East Carolina University
Department of Psychology
Skewness and the Relative Positions of Mean, Median, and Mode
Dennis Roberts showed me this question from a statistics
exam.
6. In a negatively skewed distribution, what would be the correct order of the 3
measures of central tendency, going from left to right?
A. Mean, Median, Mode B. Mean, Mode, Median
C. Mode, Median, Mean D. Median, Mode, Mean
One can't really answer this question given only the
skewness of the distribution. For example, focusing on the relative position of
the mode, consider the two little distributions used in the SAS program below:
data one; input x @@; cards;
100 99 99 98 97 96 95 94 93 92 91 90 5
proc univariate;
data two; input x @@; cards;
100 99 98 97 96 95 94 93 92 91 90 5 5
proc univariate;
------------------------------------------------------------------------------
Univariate
Procedure
Moments
N 13 Sum Wgts 13
******* Mean 88.38462 Sum 1149
Std Dev 25.26044 Variance 638.0897
******* Skewness -3.50235 Kurtosis 12.4657
Quantiles(Def=5)
100%
Max 100 99% 100
75% Q3 98 95% 100
******** 50% Med 95 90% 99
25% Q1 92 10% 90
0% Min 5 5% 5
1% 5
******** Mode 99
For this negatively skewed distribution, the mode is higher than the median which is higher than the mean.
But wait, you say, at least the median must always be greater than the mean in a positively skewed distribution, right? NO! At least not if you use Fisher's g measure of skewness (the most common measure, based on the third moment about the mean). I first realized this when I saw in a SAS manual PROC UNIVARIATE output for a distribution where the mean was NOT drawn in the direction of the skewness. I append here output from such a distribution from a thesis here at ECU (Jerry Stephenson's).
---------------------------------- GROUP=3 -----------------------------------
Univariate Procedure
Variable=ACIDS
Moments
N 17 Sum Wgts 17
********* Mean 40.23529 Sum 684
Std Dev 13.17919 Variance 173.6912
******* Skewness 1.164597 Kurtosis 2.757279
Quantiles(Def=5)
100%
Max 77 99% 77
75% Q3 45 95% 77
******** 50% Med 41 90% 54
25% Q1 35 10% 24
0% Min 24 5% 24
1% 24
Mode 24
***** Notice that the skewness is quite positive, but the mean is a bit less than the median!
Stem Leaf # Boxplot
7
7 1 *
6
5 04 2 |
4 112556 6 +--+--+
3 5677 4 +-----+
2 4446 4 |
----+----+----+----+
Multiply Stem.Leaf by 10**+1
Date: Mon, 09 Feb 98
09:48:47 EST
From: "Karl L. Wuensch"
Subject: Re: relative position of mean, median, mode
To: dennis roberts <dmr@psu.edu>
Dennis,
Regarding the mean, median, mode inequality, here are some details:
See MacGillivray, H.L. (1981). The mean, median, mode inequality and skewness for a class of densities. Australian Journal of Statistics, 23: 247.
Short summary:
mu = mean, m = median, M = mode, f = pdf, F = cdf
1.) If f(m+x) - f(m-x) changes sign in x>0 once and that f(M+x) - f(M-x) does not change sign, then the mean, median, mode inequality holds. (not necessarily restricted to non-negative random variables).
2.) A more general sufficient condition is that:
1-F(m-x) - F(m+x) <= 0, x > 0
>= 0, x > 0, i.e. one sign over the range of x
Thanks to Alan Hutson for this summary. As I noted earlier, not the sort of material you'd likely want to present to an introductory class.
Terry Moore added that the inequality will hold (in fact, (mean - mode) will = 3(mean - median)) for distributions that are "almost normal."
This information came from the STAT-L list, March 1996.
Another interesting inequality, passed on to me (KLW) by Gary McClelland in 2003, is that the absolute difference between the mean and the median cannot exceed one standard deviation. I asked for what distribution could the difference equal 1. His reply:
it is an unusual distribution:
put k observations at -1 and (k+1) observations at +1.
the median is +1 and remains +1 as k->Infinity
the mean is 1/(2k+1) and appproaches 0 as k->Infinity
the standard deviation is 4k(k+1) /(2k+1)^2
and approaches 1 as k ->Infinity
So, at the limit, the mean and median would be 1 s.d. apart.
In June of 2005 Paul von Hippel in the Department of Sociology at Ohio State kindly pointed me to his nice article on skewness and the relative positions of mean, median, and mode:
Also see:
Contact Information for the Webmaster,
Dr. Karl L. Wuensch
This page most recently revised on
20. September 2011.