204 

American Academic Scientific Research Journal for Engineering, Technology,  and Sciences   

ISSN (Print) 2313-4410, ISSN (Online) 2313-4402 

 

http://asrjetsjournal.org/  

 

Strong Asymmetry and the Mode, Median, and Mean 

Inequality 

J. Gama
a
, C. E. Suárez

b*
 

a
UFMG, Av. Pres. Antônio Carlos, 6627, Belo Horizonte, Brazil 

b
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil 

a
Email: camplise@impa.br 

b
Email: gamatorres1000@gmail.com 

Abstract 

This paper introduces the consoled strong asymmetry and variation coefficient as generalizations of the standard 

measures of Pearson skewness coefficient and variance. Using these new measures, the concept of strong 

asymmetry is introduced. We prove that the median-mean inequality is valid for that kind of distribution, but 

even in that case, there is no relation between mode and median. A property similar to first-order stochastic 

dominance is proved for the variation coefficient. We also discuss the implications of that concept into 

economics. 

Keywords: Skewness; Mode-Median-Mean inequality; Measures of dispersion. 

1. Introduction  

There is a false belief that for every unimodal and positive skewed random variable (that is 𝐸(𝑋 − 𝐸𝑋)3 ≥  0), 

the inequality of 𝑚𝑜𝑑𝑒 ≤  𝑚𝑒𝑑𝑖𝑎𝑛 ≤  𝑚𝑒𝑎𝑛 holds. Several authors have shown that it is invalid, and they have 

found conditions under which that inequality is valid. Reference [1] showed a list of counterexamples for all the 

possible combinations of orders between the three measures. Reference [11] found a condition under which 

𝑚𝑒𝑑𝑖𝑎𝑛 ≤  𝑚𝑒𝑎𝑛 , but this condition is not related to Skewness. Instead it is a relation in the cumulative 

distribution function. Reference [2] have extensive work about the class of distributions of fixed variance and 

the possible sorts in the three central measures. Inspired by advances in economics such as Behavioral 

Economics and specifically Prospect Theory, we have extended the classical measure of asymmetry based on 

the polynomial 𝑥3  to any odd and non-decreasing function. Using this new measure, we define a class of 

distributions named strong asymmetric, which have the property that its corresponding measure of skewness 

relative to any increasing odd and convex function in 𝑅+  is positive. The inequality 𝑚𝑒𝑑𝑖𝑎𝑛 − 𝑚𝑒𝑎𝑛  and 

𝑚𝑜𝑑𝑒 − 𝑚𝑒𝑎𝑛 can be established for this new class of functions. Similarly, a generalization of the variance is 

shown and several properties related to stochastic dominance. 

------------------------------------------------------------------------ 

* Corresponding author.  

http://asrjetsjournal.org/


American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

205 

2. Strong asymmetry 

Similar to some notions introduced in [10,9] we make the next definitions. 

Definition 1: A function 𝑓: 𝑅 → 𝑅 is called odd-concave if it is odd and concave in 𝑅+  . Analogously, an odd-

convex function is odd and convex in 𝑅+ 

Definition 2: Let 𝑋 be a random variable and 𝑓a continuous, increasing, and odd function; we define  

(1) 

as the Skewness coefficient of 𝑋 relative to the function 𝑓, and  

(2) 

as the total variation coefficient of 𝑥 relative to the function 𝑓. 

Remark 3: Suppose that 𝑓  is a continuous, increasing, and odd function and 𝑋 is a real random variable with 

distribution 𝐹, such that 𝐸[𝑋] = 0. The skewness of 𝑋 relative to 𝑓 is  

(3) 

So, if 𝑆𝑘𝑓(𝑋)  is positive, 

(4) 

The right side of the distribution is under the weights given by 𝑓. (the positive side) is stronger than the left side. 

Suppose additionally that 𝑓 is an odd-concave function. Therefore. 
𝑓(𝑥)

𝑥
 is a decreasing function, and 𝑆𝑘𝑓(𝑋)  is 

considerably more sensitive to values close to zero than to values far from it, then we might expect that 𝐹′(𝑥) >

𝐹′(−𝑥) for values near to 0, i.e, intuitively, the distribution is right skewed when it is restricted to values near to 

the reference point. However, since 𝐹′(𝑥) > 𝐹′(−𝑥) for values near to 0 we might have that, for values far from 

0, 𝐹′(𝑥) < 𝐹′(−𝑥), which suggests that the distribution is left-skewed. Thus, a positive 𝑓 skewness coefficient 

with 𝑓 odd-concave suggests that the distribution is left-skewed, and that 𝑆𝑘𝑔(𝐹) might be negative if 𝑔 is a 

odd-convex function. On another side, if 𝑋 is a symmetric random variable ∀𝑥, 𝐹′(𝑥) = 𝐹′(−𝑥) 𝑠𝑜, 𝑆𝑘𝑓(𝑋) =

0. 

