3.4 - Inferential Statistics
Understanding distribution curves in statistical data
Distribution curves are graphical representations of how data is spread across a range of values. They are essential for interpreting the characteristics of data sets in psychological research, particularly in determining the suitability of statistical tests.
Types of distribution curves
- Normal distribution curve - This is a symmetrical, bell-shaped curve where the mean, median, and mode are all at the same central point. Most data points cluster around the middle, with fewer at the extremes.
- Skewed distribution curve - This curve is asymmetrical, with data points more spread out on one side. It can be:
- Positive skew - The tail extends to the right, with the mode at the peak, followed by the median, and then the mean furthest to the right.
- Negative skew - The tail extends to the left, with the mode at the peak, followed by the median, and the mean furthest to the left.
Concepts of probability and significance levels
Statistical analysis in psychology relies on understanding the likelihood of results occurring by chance and setting thresholds to determine if findings are meaningful.
Key concepts in statistical significance
- Probability - This refers to the likelihood of an event or result happening.
- Significance level - This is the probability threshold at which psychologists decide to reject the null hypothesis, indicating that the results are unlikely to have occurred by chance. Commonly, this level is set at p < 0.05.
- Tables of critical values - These are reference tables used to compare the calculated value from a data set with predetermined critical values. If the calculated value exceeds the critical value, the result is considered statistically significant.
Criteria for selecting parametric and non-parametric tests
Choosing the appropriate statistical test is crucial for accurate analysis of data in psychological studies. The decision depends on the nature of the data and its distribution.
Conditions for parametric tests
- Data level - Must be interval or ratio data.
- Distribution - The data should follow a normal distribution curve.
- Variance similarity - The variances of the groups being compared should be roughly equal.
Examples of parametric tests include t-tests and ANOVA tests, though detailed knowledge of these specific tests is not required. If data does not meet these criteria, a non-parametric test is typically used instead.
Guidelines for non-parametric tests
Non-parametric tests are used when data does not fit the requirements for parametric testing, such as when it does not show a normal distribution. The choice of test depends on the type of data and experimental design, as outlined in the table below.
| Type of data | Independent design (test of difference) | Repeated design (test of difference) | Test for correlation |
|---|---|---|---|
| Nominal | Chi-square test | Binomial Sign test | Not applicable |
| Ordinal | Mann-Whitney U test | Wilcoxon Signed-Rank test | Spearman's Rho |
| Interval | Not applicable | Not applicable | Spearman's Rho |
Specific non-parametric tests and their applications
Non-parametric tests are essential tools for analysing data that does not meet the assumptions of parametric tests. Each test is suited to specific types of data and research designs.
Overview of non-parametric tests
- Mann-Whitney U test - A test of difference used with independent data. It requires at least ordinal data but can also handle interval or ratio data converted to ordinal for analysis.
- Wilcoxon Signed-Rank test - A test of difference for related data. It also requires at least ordinal data and can use converted interval or ratio data.
- Chi-square test - A test of association for independent data at the nominal level, using frequencies. It examines relationships between categories by comparing observed frequencies with expected frequencies.
- Binomial Sign test - A test of difference for related data at the nominal level. It focuses on the direction of differences between paired scores.
- Spearman's Rho - A test of correlation for paired scores, requiring at least ordinal data. It can also be applied to interval or ratio data converted to ordinal for the test.
Types of errors in hypothesis testing
When conducting statistical tests, errors can occur in the interpretation of results, leading to incorrect conclusions about the hypotheses being tested.
Common errors in statistical analysis
- Type 1 error - This occurs when the null hypothesis is rejected when it is actually true. In other words, it involves accepting the alternative hypothesis incorrectly.
- Type 2 error - This happens when the null hypothesis is accepted when it is actually false. This means rejecting the alternative hypothesis incorrectly.