Chapter 5: z-Scores (a) (b) (c) = 82 = 70 = 70 = 12 = 12 = 3 - PDF document

Chapter 5: z-Scores (a) (b) (c) µ = 82 µ = 70 µ = 70 σ = 12 σ = 12 σ = 3 X = 76 is X = 76 is X = 76 is far slightly below slightly above above average average average 12 12 3 82 70 70 x = 76 x = 76 x = 76 Definition of z-score • A z-score specifies the precise location of each x-value within a distribution. The sign of the z-score (+ or - ) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and µ. 1

σ X µ z -2 -1 0 +1 +2 Intelligence Quota - IQ Scores µ = 100 σ = 16 σ X 68 84 100 116 132 z -2 -1 0 +1 +2 µ = 40 σ = 7 σ X 26 33 40 47 54 z -2 -1 0 +1 +2 2

SAT Scores µ = 500 σ = 100 σ X 300 400 600 700 500 z -2 -1 0 +1 +2 Average male height (in inches) for American Males (ages 20-29) – taken from Wikipedia 2014 µ = 70 σ = 2.7 σ X 64.6 67.3 70 72.7 75.4 z -2 -1 0 +1 +2 Average female height (in inches) for American Females (ages 20-29) – taken from Wikipedia 2014 µ = 64.5 σ = 2.4 σ X 62.1 66.9 69.3 59.7 64.5 z -2 -1 0 +1 +2 3

Example 5.2 • A distribution of exam scores has a mean (µ) of 50 and a standard deviation ( σ ) of 8. x − µ z = σ s = 4 s s 60 64 68 µ X 66 4

Example 5.5 • A distribution has a mean of µ = 40 and a standard deviation of s = 6. To get the raw score from the z-score: x = µ + z σ If we transform every score in a distribution by assigning a z-score, new distribution: 1. Same shape as original distribution 2. Mean for the new distribution will be zero 3. The standard deviation will be equal to 1 5

X 80 90 100 110 120 µ σ z -2 -1 0 +1 +2 µ A small population 0, 6, 5, 2, 3, 2 N=6 x x - µ (x - µ) 2 ∑ x SS 0 0 - 3 = -3 9 µ = σ = N N = 18 6 6 - 3 = +3 9 ( x − µ ) 2 ∑ = 6 6 5 5 - 3 = +2 4 = 3 24 = ( x − µ ) 2 = SS 6 ∑ 2 2 - 3 = -1 1 = 24 = 4 3 3 - 3 = 0 0 = 2 2 2 - 3 = -1 1 (a) frequency 2 s 1 X 0 1 2 3 4 5 6 µ (b) frequency 2 s 1 z -1.5 -1.0 -0.5 0 +0.5 +1.0 +1.5 µ 6

Let ’ s transform every raw score z = x − µ into a z-score using: σ z = 0 − 3 x = 0 = -1.5 2 z = 6 − 3 x = 6 = +1.5 2 z = 5 − 3 x = 5 = +1.0 2 z = 2 − 3 x = 2 = -0.5 2 z = 3 − 3 x = 3 = 0 2 z = 2 − 3 x = 2 = -0.5 2 z ∑ Mean of z-score µ z = N distribution : = ( − 1.5) + (1.5) + (1.0) + ( − 0.5) + (0) + ( − 0.5) 6 = 0 SS z ( x − µ z ) 2 ∑ Standard deviation: σ z = = N N z z - µ z (z - µ z ) 2 -1.5 -1.5 - 0 = -1.5 2.25 +1.5 +1.5 - 0 = +1.5 2.25 +1.0 +1.0 - 0 = +1.0 1.00 SS 6 σ = = 6 = 1 = 1 -0.5 -0.5 - 0 = -0.5 0.25 N 0 0 - 0 = 0 0 -0.5 -0.5 - 0 = -0.5 0.25 ( z − µ z ) 2 6.00 = ∑ What do we use z-scores for? • Can compare performance on two different scales (e.g. compare your score on the ACT to your score on SAT) by converting scores to z scores and comparing z scores • Can convert a distribution of scores with a specific mean and standard deviation to completely new distribution with a new mean and standard deviation 7

Comparing test scores on two different scales Meghan’s Semester Test Results • Psychology Exam: score of 60 • Biology Exam: score of 56 • Which was her better score, relative to the others in each class? To answer the question convert her scores to standard scores and compare—explain your answer fully in terms of z scores and standard deviations 8

Psychology Biology 10 4 X X 50 48 52 µ µ X = 56 X = 60 Converting Distributions of Scores Original Distribution Standardized Distribution 14 10 X X 57 50 µ µ Maria Joe Maria Joe X = 43 X = 64 X = 40 X = 55 z = -1.00 z = +0.50 z = -1.00 z = +0.50 9

Correlating Two Distributions of Scores with z-scores Distribution of Distribution of adult heights adult weights (in inches) (in pounds) s = 4 s = 16 µ = 68 µ = 140 Person A Person B Height = 72 inches Weight = 156 pounds 10