Relationship between mean, median and mode with grouped data

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 M_e=L+((n/2-F)/f_m)×d and the mode M_o=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= f_m - f_m-1 and D₂=f_m - f_m+1, f_m is the median/modal group frequency,  f_m-1 and f_m+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)/2^th group. Necessary and sufficient conditions are derived for each of six arrangements of mean, median and mode.

Results: Table 1. Conditions and conclusions on the relationship among M, M_e & M_o

Condition	Conclusion
(1)   fmDr≤G-F≤2fm(A-½ (k+1))	Mo≤Me≤ M
(2)   G-F≥2fm(A-½ (k+1)) or G-F≤fmDr	M≤ Me ≤Mo
(3)   G-F≥ fmDr, A≤Dr*+k/2	M≤ Mo≤Me
(4)   G-F≤ fmDr, A≥Dr*+k/2	Me ≤Mo≤ M
(5)   G-F≥2fm(A-½ (k+1)), A≥Dr*+k/2	Mo≤M≤Me
(6)   G-F≤2fm(A-½ (k+1)), A≤Dr*+k/2	Me≤ M≤Mo

Footnotes:       D_r=D₁-D₂ /D₁₊D₂,          D_r^*=D₁/D₁₊D₂

Conclusion: For grouped data, the mean-median-mode inequality can be any order of six possibilities.