Chapter 6 The Normal Distribution

Transcription

1 Chapter 6 The Normal PSY 395 Oswald Outline s and area The normal distribution The standard normal distribution Setting probable limits on a score/observation Measures related to 2 s and Area The idea of a pie chart as representing Where are the Bad Guys? See next slide Bar charts say See subsequent slide 61% 19% 11% 9% Prison Jail Percentage Parole Probation 3 4

2 Bar Chart of Bad Guys A more continuous distribution P e r c e n t a g e Frequency 10 0 Prison Jail Parole Probation Std. Dev = Mean = 49.1 N = Behavior Problem Score Behavior problem score 6 Normal Normal We study the normal distribution because many naturally occurring events yield (unimodal, symmetric) Mean Median Mode 7 8

3 Normal Percent Area Under the Normal The normal distribution is a (what are the endpoints?) Normal here does not mean ; it is a for this specific mathematical function for the curve The mean/median/mode are If we split the distribution in half, 50% of the scores of the sample lie (or median, or mode), and 50% of the scores lie (or median, or mode) 9 10 Percent Area Under the Normal Proportions of Area Under the Normal 50% of scores 50% of scores The entire area under the normal curve can be considered to be equal to a proportion of Speaking in proportions,.500 of the scores lie in the bottom half (below the mean, and Mean, Median, Mode 11 12

4 Proportions of Area Under the Normal.5000 of scores.5000 of scores f X is the Y is the Formula for the Normal 1 2π ( X µ ) ( X ) = e π and e are Mean, Median, Mode Cont. Histogram w/ Normal data, X = 0, s = 1 -scores µ = X X Std. Dev = 1.00 Mean = -.01 N = scores (or standard scores) are a way of expressing a raw score s Subtract from X Divide by This is a (every data point gets changed ) = 15 16

5 -scores A -score rather than a raw score is a better way to let you know than a raw score -scores If the average of the class grades is 71 and the (population) standard deviation is 5.2, then the - score for a grade of 85 would be: µ = 71 = 5.2 A student gets a 75/100 on a test Is this a high score or a low score? 17 = X µ = = = scores -scores This means that a mark of 85% is actually The student would have done as compared to the rest of the class represents the (which would be 71 in this example) represents a score (which would be less than 71) represents a score (which would be greater than 71) 19 20

6 -scores By definition, for any set of scores: the sum of -scores will equal ( Z = 0.00) ( µ Z = 0.00) and a standard deviation ( = 1.00) Z -scores A -score expresses the position of the raw score above or below the mean in For instance = +1 means that the raw score is 1 standard deviation = means that the raw score is 2 standard deviations Reflections on µ µ = X The mean and standard deviation are always given; they are -scores only reflect the data points position to the mean (and nothing more) You re now considering the data as a, as you re not looking to infer to ) This means for standard deviation rather than the sample formula whenever you calculate (i.e., ) -score: Another example You take two exams in Math and English, getting the following scores: Math µ 70% (class = 55%, = 10%) English µ 60% (class = 50%, = 5%) Which test score represents the better performance? 23 24

7 -score (Example cont.) Z-score Example Illustration Math mark: = (70-55)/10 = English mark: = (60-50)/5 = µ = X Math: =1.50 English: = Mean = The Answer -score: Solving for X Because: = is greater than = +1.50, the English grade of 60% reflects a The -score formula can be rearranged to solve for the raw score X: than does the Math grade of 70% even though µ = X X = ()( ) + µ 27 28

8 Z-scores: Solving for X This formula is used when you know the - score of a data point, and X Example E.g., if a class midterm exam has µ = 65 and = 5, what exam mark has a -score value of 1.25? X = (1.25)(5) + 65 = ()( ) + µ = = So, a person whose test is obtained a score of 71.25% scores -score problems ask you to solve for X or solve for The Standard Normal Transform all X values into scores, which (always) have mean = 0 and standard deviation = 1 Define the area under the curve to be

9 Tables of We use tables to find A sample table is on the next slide The following slide illustrates areas under the distribution Normal Cutoff at Area = = Area = Table Normal Mean to Larger Portion Smaller Portion Cutoff at = Area = Area =

10 Using the Tables Based on the you have, Define is, so the table negative values of Example: Area/proportion of people between = +1.5 and = -1.0 See next slide Calculating areas Area between mean and +1.5 = Area between mean and -1.0 = Sum equals Therefore about 77% of the observations would be Converting Back to X Assume µ = 30 and = 5 77% of the distribution X µ = Therefore X = µ + X = = 25 X = = Probable Limits X = µ + Our last example has µ = 30 and = 5 We want This means (5% outside the middle, half on both sides) Well, the table gives the for of the normal distribution X = µ + X = = 39.8 X = = Cont.

11 Probable Limits--cont. Measures Related to We have just shown that in this example that lies between 20.2 and 39.8 Therefore the probability is will lie between 20.2 and 39.8 Standard score Percentile score The point below which a T scores Scores with and a standard deviation of fall Review Questions Review Questions--cont. What do we gain? How is a standard normal distribution different from any other normal distribution? How do we convert X to? How do we use the tables of? Of what use are? If we know your test score, how do we? What is a T score and why do we care? 43 Cont. 44