Structures, water, computers, languages and people (not necessarily in this order)

Mean, median and mode of salaries

Pic: Young British civil engineers do not believe the ICE's average salary. Full opinion here, in New Civil Engineer Magazine.

Although the intelligence quotient of the population fits a Gaussian bell well, the salary distribution is skewed and largest salaries are much too large to match a normal pattern. Plots of the population density of income distribution for various countries are well reproduced by a gamma function. In a Gaussian process, mean, median and mode coincide, but the same cannot be applied to a gamma distribution.

The arithmetic average of the salaries does not give us the information we want. Suppose you are looking for a job. You have an interview with a company that has ten employees, and the interviewer tells you that the average daily salary is 200€. Wow, that seems good money! But for this particular company you could work for a quarter of that. Imagine that seven employees make 50€ each, two managers each make 400€, and the owner makes 850€. Yes, the average salary is 200€ but note that the median and the mode are 50€ and that is probably what you would get.

It is clear that the mean alone is a bad salary estimator and some staticians (i.e. Gini) tried to measure the dispersion of incomes. However, the mean provides optimistic expectations and no goverment or institution dares to change it.
Pic: Gross annual salary distribution in Spain (2002). Click on the image to enlarge.


  1. ¡Bravo!
    I once tried to do a post like this, but I had to discard it because I couldn't find such a graphic. It´s time to say it as loud as possible, and it´s time to say that, in Spain, Euro brougt us a general price rising, that was not compensed with salary modifications.

  2. Dear Xose Manuel,

    I liked this article and I read it many times. Can the argument of the post be applied worldwide? I think so. The distribution of richness in the world is a Gamma function?

    I will never consider anymore values about the salary in companies, but only the one I would get, especially if I need to look for a new job.


    Caro Xose Manuel,

    Questo articolo mi è piaciuto molto e l'ho letto molte volte. L'argomento del post può essere applicato dovunque? Penso di si. La distribuzione della ricchezza del mondo, è una funzione gamma?

    Non considererò mai più valori di salari delle aziende tranne quello che dovrei ricevere. Specialmente se dovrò cercare un nuovo lavoro.

  3. @ Fabrizio,

    No idea if the gamma fuction is the best to describe the distribution of wealth in the world. Some authors prefer the Pareto density law f(x)=∝*x^(−α−1), but I am afraid they are quite similar. What is clear is that it is not a normal distribution and that the mean, the median and the mode do not coincide.

  4. Interesting topic!!

    The gamma distribution has two parameteres. One of them is just the "scale" and the other (let's call it alpha) is the shape. If alpha>>1 the gamma distribution looks like a normal (it actually converges to it as alpha->inf). If alpha<=1 then the density is a strictly decreasing function, as it happens with the Pareto law (which means that the mode is the minimum value in the domain). Furthermore, the limit as alpha->0 of the gamma converges in distribution to the limit of a Pareto with alpha=0 as the support tends to 0.

  5. @imp: add 3 points to your karma because you were inspirated.

  6. I often find that many of these average salaries are way off, it makes me wonder where they are getting there information. You've really got to dig down and even evaluate what the area of specialization is. Highway engineers may be more highly compensated than land development engineers. Also, what do you consider to be an engineer? Are those folks taking the surveying verifying that they are speaking with licensed professional engineers or are they talking to a designer with 20 years of experience who personally feels that he has earned the title of engineer?

    building teams . engineering careers

  7. In Anglo-Saxon countries there is a common misuse of the term "engineer". However, in most of the European countries the title "engineer" is protected by law. Only a proper university degree and, depending on the place, some years of relevant experience allows you to be named "engineer".