# Is the mode of wage distribution a meaningful economic indicator?

The Central Statistical Office of Poland publishes a report every two years, in which it diagnoses the labor market. Among other indicators, they publish the mode of the distribution of wages, in painful detail (up to a grosz, worth around a quarter of a cent). For instance for 2017 it was 2074.03 PLN worth at the time of writing 548.6 USD (source tweet in Polish).

These statistics always resonate strongly with a lot of my friends, however I have doubts. Having a background in statistics but none in economy, I criticize this indicator as being excessively susceptible to noise. I expect very low correlation between the frequency of wage equal X PLN and X+0.01 PLN and as such conveying very little information about the labor market as a whole. I could see this indicator useful if applied to a reasonably quantized histogram (which however raise the question where should the bin edges be). On the other hand, if somebody does not wish to look at the market as a whole, I could imagine some creative uses of this value.

I would like to ask the community if it is aware of any meaningful uses of the aforementioned in economic research, and if so, at which degree of quantization.

Yes, it is a very useful indicator. You are correct that it has greater noise, but, depending upon how it was calculated it can be a very useful measure.

Statistical efficiency is a nice thing, but it implies a very specific loss function. Let us imagine you are a government. For the US government, the mean wage includes individuals like Bill Gates. The implicit loss function is quadratic loss. However, is this a politically useful measure?

You want to get over 50% of the votes, and this is a highly skewed distribution. That does not attest to the well-being of the middle of the body politic. For that, you want to look at the median, which minimizes absolute linear loss. It is less efficient. It tells you the experience of the median voter, provided votes are roughly distributed randomly across the income spectrum. However, since voting is an all-or-nothing proposition, you don't just want to know the health of the middle.

This brings us to the all-or-nothing loss function. It is the populist loss function. It is the most common experience of your people. Hillary Clinton and David Cameron managed median voter theory perfectly; they both lost. This is a problem that, in politics, used to be called the "farm-block," problem. The median is not representative if any one large body can move the vote. The body most likely to do this are the individuals at the mode. If they act together, they move an otherwise tied election, and almost all elections are close to being tied if competition is real. Voting is an all-or-nothing proposition. The mode minimizes the all-or-nothing loss function. To ignore the mode is to watch a government collapse.

Do not think of this as an attempt to find the center of location. Think of it as a gamble that no government wants to lose.

Don't use bins, find the supremum of a kernel density estimate.

I agree with you that the mode of the wage distribution is not particularly useful, especially at this detail. I wonder how many people earn this amount (5, 10?). This is actually the first time I see an institution reporting the mode of the wage distribution at all. In itself, the figure can be easily misunderstood. Journalists/Commentators often enough mix up median and mean; what would happen with mode?

The mode of the distribution is a very useful statistic. Let $$m_d$$ be the mode, and let $$d>0$$ be some value. Let $$w'\neq m_d$$ be any other value in the support of the random variable wage $$W$$. Then, in general,
$$Pr(W \in [m_d-d,m_d+d] ) > Pr(W \in [w'-d,w'+d] )$$
I also had a look at the histogram you linked to, and something is wrong (inconsistent) here: you say that the reported mode of the wage is $$2074.03$$ PLN. But in the histogram, this value does not fall inside the bin with the greatest relative empirical frequency (which has a low bound of $$2173.39$$. This creates an awkward situation, as regards interpretation. Usually the mode is in the center of the bin with the greatest relative empirical frequency.