What is the message of Figure 1? Why is the message important?What Figure 1 shows is the cumulative frequency for normal distribution, in which probabilities of each result are approximately equal and they can possibly form a straight line. In this study, the author wants to emphasis that the residuals from Equation (1) does not conform this kind of Gaussian distribution, or saying the distribution for residuals has uneven tails. In other words, in certain situations, residuals might not be completely random and possibly have some causal relationship with some behaviors, in this case, the split or the information about splitting, and thus influence the dependent variable, which is the monthly returns for the securities, so the frequencies of these residuals are not the same comparing with other situations without factitious behaviors. If this is true, the solutions and results from Equation (1) and Table (1) are not valid due to its assumptions that residuals have an expectation of zero and are consistent with normal distribution.However, the study cites from other professionals showing that the estimates from Equation (1) are “unbiased and consistent”, plus this is a large-sample regression, so the results are not completely invalid.What do um and Um refer to, in words? Briefly, what is the procedure for calculating them?um is average residual for a certain month of an individual security. It is calculated by adding all the regression residual estimates for N splits of one security, and then divide the sum by the number of splits N. The range for m is from -29 to 30, which is from 29 months before a declaration of a split to 30 months after the declaration. Since this is the residual term in Equation (1), it determines the differences of monthly returns of securities and the market level of return, which is the so-called deviation compared with normal relationship with the market.