Lecture 8: z-Score and the Lecture 8: z-Score and the Normal DistributionNormal Distribution

2011, 10, 62011, 10, 6

Learning ObjectivesLearning Objectives

Review standard deviationReview standard deviation Characteristics of standard deviationCharacteristics of standard deviation What is z-score? **What is z-score? ** The Normal Distribution**The Normal Distribution**

Standard DeviationStandard Deviation

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

StudentStudent Score Score (N=10)(N=10)

11 9696

9494

22 44

22 9595 11 11

33 9595 11 11

44 9494 00 00

55 9494 00 00

66 9494 00 00

77 9494 00 00

88 9393 -1-1 11

99 9393 -1-1 11

1010 9292 -2-2 44

TotalTotal ---- ---- 00 Sum of Squares (Sum of Squares (SSSS) = 12) = 12

Mean Mean Variance (Variance (22) = 12/10 =1.2) = 12/10 =1.2

Square Root Square Root Std. Dev. (Std. Dev. () = = 1.1) = = 1.1

2)( XX XX X

Lab 1

2.1

87

88

89

90

91

92

93

94

95

96

97

98

99

10

0

10

1

Lab2

0.0

0.5

1.0

1.5

2.0

Fre

qu

en

cy

Characteristics of Standard Characteristics of Standard DeviationDeviation

Change/add/delete a given score, then the standard Change/add/delete a given score, then the standard deviation will change.deviation will change.

Add/subtract a constant to each score, then the standard Add/subtract a constant to each score, then the standard deviation will NOT change.deviation will NOT change.

102

SAT®SAT®

National National

NormNorm = 20.9; = 20.9; = 4.9 = 4.9

National National NormNorm

= 508; = 508; = 112 = 112

30 – 20.9 = + 9.130 – 20.9 = + 9.1

3030

86.19.4

9.203011

X

Z

620 – 508 = 112620 – 508 = 112

1112

50862011

X

Z

620620

You take a SAT test (620) and a ACT test You take a SAT test (620) and a ACT test (30), which one do you want to send to (30), which one do you want to send to

college?college?

z-Score (Standard Score)z-Score (Standard Score)

A number that indicates how many standard A number that indicates how many standard deviation a raw score is from the mean of a deviation a raw score is from the mean of a distributiondistribution

For a population: For a population:

For a sample:For a sample:

X

z

xs

XXz

Compute a z-ScoreCompute a z-Score

X=87X=87SSxx=6.32=6.32

Compute a Raw ScoreCompute a Raw Score

If your z-score for PSY 138 exam If your z-score for PSY 138 exam (mean = (mean = 87, Std. Dev. = 6.32) 87, Std. Dev. = 6.32) is 1.5 (that is, your is 1.5 (that is, your score is 1.5 standard deviation higher than score is 1.5 standard deviation higher than the class mean), what is you raw score? the class mean), what is you raw score?

For sample: For sample:

For population: For population: z = +1.5z = +1.5

The Normal Distribution The Normal Distribution (The z-Distribution)(The z-Distribution)

Shape: Symmetrical and unimodalShape: Symmetrical and unimodal Mean: μ = 0 Mean: μ = 0 The 68% -- 95% -- 99.7% ruleThe 68% -- 95% -- 99.7% rule

Application: SAT Verbal score is a Application: SAT Verbal score is a normal distributionnormal distribution

Mean = 508; Std. Dev. = 112Mean = 508; Std. Dev. = 112 508 508 112 = 396 ~ 620 (68%) 112 = 396 ~ 620 (68%) What proportion or percentage scored at or above 508? 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 below 396? What proportion or percentage scored at or above 396?What proportion or percentage scored at or above 396?

+1+1-1-1 =0=0

68%68%

396396 620620508508

Lecture RecapLecture Recap

Review standard deviationReview standard deviation How to compute z score?How to compute z score? How to compute raw score?How to compute raw score? The Normal DistributionThe Normal Distribution

– Shape: Symmetrical and unimodalShape: Symmetrical and unimodal– Mean: μ = 0 Mean: μ = 0 – The 68% - 95% - 99.7% ruleThe 68% - 95% - 99.7% rule