Beyond the Comfort of Averages (HEN Editorial 361)

The mean feels intui­tive and con­ve­ni­ent, but it often hides ine­qua­lity, hete­ro­gen­eity and extre­mes, sha­ping mis­lea­ding nar­ra­ti­ves about poverty, growth and cli­mate change.

As rese­ar­chers most of us sup­po­sedly like the mean. The mean is see­mingly very kind to us: it is a simple num­ber, it (sup­po­sedly) sum­ma­ri­zes a lot of infor­ma­tion and it can serve as (some­what) solid anchor for fur­ther sta­tis­ti­cal reaso­ning. Hence, all praise to the mean and its convenience ;-)

While I think the above is lar­gely cor­rect, it is also incom­plete: pre­cis­ely because the mean com­pres­ses infor­ma­tion into a sin­gle num­ber, it can also hide important pat­terns. This mat­ters when­ever dis­tri­bu­ti­ons are mul­ti­mo­dal, when mul­ti­pli­ca­tive dyna­mics are at work, or when chan­ges in inputs pro­duce qua­li­ta­tive shifts in out­puts — for exam­ple, in the rela­tion bet­ween income and sub­jec­tive well-being. In such cases, a mean-based view can con­ceal sub­stan­tial heterogeneity.

A key exam­ple of how the mean can hide signi­fi­cant aspects comes from a femi­nist per­spec­tive on mea­su­ring poverty (see here or here). Such a view empha­si­zes how using house­hold means as a base­line for con­s­truc­ting poverty esti­ma­tes con­ce­als lack of dis­posable income for house­hold mem­bers with no or mar­gi­nal employ­ment – often women. This often amounts to ren­de­ring women’s poverty invi­si­ble and, in addi­tion, impo­ses a down­ward bias on stan­dard poverty estimates.

Ano­ther exam­ple is given by GDP as a key eco­no­mic figure, or more pre­cis­ely, its growth rate. GDP growth is typi­cally given by the growth of mean inco­mes, which gives the impres­sion that the GDP growth rate resem­bles the „typi­cal“ change in indi­vi­dual out­co­mes in an eco­nomy. Howe­ver, this is, gene­rally, incor­rect: GDP growth mea­su­res growth in money terms. Hence, the under­ly­ing „mean growth“ does not refer directly to indi­vi­du­als, but weighs them by their income. As a con­se­quence, the growth of richer indi­vi­du­als mat­ters more for GDP growth than does the growth rate enjoyed by the poor.

A truly demo­cra­tic mean – more apt to repre­sent the typi­cal expe­ri­ence of an indi­vi­dual – would require us to cal­cu­late not the growth in average income (which employs a subtle weight­ing and, hence, can easily be dri­ven by a few if the dis­tri­bu­tion is une­qual), but, rather, the average growth in income (see here). Only in the second cal­cu­la­tion – where peo­ple ins­tead of money enter the deno­mi­na­tor, and hence dis­tri­bu­tio­nal issues play a role* – the mean is infor­ma­tive about the „typi­cal“ indi­vi­dual experience.

Finally, cli­mate change adds yet ano­ther case. Mean tem­pe­ra­ture chan­ges are very useful, but they can also create the impres­sion of a slow and uni­form pro­cess. Loo­king at extre­mes often tells a dif­fe­rent story: across many regi­ons, tem­pe­ra­ture extre­mes change fas­ter than the mean, and cli­mate risks are fre­quently dri­ven by rare events rather than by average con­di­ti­ons. This is espe­ci­ally visi­ble in phe­no­mena such as heat­wa­ves, cold spells, or El Niño events, whose impacts are often dis­pro­por­tio­nate to what mean tem­pe­ra­tures alone would suggest.

Of course, this shift in out­liers does not con­tra­dict the mean. It just points to the fact that very rele­vant varia­bles in eco­no­mics – like chan­ges in urban tem­pe­ra­ture set­tings, but also key varia­bles like income, wealth or firm size – intrin­si­cally require con­side­ring extreme cases if they are to be pro­perly asses­sed and understood.

In this spi­rit: stay cool in the upco­ming heat­wa­ves, and all the best,

PS: If you are now inspi­red to think more deeply about what it could mean to do „sta­tis­tics bey­ond the mean“ (what a pun!), I can warmly recom­mend to cont­act this fan­ta­stic and inspi­ring col­le­ague of mine.

*In gene­ral, the demo­cra­tic growth rate will always be grea­ter than the stan­dard growth rate if the dis­tri­bu­tion beco­mes more equal and vice versa.