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

observed score - meanZ = _____________________

Standard deviation

Normal curve and z scores

• Calculate z-score

Y – Y “Hat”Z = _____________________

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/standard-normal-distribution-table.html

Normal curve and z-scores

• 1,200 students in stats class, 1983-1993

• Mean 70.07• Median 70• Mode 70• Std. deviation 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)

B C.500 of total area

-Z MEAN + Z = 1.94

B C.500 of total area

-Z MEAN + Z = 1.94 70 90

Check table to determine area of B; or area of C

.4738 .0262.500 of total area

-Z MEAN + Z = 1.94 70 90

Check table to determine area of B

.4738 .0262.500 of total area

-Z MEAN + Z = 1.94 70 90

.50 + .4738 = .9738. 97.38% scored lower than 90

.4738 .0262.500 of total area

-Z MEAN + Z = 1.94 70 90

.50 + .4738 = .9738. 2.62% scored higher than 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

B

C

.500 of total area

-Z MEAN + Z40 70

B: .4983

C:0.0017

.500 of total area

-Z MEAN + Z40 70

Area C = 0.0017 of area; Area B = .4983 0.17% scored lower than 40

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

•

.0262, or 2.62%

.500 + .4738 of total area = 97.38%

-Z MEAN + Z = 1.94 70 90

Score of 90 higher than 97.38% who took stats test97.38th percentile

B

C

.0735, or 7.35%

.500 + .4265 of total area = 92.65%

-Z MEAN + Z = 1.45 70 85

Score of 80 higher than 92.65% who took stats test92.65th percentile

B C

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

.

Z = -0.98 MEAN + Z 60 70

Score of 60 higher than 16.35% who took stats test16.35th percentile

B:.3365; 33.65%

C: .1635,16.35%

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