Types of Inferential Statistics• Inferential Statistics: estimate the value of a population

parameter from the characteristics of a sample• Parametric Statistics:

– Assumes the values in a sample are normally distributed

– Interval/Ratio level data required• Nonparametric Statistics:

– No assumptions about the underlying distribution of the sample

– Used when the data do not meet the assumption for a nonparametric test (ordinal and nominal data)

Choosing Statistical Procedures

Two Independent Variables

Interval or RatioIndependent

t-testDependent

t-testOne-Way ANOVA

Repeated Measures ANOVA

Two -Factor ANOVA

Two-Factor ANOVA

Repeated Measures

OrdinalMann-

Whitney UWilcoxon

Kruskal-Wallis

Friedman

Nominal Chi-Square Chi-Square Chi-Square

Factorial Designs

Independent Groups

Dependent Groups

Measurement Scale of the Dependent

Variable

One Independent Variable

Two Levels More than 2 Levels

Two Independent

Groups

Two Dependent

Groups

Multiple Independent

Groups

Multiple Dependent

Groups

Mann Whitney U Test• Nonparametric equivalent

of the independent t test– Two independent groups– Ordinal measurement of the

DV– The sampling distribution of

U is known and is used to test hypotheses in the same way as the t distribution.

Mann Whitney U Test• To compute the Mann

Whitney U:– Rank the scores in both

groups (together) from highest to lowest.

– Sum the ranks of the scores for each group.

– The sum of ranks for each group are used to make the statistical comparison.

Income Rank No Income Rank25 12 27 1032 5 19 1736 3 16 2040 1 33 422 14 30 737 2 17 1920 16 21 1518 18 23 1331 6 26 1129 8 28 9

85 125

Non-Directional Hypotheses

• Null Hypothesis: There is no difference in scores of the two groups (i.e. the sum of ranks for group 1 is no different than the sum of ranks for group 2).

• Alternative Hypothesis: There is a difference between the scores of the two groups (i.e. the sum of ranks for group 1 is significantly different from the sum of ranks for group 2).

Computing the Mann Whitney U Using SPSS

• Enter data into SPSS spreadsheet; two columns 1st column: groups; 2nd column: scores (ratings)

• Analyze Nonparametric 2 Independent Samples

• Select the independent variable and move it to the Grouping Variable box Click Define Groups Enter 1 for group 1 and 2 for group 2

• Select the dependent variable and move it to the Test Variable box Make sure Mann Whitney is selected Click OK

Interpreting the OutputRanks

10 12.50 125.00

10 8.50 85.00

20

Income StatusIncome Producing

No Income

Total

Equal Rights AttitudesN Mean Rank Sum of Ranks

Test Statisticsb

30.000

85.000

-1.512

.131

.143a

Mann-Whitney U

Wilcoxon W

Z

Asymp. Sig. (2-tailed)

Exact Sig. [2*(1-tailedSig.)]

Equal RightsAttitudes

Not corrected for ties.a.

Grouping Variable: Income Statusb.

The output provides a z score equivalent of the Mann Whitney U statistic.

It also gives significance levels for both a one-tailed and a two-tailed hypothesis.

Generating Descriptives for Both Groups

• Analyze Descriptive Statistics Explore

• Independent variable Factors box

• Dependent variable Dependent box

• Click Statistics Make sure Descriptives is checked Click OK

Wilcoxon Signed-Rank Test• Nonparametric equivalent of

the dependent (paired-samples) t test– Two dependent groups

(within design)– Ordinal level measurement of

the DV.– The test statistic is T, and the

sampling distribution is the T distribution.

Wilcoxon Test• To compute the Wilcoxon T:

– Determine the differences between scores.

– Rank the absolute values of the differences.

– Place the appropriate sign with the rank (each rank retains the positive or negative value of its corresponding difference)

– T = the sum of the ranks with the less frequent sign

Pretest Posttest Difference Rank36 21 15 1123 24 -1 -148 36 12 1054 30 24 1240 32 8 732 35 -3 -350 43 7 644 40 4 436 30 6 529 27 2 233 22 11 945 36 9 8

Non-Directional Hypotheses

• Null Hypothesis: There is no difference in scores before and after an intervention (i.e. the sums of the positive and negative ranks will be similar).

• Non-Directional Research Hypothesis: There is a difference in scores before and after an intervention (i.e. the sums of the positive and negative ranks will be different).

Computing the Wilcoxon Test Using SPSS

• Enter data into SPSS spreadsheet; two columns 1st column: pretest scores; 2nd column: posttest scores

• Analyze Nonparametric 2 Related Samples• Highlight both variables move to the Test Pair(s) List

Click OKTo Generate Descriptives:• Analyze Descriptive Statistics Explore• Both variables go in the Dependent box • Click Statistics Make sure Descriptives is checked

Click OK

Interpreting the OutputRanks

10a 7.40 74.00

2b 2.00 4.00

0c

12

Negative Ranks

Positive Ranks

Ties

Total

POSTTEST - PRETESTN Mean Rank Sum of Ranks

POSTTEST < PRETESTa.

POSTTEST > PRETESTb.

POSTTEST = PRETESTc.

Test Statisticsb

-2.746a

.006

Z

Asymp. Sig. (2-tailed)

POSTTEST -PRETEST

Based on positive ranks.a.

Wilcoxon Signed Ranks Testb.

The T test statistic is the sum of the ranks with the less frequent sign.

The output provides the equivalent z score for the test statistic.

Two-Tailed significance is given.