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:

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