Remark 4:  Some properties of the Skewness coefficient relative to 𝑓 are:  



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

206 

1. if 𝐹  has a symmetric distribution then 𝑆𝑘𝑓(𝐹) = 0 for every odd function 𝑓 . If for every odd and 

increasing function 𝑓, 𝑆𝑘𝑓(𝑋) = 0 then 𝑋 is symmetric.  

2. If 𝑓(𝑥) = 𝑥3 then 𝑆𝑘𝑓(𝑥) correspond with the third moment around the mean.  

3. If 𝑓(𝑥) = 𝜂𝑥 then for all 𝑥, 𝑆𝑘𝑓(𝑥) = 0.  

Example 5: Consider 𝐹 the lottery given by (−𝐿, 1 − 𝑝; 𝐺, 𝑝) with 𝐺, 𝐿 > 0, that is, there is a probability 𝑝 to 

gain 𝐺 and probability 1 − 𝑝 of loss L. Suppose that 𝐸(𝐿) = 0 so 𝑝 =
𝐿

𝐺+𝐿
 and 1 − 𝑝 =

𝐺

𝐺+𝐿
. Let 𝑓 strictly odd-

convex   

(5) 

then 𝑆𝑘𝑓(𝐹) > 0 iff  

(6) 

Because 𝑓(0) = 0 and 𝑓 is strictly odd-convex, the last relation is valid iff 𝐺 > 𝐿. Analogously if 𝑓 is strictly 

odd-concave 𝑆𝑘𝑓(𝐹) > 0 if 𝐺 < 𝐿.  

The last example inspires the following definition: 

Definition 6:  A random variable 𝑋 is strongly asymmetric to the right if for all 𝑓 increasing and odd-convex, 

𝑆𝑘𝑓(𝑋) ≥ 0. Analogously 𝑋 is strongly asymmetric to the left if for all 𝑓 increasing and odd-convex, 𝑆𝑘𝑓(𝑋) ≤

0. 

Example 7: Let 𝑋 a random variable with exponential distribution and density function is given by 

(7) 

This distribution is strongly asymmetric to the right. Let 𝑓  be a continuous and odd-convex function then 

(8) 

Because of the convexity of 𝑓, for each 0 ≤ 𝑦 ≤ 1, 𝑓(𝑦) ≤ 𝑦𝑓(1), we have that  



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

207 

(9) 

On other side for each 𝑦 > 1, 𝑓(𝑦) > 𝑓′(1)(𝑦 − 1) + 𝑓(1), and we have that 

 

Therefore, we have that. 

(10) 

In particular, note that if 𝑓 is strictly odd-convex, the inequalities above are strict. 

Motivated by Example 7, we have the following result. 

Theorem 8: Let 𝑋: (−∞, ∞) → 𝑅 a random variable with finite mean and density function given by φ(𝑥). 

Suppose that exist 𝑎 > 𝐸(𝑋) such that  

1.∀𝑥 ∈ [𝐸[𝑋], 𝑎), φ(𝐸[𝑋] + 𝑥) − φ(𝐸[𝑋] − 𝑥) ≤ 0 

2.∀𝑥 ∈ [𝑎, ∞), φ(𝐸[𝑋] + 𝑥) − φ(𝐸[𝑋] − 𝑥) ≥ 0 

Then, the random variable is𝑋 is strongly asymmetric to the right. 

Proof.  Suppose without loss of generality that 𝐸(𝑋) = 0. Let be 𝑔(𝑥) = φ(𝑥) − φ(−𝑥),we want to prove that 

for every odd-convex function 𝑓 

(11) 

Due to the parity of 𝑓,, we have 

(12) 

Since 𝑓(𝑥) is convex in 𝑅+ 𝑓(𝑥) ≤
𝑓(𝑎)

𝑎
𝑥, for all 0 ≤ 𝑥 ≤ 𝑎  and  𝑓(𝑥) ≥ 𝑓′(𝑎)(𝑥 − 𝑎) + 𝑓(𝑎) ,  for all  𝑥 > 𝑎 

therefore. 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

208 

(13) 

We conclude that: 

 

 

(14) 

Remark 9: Because of Theorem 8, distributions such as the gamma family are strongly asymmetric to the right. 

Theorem 10: A random variable 𝑋 is strongly asymmetric to the right if and only if for all 𝑔 increasing and 

odd-concave, 𝑆𝑘𝑔(𝑋) ≤ 0. 

Proof.  First, suppose that there exists a constant η > 0 such that for all 𝑥 ∈ 𝑅+ , η𝑥 ≥ 𝑔(𝑥)|𝑅+ , then the 

function ℎ(𝑥) = η𝑥 − 𝑔(𝑥) is increasing, odd and convex, so 𝑆𝑘ℎ(𝑥) ≥ 0. 

