# Examining one-sample t test results to detect assumption violations

All the following results are provided as part of a one-sample t test analysis.

#### Result for residuals:

Residuals plotted against fitted values: detecting incorrectness of the t test model

• Normality test:
• If the assumptions for the t test hold, the sample values should come from a normal distribution. Departures from normality can suggest the presence of outliers in the data, or of a nonnormal distribution in one or more of the samples. The normality test will give an indication of whether the population from which the sample was drawn appears to be normally distributed, but will not indicate the cause(s) of the nonnormality. The smaller the sample size, the less likely the normality test will be able to detect nonnormality. Because the residuals from a one-sample t test are simply the original observed sample values minus the sample mean, the normality test P value for the residuals is identical to that for the sample values.
• Histogram:
• The histogram for the sample has a reference normal distribution curve for a normal distribution with the same mean and variance as the sample. This provides a reference for detecting gross nonnormality when the sample size is large. Because the residuals from a one-sample t test are simply the original observed sample values minus the sample mean, the histogram for the residuals would be identical to that for the sample values, except for the range of the X axis.
• Boxplot:
• Suspected outliers appear in a boxplot as individual points o or x outside the box. If these appear on both sides of the box, they also suggest the possibility of a heavy-tailed distribution. If they appear on only one side, they also suggest the possibility of a skewed distribution. Skewness is also suggested if the mean (+) does not lie on or near the central line of the boxplot, or if the central line of the boxplot does not evenly divide the box. Examples of these plots will help illustrate the various situations. Because the residuals from a one-sample t test are simply the original observed sample values minus the sample mean, the boxplot for the residuals would be dentical to that for the sample values, except for the range of the Y axis.
• Normal probability plot:
• For values sampled from a normal distribution, the normal probability plot, (normal Q-Q plot) has the points all lying on or near the straight line drawn through the middle half of the points. Scattered points lying away from the line are suspected outliers. Examples of these plots will help illustrate the various situations. Because the residuals from a one-sample t test are simply the original observed sample values minus the sample mean, the normal probability plot for the residuals would be identical to that for the sample values.
• Residuals plotted against fitted values:
• Because there is only one unique fitted value, the sample mean, the graph of residuals against fitted values will consist of a vertical "stack" of residuals. Except for the subtraction of a constant value (the mean) from all the sample values, this graph is identical in appearance to a distribution graph of the sample values. Outliers may appear as anomalous points in the graph (although an outlier may not turn up in the residuals plot by virtue of affecting the mean so that its fitted value lies near it).