Much of science, including public health research, focuses on means (averages); the purpose of the present paper is to reinforce the idea that variability matters just as well

Variability Matters. Maarten Jan Wensink, Linda Juel Ahrenfeldt and Sören Möller. Int. J. Environ. Res. Public Health 2021, 18(1), 157; December 28 2020. https://doi.org/10.3390/ijerph18010157

Abstract: Much of science, including public health research, focuses on means (averages). The purpose of the present paper is to reinforce the idea that variability matters just as well. At the hand of four examples, we highlight four classes of situations where the conclusion drawn on the basis of the mean alone is qualitatively altered when variability is also considered. We suggest that some of the more serendipitous results have their origin in variability.

Keywords: inequality; statistical inference; forecasting; lifespan; socioeconomic status; academic performance

6. Discussion

However witty, the accusation levelled against statisticians in the epigraph of this article is unfair. A statistician would always begin by listing some summary statistics of the data (the river, in this case), including variance, minimum and maximum values, and perhaps some quantiles. A statistician would also have some concern with the consequences of making a wrong decision: if these consequences are grave, such as drowning, a statistician would tolerate a smaller likelihood of a wrong decision than when consequences are trivial. As a result, no statistician would set out to wade confidently through a river based solely on information about its average depth.

Nevertheless, the oversimplistic approach of considering the mean alone, and not the distribution around it, is taken all too often. We have given four examples of situations where we needed to augment the information provided by the mean with information about variability in order to draw more informed conclusions. These examples illustrated four classes of situations: (1) if variability is different between groups, means alone give little insight into the share of people that reach a certain threshold and are hence selected into some group (say, high-achieving mathematicians, first example); (2) failing to account for variability gives wrong predictions of future trends (second example); (3) increasing variability implies that not all in a population capitalize on positive health trends to the same extent (third example); and (4) means without variability do not reveal the potential extent of public health issues (fourth example). A mean alone says little about the proportion of a population that crosses a threshold, while different subgroups of a population may follow different trends. This phenomenon also seems to be evident in spread of infectious diseases, most recently the Coronavirus Disease 2019 (COVID-19), where there is large heterogeneity in disease spreading [28].

There are limits to the amount of information that a single number can contain. From that perspective, it is unsurprising that we found situations where averages alone (single numbers) were insufficient to describe relevant aspects of the situation. This immediately begs the question whether a single number like the variance or standard deviation can contain all the relevant aspects of variability. Of course, if data are described sufficiently well by a two-parameter distribution, such as the normal distribution, two numbers say it all. However, in more complex situations, we may need more. Just like an average may not always suffice, descriptions of variability in a single number may be insufficient as well depending on the question we are trying to answer. We then need to complement this information with more numbers, for example skew, which help determine which part of a population crosses a threshold, or which part of a population experience increases in life expectancy.

Piketty [2] finds the Gini coefficient (a single number between 0 and 1) insufficient to describe economic inequalities and their evolution, and instead resorts to quantiles, such as the proportion of the overall wealth possessed by the poorest half of the population, the upper 10%, the upper 5%, the upper 1%, or indeed the upper 0.1% or even 0.01%. Depending on the kind of inequality and the period that he studies, four or five of such quantiles are deemed sufficient to grasp the main elements of economic inequality and its evolution. In their approach of forecasting statistical moments of the age-at-death distribution, Pascariu et al. [29] found that using seven statistical moments tends to strike a good balance between simplicity and computational tractability. Standard statistical courses often teach the first four moments (mean, variance, skew, kurtosis), all of which can be explained intelligibly with pictures, which becomes increasingly difficult for higher moments. Variability, then, is multidimensional as well.

When an inequality is detected, the level of inequality determines whether intervention is indicated, and how. For example, the insurance element of pension systems implies that some people will accumulate more pension payments than others. Indeed, the guarantee of a monthly income is precisely the point of pension systems, so inequality in the total amount of pension received is not necessarily problematic. An issue may arise when certain subgroups can systematically expect to receive a higher pension payout for every Euro they pay into the pension system. This is a matter of some concern, as pension systems often redistribute from poor to rich. Life expectancy of the rich is higher, while in-payments tend to be proportional to monthly benefits. Thus, for every Euro paid in, the rich and educated can expect to receive much more pension than the poor and uneducated, which raises questions about fairness [30].

In the same vein, consider a simple model where every human consists of the same number of cells that are all equally prone to become malignant due to genetic mutations that all occur at the same rate [31]. In such a model, there are no differences between humans at age 0. Yet the age at which cancer occurs, or indeed whether cancer occurs at all before death by competing causes [32], tends to be greatly distinct. Here we have inequality in consequences without any inequality whatsoever in baseline conditions. Yet, we wish to address them, so we should do so as inequality in the outcome emerges. Notice how chances are even at birth, but we still feel we should mend inequality in the outcome.