What is the primary difference between independent samples and related samples?

Independent samples consist of different individuals in each group with no overlap, while related samples involve the same individuals measured twice or individuals matched on specific traits.

When should a researcher choose a non-parametric test over a parametric test?

A non-parametric test should be chosen when the data are ordinal or nominal, when the sample size is very small and normality cannot be assumed, or when the data significantly violate the assumption of normality or homogeneity of variance.

How does the choice of test change when moving from comparing two groups to comparing three groups?

When comparing two groups, a t-test is appropriate; however, for three or more groups, an ANOVA (parametric) or Kruskal-Wallis (non-parametric) test must be used to avoid inflating the Type I error rate.

What is the risk of performing multiple t-tests instead of a single ANOVA?

Performing multiple t-tests increases the 'family-wise error rate,' making it much more likely that the researcher will find a 'statistically significant' result purely by chance (Type I Error).

Why is it incorrect to use a parametric t-test on nominal data?

Parametric tests require the calculation of means and variances, which are mathematically meaningless for nominal data (categories like 'Type A' or 'Type B') that have no numerical value or order.

What error is made if a researcher ignores the 'related' nature of a pre-test/post-test design?

Ignoring the relationship between samples fails to account for the correlation between scores, which usually results in a loss of statistical power and an inaccurate estimate of the treatment effect.

Define 'Homogeneity of Variance'.

It is the assumption that the different groups being compared have approximately the same variance or spread in their scores, which is a requirement for most parametric tests.

What is 'Ordinal Data'?

Ordinal data is a level of measurement where data points can be placed in a logical order or rank, but the exact distance between the points is unknown or inconsistent.

What does it mean for a test to be 'Distribution-Free'?

A distribution-free (non-parametric) test does not assume that the data follow a specific probability distribution, such as the normal distribution, to produce valid results.

How does the Central Limit Theorem influence the choice of statistical tests?

It suggests that for large samples, the sampling distribution of the mean will be normal regardless of the population's shape, often justifying the use of parametric tests for large, non-normal datasets.

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Factors Affecting the Choice of Statistical Test

Summary

Selecting the appropriate statistical test is a critical step in data analysis that ensures the validity of research conclusions. This choice is governed by the level of measurement of the variables, the distribution of the data, the relationship between samples, and the specific research question being addressed.

1. Level of Measurement

Nominal Data: This represents the simplest level of measurement where data are categorized into mutually exclusive groups without any inherent order (e.g., gender, eye color). Statistical tests for nominal data often involve frequencies and proportions, such as the Chi-square test.
Ordinal Data: This level involves data that can be ranked or ordered, but the mathematical difference between ranks is not consistent or measurable (e.g., Likert scales, race finishing positions). Non-parametric tests like the Spearman's Rho or Mann-Whitney U are typically used.
Interval and Ratio Data: Often grouped as 'Scale' or 'Continuous' data, these levels have consistent intervals between values. Ratio data also possesses a true zero point (e.g., height, weight, temperature in Kelvin). These data types are candidates for Parametric tests if other assumptions are met.

2. Distributional Assumptions

Normality: Parametric tests assume that the data follow a Normal Distribution (a bell-shaped curve). If the data are significantly skewed or have heavy tails, non-parametric alternatives are preferred because they do not rely on distribution shape.
Homogeneity of Variance: Also known as Homoscedasticity, this principle requires that the variance (spread) of the data is roughly equal across all groups being compared. Significant differences in variance can lead to an increased Type I error rate in tests like the ANOVA.
Sample Size: Large sample sizes ( $n > 30$ ) often allow for the use of parametric tests even with slight non-normality due to the Central Limit Theorem, which states that the sampling distribution of the mean approaches normality as $n$ increases.

Comparison of Normal vs Skewed distributions for test selection.

3. Research Design and Sample Relationship

4. Key Distinctions: Parametric vs. Non-Parametric

5. Exam Strategy & Decision Framework

6. Common Pitfalls & Misconceptions