Chapter 4 z scores and Normal Distributions Computing

Chapter 4 z scores and Normal Distributions

Computing a z score Example: X = 400 μ = 500 σ = 100 what is z?

The score of 400 is -1. 00 standard deviations from the mean

Comparing/Combining with z scores Comparison - Joe has a measured IQ of 105, and received a 700 on the SAT Verbal, how do these scores compare? l IQ scores: μIQ = 100 σIQ = 15 l SAT scores: μSATV = 500 σSATV = 100 l

Comparing Scores using z transformations l These scores suggest that Joe’s SAT performance was better than would be expected by his general intellectual ability

Comparing Scores using z transformations l Matt’s scores on three tests in Stats: MX = X – MX = s = zi = = Test 1 Test 2 Test 3 31 22. 2 8. 8 12. 5 21 19. 5 1. 5 2. 1 35 32. 0 3. 0 1. 8 (31 -22. 2)/12. 5 (21 -19. 5)/2. 1 (35 -32)/1. 8 +0. 70 +0. 71 +1. 67

Back to Distributions l What if we took a distribution of raw scores and transformed all of them to z -scores?

Positive skewed Distribution Of z-scores Positive skewed Distribution Of Raw scores 5 4 3. 5 3 2. 5 2 1. 5 1 0. 5 0 scores z-scores

Bimodal, Negatively Skewed, Asymmetric Distribution Of Raw Scores Bimodal, Negatively Skewed, Asymmetric Distribution Of z-Scores 5 5 4 4 3 3 2 2 1 1 0 0 scores z-scores

Normal Distribution Of z Scores Normal Distribution Of Raw Scores 0. 45 0. 4 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 scores z-scores

A VERY Special Distribution: Standard Unit. Normal Distribution l l l A Normal Distribution of z-scores Popular member of the family where: μ = 0 and σ = 1 It is also known as – Unit-Normal Distribution or – The Gaussian – Often Symbolized “z. UN”

Transforming Normal Distributions l ANY normal distribution can be transformed into a unit-normal distribution by transforming the raw scores to z scores:

Unit-Normal Distributions (z. UN) . 02. 14 -3 -2 . 34 -1 . 34 0 . 14 1 2 3

Using Table A (and a z. UN score) to find a %tile Rank To find the corresponding percentile rank of a z = 1. 87, Table A from your text book is used l Find z = 1. 87 l The area between z. UN = 0 and z. UN = 1. 87 is. 9693 l

Using Table A to determine Percentile Rank . 9693. 0307 -3 -2 -1 0 1 z. UN = 1. 87 =. 0307 Percentile rank = 1 -. 0307 = 96. 93% 2 3

Procedure (in words) (raw score to z to %tile rank) l Transform raw score to z. UN (scores must be normally distributed) Look up the proportion (p) of scores between -∞ and the z. UN of interest l Multiply by 100 l