standard deviations 470 3 2 1 0 1 2 3 For example: To solve the problem below, convert an SAT Score to a Z score, then use the Z table as usual. Proportion mu to Z for Z score of -.30 = .1179 F r e q u e n c y Z score = ( 470 - 500) / 100 = -30 / 100 Proportion below score = .5000 - .1179 = . 3821 = 38.21% = -0.30 score 200 800 300 700 400 500 600 What percentage of test takers obtain a verbal score of 470 or less, given a mean of 500 and a standard deviation of 100?

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

  13. Z score = (85 - 100) / 15 = -15 / 15 = -1.00 Z score = (115 - 100) / 15 = 15 / 15 = 1.00 Proportion = .3413 + .3413 = .6826 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? 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

  14. Z score = (95 - 100) / 15 = -5 / 15 = -.33 Z score = (115 - 100) / 15 = 5 / 15 = .33 Proportion = .1293 + .1293 = .2586 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? 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

  15. X  (X-)  (X-)/  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 Convert an IQ score of 100 to a percentile. An IQ of 100 is right at the mean and that’s the 50th percentile.

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

  17. Convert Z scores to IQ scores: Individual scale scores get rounded to nearest integer. Z  (Z*) IQ= + (Z * ) +2.67 +2.67 15 40.05 +2.67 15 +2.67 15 40.05 100 +2.67 15 40.05 100 140 -.060 15 -9.00 100 91

  18. Tougher problems – like online quiz or midterm

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

  20. Z  (Z*) SAT= + (Z * ) Percentile to scale score If someone scores at the 58th percentile on the SAT-verbal, what SAT-verbal score did he receive? 58th Percentile is above the mean. This will be a positive Z score. The mean is the 50th percentile. So the 58th 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. Look at Column 2 of the Z table on page 54. Closest Z score for area of .0800 is 0.20 0.20 100 20 500 520

  21. Slightly tougher –below the mean

  22. Z  (Z*) SAT= + (Z * ) Percentile to scale score If someone scores at the 38th percentile on the SAT-verbal, what SAT-verbal score did he receive? 38th percentile is below the mean. This will be a negative Z score. The mean is the 50th percentile. So the 38th 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. Look at Column 2 of the Z table on page 54. Closest Z score for area of .1200 is 0.31. Z is negative -0.31 100 -31 500 469

  23. Raw  (X- ) Scale Scale Scale score (raw) (raw)  Z   score Double translations 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? 35 25.00 10.00 6.00 1.67 6.00 1.67 100 15 125 Z score = 10.00 / 6.00 = 1.67 IQ score = 100 + (1.67 * 15) = 125

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