Lecture 8: z-Score and the Normal Distribution 2011, 10, 6

Learning Objectives  Review standard deviation  Characteristics of standard deviation  What is z-score? **  The Normal Distribution**

Standard Deviation  In essence, the standard deviation measures how far off all of the individuals in the distribution are from the mean of the distribution. Essentially, the average of the deviations. 

Student Score (N=10) Total----0 Sum of Squares (SS) = 12 Mean Mean Variance (  2 ) = 12/10 =1.2 Square Root Square Root Std. Dev. (  ) = = 1.1 Lab 1

Characteristics of Standard Deviation  Change/add/delete a given score, then the standard deviation will change.  Add/subtract a constant to each score, then the standard deviation will NOT change. 102

SAT® NationalNorm  = 20.9;  = 4.9 National NationalNorm  = 508;  = – 20.9 = – 508 = You take a SAT test (620) and a ACT test (30), which one do you want to send to college?

z-Score (Standard Score)  A number that indicates how many standard deviation a raw score is from the mean of a distribution  For a population:  For a sample:

Compute a z-Score

X=87 S x =6.32 Compute a Raw Score  If your z-score for PSY 138 exam (mean = 87, Std. Dev. = 6.32) is 1.5 (that is, your score is 1.5 standard deviation higher than the class mean), what is you raw score?  For sample:  For population: z = +1.5

The Normal Distribution (The z-Distribution)  Shape: Symmetrical and unimodal  Mean: μ = 0  The 68% -- 95% % rule

Application: SAT Verbal score is a normal distribution  Mean = 508; Std. Dev. = 112  508  112 = 396 ~ 620 (68%)  What proportion or percentage scored at or above 508?  What proportion or percentage scored at or below 396?  What proportion or percentage scored at or above 396? +1  -1   =0 68%

Lecture Recap  Review standard deviation  How to compute z score?  How to compute raw score?  The Normal Distribution –Shape: Symmetrical and unimodal –Mean: μ = 0 –The 68% - 95% % rule