• Recall that for estimating and testing population means it is assumed that the variable has an approximately normal distribution
• Recall the two-sample t-test requires that the standard deviations (and variances) are homogenous

Detecting deviations from normality

• Histograms (frequency distributions) can be a useful tool to quickly visualise the shape of the distribution

• Normal quantile plot (Q-Q plot) is sometimes more powerful
  • Compares each observation in the sample with the corresponding quantile expected from the standard normal distribution
  • If the sample is normally distributed, the points will follow a straight line roughly

Formal test of normality

• Shapiro-Wilk test is used to test departures from normality
  • $H_0$: Data are sampled from a population having a normal distribution
  • $H_1$: Data are sampled from a population not having a normal distribution
  • First estimates the mean and standard deviation of the population using the sample
  • Then tests the goodness of fit to the data of the normal distribution having this same mean and standard deviation

Ignoring violations

• Feasible when we know that a method for estimating and testing means are not highly sensitive to violations of the assumptions of normality and equality of standard deviations
  • Usually because of the central limit theorem
• Also not sensitive when comparing two groups with similar distribution, even deviated from normality
• Frequency distribution with extreme outliers cannot be used
• Two sample t-test will still hold with even a threefold difference in standard deviation, as long as the sample sizes are approximately equal
  • Otherwise Welch’s t-test is a much better alternative

Data transformation