 

 

Now suppose that there is no $\eta>0$ satisfying that, for all 𝑥 ∈ 𝑅+, η𝑥 ≥ 𝑔(𝑥)|𝑅+, define 

(15) 

where 𝑟𝑛 > 0 satisfies that 𝑛𝑟𝑛 = 𝑔(𝑟𝑛), such 𝑟𝑛exist because of the concavity of 𝑔. We have that 𝑟𝑛 → 0 and 

that 𝑔𝑛(𝑥) converges uniformly to 𝑔(𝑥). Applying the first part we have that 𝑆𝑘𝑔𝑛
(𝑋) ≤ 0 and by convergence 

𝑆𝑘𝑔(𝑋) ≤ 0 . Conversely consider 𝑓(𝑥)  odd concave and increasing function. We will assume that 𝑓  is 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

209 

differentiable (an analogous analysis is made using subgradient) and that 𝐸(𝑋) = 0. For any α > 0 consider the 

function 

(16) 

Then  𝑔α  is oan dd-concave and increasing function. By hypothesis, if 𝐹  is the distribution function of the 

random variable 𝑋, 

 

adding ∫ 𝑓′(α)𝑥𝑑𝐹(𝑥)
|𝑥|>α

− ∫ 𝑓′(α)𝑥𝑑𝐹(𝑥)
|𝑥|>α

+ ∫ 𝑓(𝑥)𝑑𝐹(𝑥)
|𝑥|>α

− ∫ 𝑓(𝑥)𝑑𝐹(𝑥)
|𝑥|>α

, and organizing the 

terms we obtain that for all α > 0, 

 

And finally, we have that. 

 

so ∫ 𝑓(𝑥)𝑑𝐹(𝑥) ≥ 0 

2.1. Strong asymmetry 

A common conceptual mistake is to say that if a distribution $F$ is unimodal and the Pearson skewness 

coefficient is positive, then it is true that 𝑚𝑜𝑑𝑒(𝐹) ≤ 𝑚𝑒𝑑𝑖𝑎𝑛(𝐹) ≤ 𝑚𝑒𝑎𝑛(𝐹). \cite{abadir2005mean} showed 

several examples of the violations of each one of the inequalities. When a strongly asymmetric is considered, it 

is possible to show that 𝑚𝑒𝑑𝑖𝑎𝑛(𝐹) ≤ 𝑚𝑒𝑎𝑛(𝐹) 

Theorem 11. If 𝑋 is a random variable strongly asymmetric to the right, then the median of 𝑋 is less or equal to 

the mean of 𝑋. 

Proof.  Let ℎ(𝑥) = 1𝑥>0 − 1𝑥<0 and  𝑔𝑛(𝑥) a sequence of functions converging to ℎ(𝑥) such that each 𝑔𝑛(𝑥) is 

odd concave and increasing (note that there is no sequence of odd-convex functions converging to ℎ(𝑥)), then 

𝑆𝑘𝑔𝑛
(𝑥) ≤ 0. Let 𝐹 the distribution function of the random variable 𝑋 then,  



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

210 

 

 

(17) 

Because |𝑔𝑛(𝑥)| ≤ |ℎ(𝑥)| = 1, we can apply the dominated convergence theorem. Finally, we conclude that the 

median must be lesser or equal to 𝐸[𝑋]. 

Theorem 12. If 𝑋 is a uni-modal random variable strongly asymmetric to the right, then the mode of 𝑋 is less or 

equal to its mean. 

Proof.  We will proceed by contradiction supposing that 𝐸[𝑋] = 0 and 𝑀𝑜𝑑𝑒(𝑋) = 𝑎 > 0. Let be ℎ𝑥(𝑥) the 

correspondent density function and define 𝑔(𝑥) = ℎ𝑥(𝑥) − ℎ𝑥(−𝑥). We have that ℎ(𝑥) is increasing close to 0, 

then there exist ε > 0 such that 𝑔(𝑥) > 0 for all 𝑥 ∈ (0, ε]. Because ∫ 𝑥𝑔(𝑥)𝑑𝑥
∞

0
= 0 there exist the minimum 

positive root of 𝑟 ≠ 0 of 𝑔(𝑥).  Let 𝑓(𝑥) = (𝑥 − 𝑟)1[𝑟,∞), clearly f is non decreasing, positive and convex. 

(18) 

which contradicts the hypothesis of strong asymmetry to the right. 

Remark 13. Even in the case of strong asymmetry, there is no relation between mode and median. In fact for 

(see figure 1) 

(19) 

We have that the distribution satisfies the criteria of theorem 8 for strong asymmetry and  

𝑚𝑒𝑑𝑖𝑎𝑛 = −1.0167 < 𝑚𝑜𝑑𝑒 = −1 < 𝑚𝑒𝑎𝑛 = −0.9983 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

211 

 

Figure 1: Strong distribution to the right with median < mode < mean. 

