THE NORMAL DISTRIBUTION AND ZSCORES Areas Under the

THE NORMAL DISTRIBUTION AND ZSCORES Areas Under the Curve

Let’s Practice! x: { 3, 8, 1} Find s: x 3 8 1 m (x-m)2 4 -1 1 4 4 16 4 -3 9 S(x-m)2 = SS S(x-m)2 = 26 √ S(x-m)2 N =√(26/3) = 2. 94 OR SS = Sx 2 2 (Sx) __ N x 3 8 1 x 2 9 64 1 Sx = 12 Sx 2 = 74 74 - (144/3) = 26 Then √(26/3) = 2. 94

The Philosophy of Statistics & Standard Deviation N=50

The Philosophy of Statistics & Standard Deviation. 24 Proportion . 20. 16. 12. 08. 04

Standard Deviation and Distribution Shape IQ

Example: IQs of Sample of Psychologists ID IQ 1 128 2 155 3 135 x= 140. 33 4 134 s= 27. 91 5 144 6 101 7 167 8 198 9 94 10 128 11 155 12 145 With some simple calculation we find: x-x z= s z(144) = [ 144 – 140. 33]/ 27. 91 = +0. 13, “normal” z(198) = [ 198 – 140. 33]/ 27. 91 = +2. 07, abnormally high z(94) = [ 94 – 140. 33]/ 27. 91 = -1. 66, low side of normal

Forward and reverse transforms “forward” population sample Raw score Z-score Z- score Raw Score x =m+zs z= x-m s z= x-x s x =x+zs Example: If population μ = 120 and σ =20 Find the raw score associated with a z-score of 2. 5 x = 120 + 2. 5(20) x = 120 + 50 x = 170 “reverse”

Why are z-scores important? • z-scores can be used to describe how normal/abnormal scores within a distribution are • With a normal distribution, there are certain relationships between z-scores and the proportion of scores contained in the distribution that are ALWAYS true. 1. The entire distribution contains 100% of the scores 2. 68% of the scores are contained within 1 standard deviation below and above the mean 3. 95% of the scores are contained within 2 standard deviations below and above the mean

m= 128 s = 32 Z-score 95% 68% -4 -3 -2 -1 0 1 2 3 4 • What percentage of scores are contained between 96 and 160? • What percentage of scores are between 128 and 160? • If I have a total of 200 scores, how many of them are less than 128?

But how do we find areas associated with z-scores that are not simply 0, 1, or 2? Table A in appendix D contains the areas under the normal curve indexed by Z-score. m= 128 s = 32 Z-score What proportion of people got a z score of 1. 5 or higher? -4 -3 -2 -1 0 1 From these tables you can determine the number of individuals on either side of any z-score. 2 3 4

z-score 1. 5

Examples of AREA C 2. 3 -1. 7

What percentage of people have a z-score of 0 or greater? 50% What percentage of people have a z-score of 1 or greater? 15. 87% What percentage of people have a z-score of -2. 5 or less? . 62% What percentage of people have a z-score of 2. 3 or greater? 1. 07% What percentage of people have a z-score of -1. 7 or less? 4. 46%

Examples of AREA B

What percentage of people have a z-score between 0 and 1? 34. 13% What percentage of people have a z-score between 0 and 2. 3? 48. 93% What percentage of people have a z-score between 0 and -2. 4? 49. 18% What percentage of people have a z-score between 0 and 1. 27? 39. 80% What percentage of people have a z-score between 0 and 1. 79? 46. 33% What percentage of people have a z-score between 0 and -3. 24? 49. 94%

Areas which require a COMBINATION of z-scores What percentage of people have a z-score of 1 or less? 84. 13%

Areas which require a COMBINATION of z-scores What percentage of people have a z-score between -1 and 2. 3? 84. 13%

Areas which require a COMBINATION of z-scores What percentage of people have a z-score of 1 or less? 84. 13%

m= 128 s = 32 Raw Score 0 32 64 96 128 160 192 224 256 What percentage of people have a z-score of -1. 7 or less? 4. 46% What percentage of people have a score of 73. 6 or less? 4. 46%? What z-score is required for someone to be in the bottom 4. 46%? -1. 7 What score is required for someone to be in the bottom 4. 46%? 128 + (-1. 7)32 128 - 54. 4 73. 6 or below

What z-score is required for someone to be in the top 25%? . 68 What z-score is required for someone to be in the top 5%? 1. 65 What z-score is required for someone to be in the bottom 10%? -1. 29 What z-score is required for someone to be in the bottom 70%? . 52 What z-score is required for someone to be in the top 50%? 0 What z-score is required for someone to be in the bottom 30%? -. 53

m= 128 s = 32 What percentage of scores fall between the mean and a score of 132? Here, we must first convert this raw score to a z-score in order to be able to use what we know about the normal distribution. (132 -128)/32 = 0. 125, or rounded, 0. 13. Area B in the z-table indicates that the area contained between the mean and a z -score of. 13 is. 0517, which is 5. 17%

m= 128 s = 32 What percentage of scores fall between a z-score of -1 and 1. 5? If we refer to the illustration above, it will require two separate areas added together in order to obtain the total area: Area B for a z-score of -1: . 3413 Area B for a z-score of 1. 5: . 4332 Added together, we get. 7745, or 77. 45%

m= 128 s = 32 What percentage of scores fall between a z-score of 1. 2 and 2. 4? Notice that this area is not directly defined in the z-table. Again, we must use two different areas to come up with the area we need. This time, however, we will use subtraction. Area B for a z-score of 2. 4: . 4918 Area B for a z-score of 1. 2: . 3849 When we subtract, we get. 1069, which is 10. 69%

m= 128 s = 32 If my population has 200 people in it, how many people have an IQ below a 65? First, we must convert 65 into a z-score: (65 -128)/32 = -1. 96875, rounded = -1. 97 Since we want the proportion BELOW -1. 97, we are looking for Area C of a zscore of 1. 97 (remember, the distribution is symmetrical!) : . 0244 = 2. 44% Last step: What is 2. 44% of 200? 200(. 0244) = 4. 88

m= 128 s = 32 What IQ score would I need to have in order to make it to the top 5%? Since we’re interested in the ‘top’ or the high end of the distribution, we want to find an Area C that is closest to. 0500, then find the z-score associated with it. The closest we can come is. 0495 (always better to go under). The z-score associated with this area is 1. 65. Let’s turn this z-score into a raw score: 128 + 1. 65(32) = 180. 8

A possible type of test question: 80 - 65 A class of 30 students takes a difficult statistics exam. The average grade turns out to be 65. Michael is a student in this class. His grade on the exam is 80. The following is known: 9. 80 SS = 2883. 2 Assuming that these 30 students make up the population of interest, what is the approximate number of people that did better than Michael on the exam? SS= 2883. 2 m = 65 N = 30 s= √ SS N = √ 2883. 2 30 = √ 96. 11 = 9. 80 z(80) = (80 -65)/9. 80 = 1. 53 Area C for a z score of 1. 53 =. 0630, so about 6. 3%, or 1. 89 people