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 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? IQ scores: μ IQ = 100 σ IQ = 15 SAT scores: μ SATV = 500 σ SATV = 100

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

Comparing Scores using z transformations Matt’s scores on three tests in Stats: Test 1Test 2Test M X = X – M X = s= z i = ( )/12.5 ( )/2.1 (35-32)/1.8 =

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

Positive skewed Distribution Of Raw scores Positive skewed Distribution Of z-scores

Bimodal, Negatively Skewed, Asymmetric Distribution Of Raw Scores Bimodal, Negatively Skewed, Asymmetric Distribution Of z-Scores

Normal Distribution Of Raw Scores Normal Distribution Of z Scores

A VERY VERY VERY Special Distribution: Standard Unit- Normal Distribution 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 ANY normal distribution can be transformed into a unit-normal distribution by transforming the raw scores to z scores:

Unit-Normal Distributions (z UN )

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 Find z = 1.87 The area between z UN = 0 and z UN = 1.87 is.9693

Using Table A to determine Percentile Rank z UN = 1.87 =.0307 Percentile rank = = 96.93%.9693

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