Background: It is widely believed that the median of a unimodal distribution is "usually" between the mean and the mode for right skewed or left skewed distributions. However, this is not always true, especially with grouped data. For some research, analyses must be conducted based on grouped data since complete raw data are not always available. A gap exists in the body of research on the mean-median-mode inequality for grouped data. Methods: For grouped data, the median Me=L+((n/2-F)/fm)×d and the mode Mo=L+(D1/(D1+D2))×d, where L is the median/modal group lower boundary, n is the total frequency, F and G are the cumulative frequencies of the groups before and after the median/modal group respectively, D1= fm - fm-1 and D2=fm - fm+1, fmis the median/modal group frequency, fm-1 and fm+1 are the premodal and postmodal group frequency respectively. Assuming there are k groups and k is odd, group width d is the same for each group and the mode and median are within (k+1)/2th group. Necessary and sufficient conditions are derived for each of six arrangements of mean, median and mode. Results: Table available at https://apha.confex.com/apha/143am/webprogram/Paper326538.html Conclusion: For grouped data, the mean-median-mode inequality can be any order of six possibilities.

It is widely believed that the median is “usually” between the mean and the mode for skewed unimodal distributions. However, this inequality is not always true, especially with grouped data. Unavailability of complete raw data further necessitates the importance of evaluating this characteristic in grouped data. There is a gap in the current statistical literature on assessing mean–median–mode inequality for grouped data. The study aims to evaluate the relationship between the mean, median, and mode with unimodal grouped data; derive conditions for their inequalities; and present their application.