http://faculty.wwu.edu/~donovat/ps366/
Review
• Chapter 2, p. 56
– SPSS MARITAL
– How would you describe where most students in
the sample were raised?
– What percent of the sample is divorced?
– What percent of the sample is married?
– What percent would you describe as currently
being single?
Chapter 5 Homework
• # 7, p. 171-72 (in slides last week)
• #9 p. 172-73
• #12 p. 174-75
Projects
• 1) Perceptions of Western Washington Univ.
• 2) Presidential campaign
– Which candidates mobilize?
• 3) Opinion on national issues
– Which issues of greatest concern?, Why?
• Guns, Immigration, House GOP, etc.
Review Friday’s Lab
• How do we measure a country’s level of
development?
– Define the concept
Review Friday’s Lab
• How do we measure a country’s level of
development?
Some measures:
– Human Development Index
– GDP
– ??
• Describe the graph
– Range, standard deviation
– Mean, median, mode
• Which country at center of distribution?
• Which countries at the extremes?
GDP
HDI vs GDP
• What differences?
• Median countries?
• Shape of distribution
• Correlated at .79
Friday, review
• 3rd factor that measures ‘development’
• Discuss: which measure is best / most valid?
Why?
Chapter 6: Normal Distribution
• Normal curve
– Theoretical, not an empirical distribution
– Mean = median = mode
– Constant proportions of area under normal curve
– Standard deviation = fixed relationship between
distance from mean and area under the curve
Std Dev & Normal Curve
Std Dev & Normal Curve
Std Dev & Normal Curve
Std Dev & Normal Curve
Normal Curve and z-scores
• Difference between and observation and the
mean can be expressed in standard scores
• Z scores
Normal curve and z scores
• Calculate z-score
Z=
observed score - mean
_____________________
Standard deviation
Normal curve and z scores
• Calculate z-score
Z=
Y – Y “Hat”
_____________________
Standard deviation
Z scores & normal distribution
• Where is a country with an HDI of .75?
– mean = .696
– sd = .186
– Z = (.75-.696 ) / .186 = .06/.186 = .32
– 0.32 deviations beyond the mean
Z scores & normal distribution
• What is the raw score for a country with a z
score of 1.5 on HDI
– Y = Y“hat” + Z(std dev.)
– Y= .696 + (1.5*.186) = .696 + .279 = .975
– so, a country with HDI at .975 = 1.5 standard
deviations beyond the mean
Standard Normal Distribution
• Appendix B in text, p. 480
Z scores & Normal curve
• Standard normal distribution
– Normal distribution represented by z scores
Normal curve and z scores
Example:
What proportion of countries would we expect
to find between the mean and 1.45 std dev. (if
normal distribution?)
What proportion below the mean?
What proportion between mean and Z = +1.45
• http://www.mathsisfun.com/data/standardnormal-distribution-table.html
Normal curve and z-scores
• 1,200 students in stats class, 1983-1993
•
•
•
•
Mean
Median
Mode
Std. deviation
70.07
70
70
10.27
Translate scores into Zs
• Score of 40: (40-70.07)/10.27 = -2.93
• Score of 70: (70-70.07) /10.27 = -0.01
• Score of 90: (90-70.07) / 10.27 = 1.94
Z scores & normal distribution
• What % of students scored above 90?
– Z for 90 is 1.94
– Use standard normal table (p. 480)
.500 of total
area
-Z
B
MEAN
C
+ Z = 1.94
Check table to determine area of B; or area of C
.500 of total
area
-Z
B
MEAN
70
C
+ Z = 1.94
90
Check table to determine area of B
.500 of total
area
-Z
.4738
MEAN
70
.0262
+ Z = 1.94
90
.50 + .4738 = .9738. 97.38% scored lower than 90
.500 of total
area
-Z
.4738
MEAN
70
.0262
+ Z = 1.94
90
.50 + .4738 = .9738. 2.62% scored higher than 90
.500 of total
area
-Z
.4738
MEAN
70
.0262
+ Z = 1.94
90
Translate scores into Zs
• Score of 40: (40-70.07)/10.27 = -2.93
• Score of 70: (70-70.07) /10.27 = -0.01
• Score of 90: (90-70.07) / 10.27 = 1.94
Z score and normal curve
• What percent scored below 40 on the stats
exam?
– Z for 40 = -2.93
– use standard normal table
.500 of total
area
B
C
-Z
40
MEAN
70
+Z
Area C = 0.0017 of area; Area B = .4983 0.17% scored lower than 40
B: .4983
.500 of total
area
C:
0.0017
-Z
40
MEAN
70
+Z
Z scores and normal curve
• Standard Normal Table expressed in
proportions
• Easily translated into percentages
– multiply by 100
• Easily translated into percentiles
Z scores and normal curve
• Find the percentile rank of a score of 85:
– Z = (score-mean) / std. deviation
– Z = (85-70.07) / 10.27 = 1.45
• Find the percentile rank of a score of 90
– Z = (score-mean) / std. deviation
– Z = (90-70.07) / 10.27 = 1.94
•
Score of 90 higher than 97.38% who took stats test
97.38th percentile
B
C
-Z
.500 + .4738 of total
.0262,
area = 97.38%
or 2.62%
MEAN
70
+ Z = 1.94
90
Score of 80 higher than 92.65% who took stats test
92.65th percentile
B
-Z
C
.500 + .4265 of total
.0735,
area = 92.65%
or 7.35%
MEAN
70
+ Z = 1.45
85
Normal curve: percentiles
• OK, a score of 70 (mean = 70.07)
– Z = ??
• A score of 60 (below the mean, sd = 10.27)
– Z = ??
• positive or negative
• guess
Normal curve: percentiles
• A score of 70
– Z = -0.01
• A score of 60 (below the mean)
– Z = (score – mean) / st dev.
• (60-70.07) / 10.27 = - 0.98
Score of 60 higher than 16.35% who took stats test
16.35th percentile
C: .1635,
16.35%
Z = -0.98
60
B:
.3365;
33.65%
.
MEAN
70
+Z
Percentiles
• Range from 0 to 100
• Percent of observations above a point
• Example
– SAT math score in 82nd percentile
– SAT writing score in 88th percentile
– SAT vocabulary score in 75th percentile
Percentiles
• SAT scores
– mean 500
– st dev 100
• What % score above 625?
Percentiles
• SAT scores (p. 203 Q 8)
– mean 500
– st dev 100
• What % score above 625?
• Translate 625 into z score (625-500) / 100 = 1.25
• Use table: Z 1.25
– Area B .3944 (.5 + .3994 = .8944 = 89.44th percentile
– Area C .1056 (10.56% of scores higher)
Percentiles
• SAT scores
– mean 500
– st dev 100
• What percent between 400 and 600?
– Find Z for 400
– Find Z for 600
– Use table