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