To plot a probability plot by hand, start by organizing your data set, which in this case reflects the warping of a long metal pipe. Use the observed values as the x-axis and calculate the rank-based z-scores for the y-axis to represent the theoretical quantiles. The process involves determining percentiles from your data, which helps in assessing normal distribution. A QQ plot can be created to visualize this relationship, allowing you to identify if the data aligns closely with a straight line, indicating normality. Understanding these concepts will enable you to manually create a probability plot without statistical software.