Chapter 2 Describing Variables 2.7 Standardized Scores (Z Scores)

Mean and Standard Deviation Two basic descriptive statistics for the distribution of a continuous variable with N observations: Mean (central tendency): Standard deviation (dispersion):

EDUC Years of Schooling

Standardized Scores (Z Scores) Transform each score in the frequency distribution of a continuous Y-variable into a standard or Z score: (1) from Yi subtract the mean (= deviation di) (2) divide case i deviation by the standard deviation Z scores can be positive or negative, indicating the number of standard deviation units that Yi lies above or below the distribution mean

Yi scores into Zi scores 5 10 15 20 25 30 35 40 45 Yi Zi -2.0 -1.0 +.67 = 30 = 15

Change some EDUC Y scores into Z scores: 1. What is the Z score for a person with16 years? 2. Find the Z score for a person with 8 years:

Now change some EDUC Z scores into Y scores: What years of EDUC separate two persons at the upper & lower limits of the range Z = ± 2.2 standard deviations? Find upper limit of range, Z = +2.20 std. dev. units : Find lower limit of range, Z = -2.20 std. dev. units :

Find the Z scores for these ungrouped data: First calculate mean & standard deviation, then the Z scores Y1: 6 - _____________________ Y2: 4 - _____________________ Y3: 5 - _____________________ Y4: 4 - _____________________ Y5: 3 - _____________________ Y6: 8 - _____________________ ZY=3 = ZY=6 = ________ ZY=4 = ________ ZY=8 = ________

Find Z scores for grouped data on Pres Find Z scores for grouped data on Pres. Obama’s handling of the global war on terror, where Mean = 3.30 and N = 158 Rating Yi fi (di)2(fi) Poor 1 10 ___________________________ Fair 2 20 ___________________________ Good 3 40 ___________________________ Excellent 4 88 ___________________________

Calculate the mean for ungrouped data 1 2 2 3 3 3 4 4 5 5 6 5 7 6 8 7 9 7 10 8

Calculate variance & std. dev. for 10 scores 2 - = _______ _______ 3 - = _______ _______ 4 - = _______ _______ 5 - = _______ _______ 6 - = _______ _______ 7 - = _______ _______ 8 - = _______ _______

Find All the Z Scores (2 - ) / _______ = _______ (2 - ) / _______ = _______ (3 - ) / _______ = _______ (4 - ) / _______ = _______ (5 - ) / _______ = _______ (6 - ) / _______ = _______ (7 - ) / _______ = _______ (8 - ) / _______ = _______