Statistical Analyses:

Chi-square test

Psych 250Winter 2013

Types of Measures / Variables

• Nominal / categorical– Gender, major, blood type, eye color

• Ordinal– Rank-order of favorite films; Likert scales?

• Interval / scale– Time, money, age, GPA

Variable Type Example Commonly-used Statistical

MethodNominal by Nominal blood type by

genderChi-square

Scale by Nominal GPA by gender

GPA by major

t-test

Analysis of Variance

Scale by Scale weight by heightGPA by SAT

RegressionCorrelation

Main Analysis Techniques

Question

Do men and women differ in the % that choose jail time vs. probation only?

Variable Type Example Commonly-used Statistical

MethodNominal by Nominal(categorical by categorical)

blood type by gender

Chi-square

Scale by Nominal GPA by gender

GPA by major

t-test

Analysis of Variance

Scale by Scale weight by heightGPA by SAT

RegressionCorrelation

Main Analysis Techniques

Stat Analysis / Hypothesis Testing

1. Form of the relationship

2. Statistical significance

Variables:Categorical by Categorical

• Form of the relationship: Cross-tab= two-way table

• Statistical Significance: Chi Square[ if n very small Fisher’s exact test ]

Example: cross-tab

Probationn = 24

Jailn = 16

Males n = 20

16 4

Females n = 20

8 12

Example: cross-tab

Probationn = 24

Jailn = 16

Males n = 20

16 80%

4 20%

Females n = 20

8 40%

12 60%

Example

• Men more likely to choose probation in the sample

• Can we infer men in general more likely to choose probation?

Statistical Significance

Statistical Significance

• Q: Is this a “statistically significant” difference?

• Can the “null hypothesis” be rejected?

Null hypothesis: there are NO differences between men and women in sentencing

Universen = ∞

M: ?% probationF: ?% probation

Samplen = 40

M: 80% probationF: 40% probation

sample

inference

Universen = ∞

Null Hypothesis:M% = F%

Samplen = 40

M: 80% probationF: 40% probation

sample

inference

Logic of Statistical Inference

1. If the Null Hypothesis is True…

… what are the expected frequencies for Men and Women in any sample?

2. Do the frequencies in my sample (n = 40) differ from the expected frequencies?

Testing Null Hypothesis:Expected Frequencies

Probationn = 24

Jailn = 16

Males n = 20

exp: 12 exp: 8

Females n = 20

exp: 12 exp: 8

Probationn = 24

Jailn = 16

Males n = 20

16 exp: 12

4 exp: 8

Females n = 20

8 exp: 12

12 exp: 8

Observed & Expected Frequencies

Logic of Statistical Inference• What is the probability of drawing the

observed sample (M = 16 probation vs. F = 8 probation) from a universe with no differences?

• If probability very low, then differences in sample likely reflect differences in universe

• Then null hypothesis can be rejected; difference in sample is statistically significant

Statistical Significance

• If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe.

p < .05

• The finding from the sample is statistically significant

Strategy

• Draw an infinite number of samples of n = 40, and graph the distribution of their male vs. female probation %-s

Null Hyp:M = 60% probationF = 60% probation

M: 60%F: 50%

Samples of n = 40 Universe n = ∞

M: 80%F: 40%

M: 70%F: 70%

M: 50%F: 65%

Chi Square Distribution

2.5% of area M % > F %

2.5% of area F % > M %

M % = F %

Statistical Significance

• If probability of obtaining my sample is less than 5 in 100, the null hypothesis can be rejected, and it can be concluded that the difference also exists in the universe.

p < .05

• The finding from the sample is statistically significant

Testing Null Hypothesis:Sample with small difference

Probationn = 24

Jailn = 16

Males n = 20

13 exp: 12

7 exp: 8

Females n = 20

11 exp: 12

9 exp: 8

UniverseN = ∞

M = 60% probationF = 60% probation

SampleN = 40

M = 65%F = 55%

sample

p < .05 ?

Chi Square p = .519

Probationn = 24

Jailn = 16

Males n = 20

13 exp: 12

7 exp: 8

Females n = 20

11 exp: 12

9 exp: 8

Small Difference• p = .519

• Over 50% of samples drawn from null hypothesis universe will have differences this large (65% vs. 55%)

• Difference is not statistically significant

Probationn = 24

Jailn = 16

Males n = 20

16 exp: 12

4 exp: 8

Females n = 20

8 exp: 12

12 exp: 8

Testing Null Hypothesis:Sample with large differences

UniverseN = ∞

M = 60% probationF = 60% probation

SampleN = 40

M = 80%F = 40%

sample

p < .05 ?

Chi Square p = .010

Probationn = 24

Jailn = 16

Males n = 20

16 exp: 12

4 exp: 8

Females n = 20

8 exp: 12

12 exp: 8

Report Findings

• “Men were found to choose probation more frequently than women: 80% of the time vs. 40% of the time (df = 1, χ2 = 6.67, p. < ,05).”

• “Men chose probation 80% of the time, and women only 40% of the time, a difference which was statistically significant (df = 1, χ2 = 6.67, p. < ,05).”