1. Analysing and Presenting Quantitative Data:
Inferential Statistics

2. Objectives After this session you will be able to:
Choose and apply the most appropriate statistical techniques for exploring relationships and trends in data (correlation and inferential statistics).

3. Stages in hypothesis testing
Hypothesis formulation.
Specification of significance level (to see how safe it is to accept or reject the hypothesis).
Identification of the probability distribution and definition of the region of rejection.
Selection of appropriate statistical tests.
Calculation of the test statistic and acceptance or rejection of the hypothesis.

4. Hypothesis formulation
Hypotheses come in essentially three forms.Those that:
Examine the characteristics of a single population (and may involve calculating the mean, median and standard deviation and the shape of the distribution).
Explore contrasts and comparisons between groups.
Examine associations and relationships between groups.

5. Specification of significance level – potential errors
Significance level is not about importance – it is how likely a result is to be probably true (not by chance alone).
Typical significance levels:
p = 0.05 (findings have a 5% chance of being untrue)
p = 0.01 (findings have a 1% chance of being untrue) [

6. Identification of the probability distribution

7. Selection of statistical tests –examples
Research question
Independent variable
Dependent variable
Statistical test
Is stress counselling effective in reducing stress levels?
Nominal groups (experimental and control)
Attitude scores (stress levels)
Paired t-test
Do women prefer skin care products more than men?
Nominal (gender)
Attitude scores (product preference levels)
Mann Whitney U (data not normally distributed)
Does gender influence choice of coach?
Nominal (choice of coach)
Chi-square
Do two interviewers judge candidates the same?
Nominal
Rank order scores
Spearman’s rho (data not normally distributed)
Is there an association between rainfall and sales of face creams?
Rainfall (ratio data)
Ratio data (sales)
Pearson Product Moment (data normally distributed)

8. Nominal groups and quantifiable data (normally distributed)
To compare the performance/attitudes of two groups, or to compare the performance/attitudes of one group over a period of time using quantifiable variables such as scores.
Use paired t-test which compares the means of the two groups to see if any differences between them are significant.
Assumption: data are normally distributed.

9. Paired t-test data set

10. Data outputs: test for normality
Case Processing Summary
Cases
Valid
Missing
Total N
Percent
StressTime1 92
98.9% 1
1.1% 93
100.0%
StressTime2
Tests of Normality
Kolmogorov-Smirnov(a)
Shapiro-Wilk
Statistic df
Sig.
StressTime1
.095 92
.041
.983
.289
StressTime2
.096
.034
.985
.363
a Lilliefors Significance Correction

11. Data outputs: visual test for normality

12. 95% Confidence Interval of the Difference
Statistical output
Paired Samples Statistics
Mean N
Std. Deviation
Std. Error Mean
Pair 1
StressTime1 92
.36366
StressTime2
8.7500
.33316
Paired Samples Test
Paired Differences df
Sig. (2-tailed)
Mean
Std. Deviation
Std. Error Mean
95% Confidence Interval of the Difference t
Lower
Upper
Pair 1
Stress Time 1 Stress Time 2
.22127
7.270 91
.000

13. Nominal groups and quantifiable data (normally distributed)
To compare the performance/attitudes of two groups, or to compare the performance/attitudes of one group over a period of time using quantifiable variables such as scores.
Use Mann-Whitney U.
Assumption: data are not normally distributed.

14. Example of data gathering instrument

15. Mann-Whitney U data set

16. Kolmogorov-Smirnov(a)
Statistical output
Tests of Normality
Sex
Kolmogorov-Smirnov(a)
Shapiro-Wilk
Statistic df
Sig.
Attitude 1
.298 32
.000
.815 2
.167 68
.909
Ranks
a Lilliefors Significance Correction
Test Statistics(a)
Attitude
Mann-Whitney U
Wilcoxon W Z
-4.419
Asymp. Sig. (2-tailed)
.000
a Grouping Variable: Sex
Ranks
Ranks
Sex N
Mean Rank
Sum of Ranks
Attitude 1 32
31.89 2 68
59.26
Total
100

17. Association between two nominal variables
We may want to investigate relationships between two nominal variables – for example:
Educational attainment and choice of career.
Type of recruit (graduate/non-graduate) and level of responsibility in an organization.
Use chi-square when you have two or more variables each of which contains at least two or more categories.

18. Chi-square data set

19. Statistical output Chi-Square Tests Value df Asymp. Sig. (2-sided)
Exact Sig. (2-sided)
Exact Sig. (1-sided)
Pearson Chi-Square
.382(b) 1
.536
Continuity Correction(a)
.221
.638
Likelihood Ratio
.383
Fisher's Exact Test
.556
.320
Linear-by-Linear Association
.380
.537
N of Valid Cases
201
a Computed only for a 2x2 table
b 0 cells (.0%) have expected count less than 5. The minimum expected count is
Symmetric Measures
Value
Approx. Sig.
Nominal by Nominal
Phi
.044
.536
Cramer's V
N of Valid Cases
201
a Not assuming the null hypothesis.
b Using the asymptotic standard error assuming the null hypothesis.

20. Correlation analysis
Correlation analysis is concerned with associations between variables, for example:
Does the introduction of performance management techniques to specific groups of workers improve morale compared to other groups? (Relationship: performance management/morale.)
Is there a relationship between size of company (measured by size of workforce) and efficiency (measured by output per worker)? (Relationship: company size/efficiency.)
Do measures to improve health and safety inevitably reduce output? (Relationship: health and safety procedures/output.)

21. Perfect positive and perfect negative correlations

22. Highly positive correlation

23. Strength of association based upon the value of a coefficient
Correlation figure
Description
None Negligible Weak Moderate Strong Very strong Perfect

24. Calculating a correlation for a set of data
We may wish to explore a relationship when:
The subjects are independent and not chosen from the same group.
The values for X and Y are measured independently.
X and Y values are sampled from populations that are normally distributed.
Neither of the values for X or Y is controlled (in which case, linear regression, not correlation, should be calculated).

25. Associations between two ordinal variables
For data that is ranked, or in circumstances where relationships are non-linear, Spearman’s rank-order correlation (Spearman’s rho), can be used.

26. Spearman’s rho data set

27. Statistical output Correlations MrJones MrsSmith Spearman's rho
Correlation Coefficient
1.000
.779(**)
Sig. (2-tailed) .
.000 N 30
** Correlation is significant at the 0.01 level (2-tailed).

28. Association between numerical variables
We may wish to explore a relationship when there are potential associations between, for example:
Income and age.
Spending patterns and happiness.
Motivation and job performance.
Use Pearson Product-Moment (if the relationships between variables are linear).
If the relationship is  or -shaped, use Spearman’s rho.

29. Pearson Product-Moment data set

30. Relationship between variables

31. Statistical output Descriptive Statistics Mean Std. Deviation N
Rainfall
48.17
11.228 30
Sales
132.47
28.311
Correlations
Rainfall
Sales
Pearson Correlation 1
-.813(**)
Sig. (2-tailed)
.000 N 30
** Correlation is significant at the 0.01 level (2-tailed).

32. Summary
Inferential statistics are used to draw conclusions from the data and involve the specification of a hypothesis and the selection of appropriate statistical tests.
Some of the inherent danger in hypothesis testing is in making Type I errors (rejecting a hypothesis when it is, in fact, true) and Type II errors (accepting a hypothesis when it is false).
For categorical data, non-parametric statistical tests can be used, but for quantifiable data, more powerful parametric tests need to be applied. Parametric tests usually require that the data are normally distributed.