DQ 2 : Compare parametric and nonparametric statistical tests.

Parametric tests assume the data come from a distribution with estimable parameters, usually a normal one, and they use those assumptions to gain power — a t-test or ANOVA extracts more information from the same sample than its rank-based counterpart because it uses the actual values rather than their order. Nonparametric tests such as Mann-Whitney, Wilcoxon signed-rank and Kruskal-Wallis make far weaker assumptions, work on ranks, and are robust to skew and to outliers. Two things follow that most posts get wrong. First, nonparametric does not mean assumption-free: Mann-Whitney still assumes independent observations and similarly shaped distributions if you want to interpret it as a comparison of medians. Second, the normality assumption in a t-test is about the sampling distribution of the mean rather than about the raw data, which is why sample size matters so much. Getting that straight is what makes the third question answerable in more than one way.

That distinction is what makes the third question answerable properly. If the data are not normally distributed you have at least three options and they are not equivalent. With a reasonably large sample the central limit theorem means the sampling distribution of the mean is approximately normal anyway, and a parametric test remains appropriate — this is the route most often forgotten. With a small sample and a fixable shape, a transformation such as a log transform may restore normality, at the cost of changing what your effect estimate means. With a small sample, an irreducibly skewed distribution or ordinal data, a nonparametric equivalent is the right choice and you accept some loss of power. Say which you would use for your own DPI Project data and why, and mention how you would check normality — a histogram and a Q-Q plot rather than a significance test alone, since normality tests are underpowered in small samples and overpowered in large ones.