Online Program

Relationship between mean, median and mode with grouped data

Tuesday, November 3, 2015

Shimin Zheng, PhD, Department of Biostatistics and Epidemiology, East Tennessee State University College of Public Health, Johnson City, TN
Eunice Mogusu, Department of Biostatistics and Epidemiology, East Tennessee State University, Johnson City, TN
Sreenivas P. Veeranki, MBBS, DrPH, MPH, Department of Preventive Medicine and Community Health, University of Texas Medical Branch, Galveston, TX
Megan Quinn, DrPH, MSc, Department of Biostatistics and Epidemiology, East Tennessee State University College of Public Health, Johnson City, TN
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, fm is 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 1. Conditions and conclusions on the relationship among M, Me & Mo



(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


(6)   G-F≤2fm(A-½ (k+1)), A≤Dr*+k/2

  Me≤ M≤Mo

Footnotes:       Dr=D1-D2 /D1+D2,          Dr*=D1/D1+D2

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

Learning Areas:

Biostatistics, economics
Other professions or practice related to public health
Public health or related research

Learning Objectives:
Define the necessary and sufficient conditions for each of six inequalities of mean, median and mode for grouped data Demonstrate the relationship between the mean, median and mode for grouped data

Keyword(s): Statistics, Methodology

Presenting author's disclosure statement:

Qualified on the content I am responsible for because: I am a co-author of the study and have participated in the development of the paper. I also have commendable skills in both qualitative and quantitative research and a solid background in statistics, coupled with meritorious skills in applying various statistical software. I am currently an MPH student, Biostatistics concentration and have a passion for research and the application of statistics in analyzing the diverse issues of public health importance, both locally and internationally.
Any relevant financial relationships? No

I agree to comply with the American Public Health Association Conflict of Interest and Commercial Support Guidelines, and to disclose to the participants any off-label or experimental uses of a commercial product or service discussed in my presentation.