Z-Scores

Find the mean and standard deviation of the following Yr 12 Verbal Reasoning scores: 100 100 100 99 96 101 90 99 107 107 108 112 97 104 104 116 85 119 98 116 84 96 101 104 105 103 80 84 93 93 89 87 150 121 120 117 83 96 104 104 102 111 92 81 95 101 79 96 65 90

Mean = 99.68 Population Standard = Deviation 13.61 x=99.68 -1SD +1SD How many in between 87 113 35 or 70%

Mean x x±1S.D. 68% of all scores in a normal dist n lie within 1 S.D. of the mean

Mean x x±2S.D. 95% of all scores in a normal dist n lie within 2 S.D. of the mean

Mean x x±3S.D. 99.7% of all scores in a normal dist n lie within 3 S.D. of the mean

Example: The Yr 12 Science test results were normally distributed with a mean of 45% and a standard deviation of ±5. What percentage of scores fell between 35 and 55? 35 = mean – 2 SD 55 = mean + 2 SD 95% of scores fell between 35 and 55

Example: The thickness of sheet metal at a mill is normally distributed with x=2mm and σ n =0.1mm. What percentage of sheets will have a thickness exceeding 2.2mm? 2.0 95% 5% outside this band 2.2 1.8 2.2mm is 2SD above mean 2 ½% > 2.2mm

The lifetimes of D-size batteries is normally distributed x=150 hours and σ n =7hours. 300 batteries are taken off the production line. How many should have a life below 143 hours? 150 157 143 68% 32% outside this band 143hrs is 1SD below the mean 16% < 143hrs 16% of 300 = 48 batteries

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 190 and 210? 68%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 200 and 210? ½ of 68% 34%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 180 and 220? 95%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 180 and 200? ½ of 95% 47½%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 180 and 210? 47½% 34% 81 ½ %

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores are greater than 180? 47½% 50% 97½%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores are greater than 210? 68% ½ of 32% = 16% > 210

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores are less than 190? 68% ½ of 32% = 16% < 190

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores are less than 230? 99.7% ½ of 0.3% = 0.15% 0.15% 99.85%

Example: In the normal dist n below x=200 and σ n =10. 200 210 220 230 190 180 170 What percentage of scores lie between 180 and 190? 68% 95% Half of the difference between 95% and 68% = 13.5%

COMPARISON OF SCORES MATHS ENGLISH x=47 σ n =8 x=59 σ n =13 47 55 63 71 23 31 39 59 72 85 98 20 33 46 Raw Mark = 63 Raw Mark = 63 Maths is better as he is more standard deviations away from the mean Maths is better as he is 2 standard deviations away from the mean compared to < 1

COMPARISON OF SCORES GEOG LATIN x=56 σ n =4 x=38 σ n =7 56 60 64 68 44 48 52 38 45 52 59 17 24 31 Raw Mark = 52 Latin is better as he is more standard deviations away from the mean Latin is better as he is 2 standard deviations away from the mean compared to 1 for Geog Raw Mark = 60

COMPARISON OF SCORES LEGAL BUSINESS x=45 σ n =11 x=53 σ n =9 45 56 67 78 12 23 34 53 62 71 80 26 35 44 Raw Mark = 71 Business is better as his score relative to the mean is +2 compared to +1 for Legal Raw Mark = 56

COMPARISON OF SCORES HISTORY SCIENCE x=59 σ n =8 x=43 σ n =12 59 67 75 83 35 43 51 43 55 67 79 7 19 31 Raw Mark = 31 Science is better as his score relative to the mean is -1 compared to -1½ for History Raw Mark = 47

COMPARISON OF SCORES WITHOUT DIAGRAMS Origami score = -1SD Bonsai score = -2SD 44 6 56 Remedial Bonsai 35 10 45 Advanced Origami Raw Score Standard Deviation Mean Subject

COMPARISON OF SCORES WITHOUT DIAGRAMS RE score = +2SD Economics score = +2SD Both equally good 77 14 49 Religious Education 72 6 60 Economics Raw Score Standard Deviation Mean Subject

COMPARISON OF SCORES WITHOUT DIAGRAMS Tullamore score = -3SD Lithgow score = -2SD 55 4 67 Knowing The Sights of Tullamore 37 8 53 Eating out in Lithgow on a $2 Budget Raw Score Standard Deviation Mean Subject

COMPARISON OF SCORES WITHOUT DIAGRAMS Cropping score = +0.7SD Welding score = +0.6SD 75 10 68 Remedial Crop Planting 78 5 75 Advanced Welding Raw Score Standard Deviation Mean Subject

Cropping score = +0.7SD Welding score = +0.6SD These are called Z-scores They can be calculated using the formula 75 10 68 Remedial Crop Planting 78 5 75 Advanced Welding Raw Score Standard Deviation Mean Subject

Cropping score = +0.7SD Welding score = +0.6SD These are called Z-scores They can be calculated using the formula z = x x s 78 75 5 = +0.6 Advanced Welding 75 10 68 Remedial Crop Planting 78 5 75 Advanced Welding Raw Score Standard Deviation Mean Subject

The z-score formula translated says Z-score = raw score minus average standard deviation

Find the z-score for Crop Planting Z-score = raw score minus average standard deviation 75 68 10 = +0.7 7 75 10 68 Remedial Crop Planting 78 5 75 Advanced Welding Raw Score Standard Deviation Mean Subject

Practice Billy scores 54% in a test for remedial origami. The mean for the class was 60% with a standard deviation of ±8%. Calculate his z-score. z = x x s 54 8 60 = -0.5

Practice Abdul scores 68% in a test for advanced basket weaving. The mean for the class was 58% with a standard deviation of ±16%. Calculate his z-score. z = x x s 68 16 58 = +0.625 The plus/minus signs are necessary! Why? They indicate above/below the mean

Z-Scores

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Z-Scores