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Describing a Score’s Position within a Distribution Lesson 5 Science & Probability Learn about populations by studying samples Introduction of error Drawing conclusions Cannot make states with certainty Probability statements Use of normal distribution Can calculate probability of a result Natural variables ≈ normal ~ Probability: Definitions Probability(P) of an event (A) Assuming each outcome equally likely P(A) = # outcomes Classified as A total # possible outcomes P(drawing ♥) = P(7 of ♥) = P(15 of ♥) = P(♥ or ♦ or ♣ or ♠) = ~ Standard Normal Distribution AKA Unit Normal Distribution Parameters m = 0, s = 1 z scores Or standard scores Distance & direction from m in units of s ~ Standard Normal Distribution f -2 -1 0 Z scores (s) 1 2 Other Standardized Distributions Many natural variables ≈ normal Standardized distributions Have defined or set parameters IQ: m = 100, s = 15 ACT: m = 18, s = 6 SAT: m = 500, s = 100 ~ IQ Scores m = 100 s = 15 f z scores -2 -1 0 1 2 IQ 70 85 100 115 130 The Normal Distribution & Probability Area under curve = frequency Area under curve represents all data Proportion (p) including all scores = 1 p for any area under curve can be calculated Proportion = probability that a score(s) is in distribution Table A.1, pg 797 ~ Probability of obtaining IQ score below the median? Greater than 115? Percentile rank of 70? Use z scores. f 0.5 0.5 IQ 70 85 100 115 130 Total area under curve = 1.0 Using z scores AKA standard scores distance from mean in units of s Uses Determining probabilities Percentile rank or scores Compare scores from different distributions *Technically must use parameters text uses sample statistics: X and s ~ z Score Equation z = X-m s Using z scores Distance and direction relative to mean Standard Normal Distribution m = 0, s = 1 Answer questions by 1st finding z score What proportion of population have IQ scores greater than 115? What is the percentile rank for IQ score of 70? What percentage of people have IQ scores between 70 and 115? z score for 115? z Xi m s z score for 70? f IQ 70 85 100 IQ Score 115 130 Handy Numbers Standard Normal Distribution z scores Proportions of distribution i.e., area under curve, table A.1 3 handy proportions Same for all normal distributions Between z = 0 and ±1 Between z = 1 and 2 (also -1 & -2) Beyond z = ±2 (area in tails) ~ Areas Under Normal Curves f .34 .34 .02 .02 .14 -2 .14 -1 0 Z scores (s) 1 2 What % of students scored b/n 18 and 24? % greater than 30? % less than 30? ACT Scores m = 18 s=6 f z scores ACT -2 6 -1 12 0 1 2 18 24 30 Comparing Scores from Different Distributions How to compare ACT to SAT? Use z scores 1. Raw ACT score z score X m z s 2. Use z score to compute Raw SAT score X zs m Areas Under Normal Curves f .34 .34 .02 .02 .14 -2 .14 -1 0 1 standard deviations 2 Percentile Rank & Percentile Percentile rank % of scores ≤ a particular score (Xi) th percentile: 84% of IQ scores ≤ 115 84 Percentile Raw score (Xi) associated with a particular percentile rank th percentile IQ score of 100 is the 50 Use z scores & table to determine ~ IQ Scores f .34 .34 .02 .02 .14 .14 IQ 70 85 100 115 130 z scores -2 -1 0 1 2 84th 98th percentile rank 2d 16th 50th