In [11] is proved that the class of distributions for which the median is located between the mode and the mean 

is characterized by the relation ∀𝑢 > 0, 𝐹(𝑚 − 𝑢) + 𝐹(𝑚 + 𝑢) ≤ 1 , but this condition is nor related with 

Skewness but instead a relation in the cumulative distribution function. 

2.2. Variation coefficient.  

As in the case of skewness, if the agent is risk-averse, they perceive more the risk for marginal changes close to 

the reference point than those who are further away. In that sense, there is a difference with the CAPM model 

since the variance gives greater weight to points far from the average. This would correspond to agents whose 

distortion function is convex. Some properties of the variation coefficient are: 𝑉𝑎𝑓(𝑥) ≥ 0, and 𝑉𝑎𝑓(𝑥) = 0 if 

and only if 𝑥 is constant.  

1. If 𝑓(𝑥) = 𝑥2 then 𝑉𝑎𝑓(𝑥) corresponds with the variance.  

2. If 𝑓(𝑥) = 𝑥 then 𝑉𝑎𝑓(𝑥) is corresponds with the mean absolute deviation 𝑀𝐴𝐷.  

3.  By Jensen inequality, 𝑉𝑎𝑓(𝑥) ≥ 𝑓(𝑀𝐴𝐷(𝑥)) if 𝑓  is odd-convex, and 𝑉𝑎𝑓(𝑥) ≤ 𝑓(𝑀𝐴𝐷(𝑥)) if 𝑓  is 

odd-concave.  

4.  𝑉𝑎𝑓(𝑥) ≥ |𝑆𝑘𝑓(𝑥)|  

5.  If 𝑓 is odd-concave, 𝑉𝑎𝑓(𝑥) satisfies triangular inequality, i.e. 𝑉𝑎𝑓(𝑥 + 𝑦) ≤ 𝑉𝑎𝑓(𝑥) + 𝑉𝑎𝑓(𝑦). On 

other side 𝑉𝑎𝑓(α𝑥) ≤ |𝑎|𝑉𝑎𝑓(𝑥) if α ≥ 1, and 𝑉𝑎𝑓(α𝑥) ≥ |𝑎|𝑉𝑎𝑓(𝑥) if α < 1. 

Remark 15: Let be 𝑋 and 𝑌 two random variables with distribution functions 𝐹 and 𝐺 respectively, then for all 

𝑓  odd and increasing function, 𝑉𝑎𝑓(𝑋) ≥ 𝑉𝑎𝑓(𝑌)  if and only if for all 𝑟 > 0 , 𝑃(|𝑋 − 𝐸𝑋| < 𝑟) ≤ 𝑃(|𝑌 −

𝐸𝑌| < 𝑟). That corresponds to the "dispersion order" notion introduced in [2].   

2.3. Relation with behavioral economics. 

Consider agents that behave according to Prospect Theory and their reference point is endogenous, similar to  

[3,8,5]. The value function is generally considered concave for gains and convex for losses, and even more [7], 



American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 90, No1, pp 204-213 

212 

[6] described how the negative side is in some way the reflection of the positive one. The losses, in general, 

correspond to the reflection of the gains with respect to the straight line 𝑦 = −𝑥 multiplied by a constant λ 

denominated loss aversion coefficient, whose value has been determined to be between 1.5 and 5. If 𝑓(𝑥) 

represent the concave value function for gains then the value function μ is equivalent to 

μ(𝑥) = min(𝑓(𝑥), λ𝑓(𝑥)) =
1

2
(𝑓(𝑥) + λ𝑓(𝑥) − |𝑓(𝑥) − λ𝑓(𝑥)|) (20) 

If we consider that outcomes have a distribution 𝐹 and the endogenous reference point is given by 𝐸(𝑋), 

(21) 

From this representation and using the properties listed previously in this paper, it is possible to measure the 

impact of agents as described in real economies. 

3. Conclusions 

 In this paper, we showed that the standard measure of skewness could be extended for the class of odd and 

increasing functions. A more robust notion of asymmetry can be defined and is consistent with all the expected 

skewness properties. 

We prove that for strong asymmetric distributions, the inequalities 𝑚𝑒𝑑𝑖𝑎𝑛 ≤ 𝑚𝑒𝑎𝑛$ 𝑎𝑛𝑑 $𝑚𝑜𝑑𝑒 ≤ 𝑚𝑒𝑎𝑛 are 

valid, but even in that case, there is no relation between the mode and the median. 

Finally, there is a strong relationship between the skewness and variation coefficients defined in this paper with 

works in regret, disappointment, and Prospect theory such as [3,8,5]. 

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