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