Unit 3: Variation and the Normal Curve Review: Standard Units z-score (std units): z = ( x x ) / the number of s above averages above average (if negative, below average) Ex: Data 3, 3, 5, 6, 7, 9: x = 5.5 differences: -2.5, -2.5, -.5, .5, 1.5, 3.5 = RMS of differences 2.15 z -1.17, -1.17, -.23, .23, .70, 1.63 NOT normally distributed Ex: A list of 100 numbers, already in standard units, begins -5.8, -4.3, 6.1, .2, 10.2, -3.7. Is something wrong? They seem large -- remember, 3 away from , which is 3 in std units, is very rare Can we check? Well, = 0, = 1, so sum of their squares should be 1 = 100/100 But (-5.8)2 + (4.3)2 + (6.1)2 + ... is adding up to more than 100 fast In fact, (10.2)2 alone is more than 100 So yes, they are too big to be in std units Normal table z Area(%) z Area(%) z Area(%) z Area(%) z Area(%) 0.0 0.0 0.9 63.19 1.8 92.81 2.7
99.953 4.4 99.9989 0.85 60.47 1.75 91.99 2.65 99.2 3.55 99.961 4.45 99.9991 Normal approx: Ex 1 Weights in the population of a city follow the normal curve, with w = 140, = 30. About what % of pop weighs over 185? In std units, 185 is (185-140)/30 = 1.5. Normal table says % > 1.5 or < -1.5 is (10086.64)% = 13.36%. We only want right half: 13.36%/2 = 6.68% Much too accurate; this is only approximation: 6.7%, or even 7% Normal approx: Ex 2 Scores on a college entrance exam follow normal curve (odd!), with x = 120 and = 40. (a) About what score is the 80th %ile? (b) About what is the IQR? In normal table, we need z that gives percent in center, not 80%, but (80 - (100-80))% = 60%, which is z = .85. So 80th %ile of scores is [undoing std units] 120 + .85(40) 154 (b) We need z so that 50% of the data is between z and -z, and thats above averages z = .70. So the 3rd quartile is 120 + 40(.70), the 1st is 120 + 40(-.70), and their difference is the IQR, 2(40(.70)) = 56 Normal approx: Ex 3 Data following the normal curve has avg 80 and std dev 10. (a) What is the 15th %ile? (b) What is the 83rd %ile? (c) What % of data is between 85 and 95? (d) What % of data is between 60 and 90?
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