## What does ‘mean’ actually mean?

#### Posted on 14 August 2018 by Kevin C

This is a re-post from Climate Lab Book

We commonly represent temperature data on a grid covering the surface of the earth. To calculate the mean temperature we can calculate the mean of all the grid cell values, with each value weighted according to the cell area – roughly the cosine of the latitude.

Now suppose you are offered two sets of temperature data, one covering 83% of the planet, and the other covering just 18%, with the coverage shown below. If you calculate the cosine weighted mean of the grid cells with observations, which set of data do you think will give the better estimate of the global mean surface temperature anomaly?

If you can’t give a statistical argument why the answer is probably (B), then now might be a good time to think a bit more about the meaning of the word “mean”. Hopefully the video below will help.

Why is this important? Over the past few years there have been more than 200 papers published on the subject of the global warming “hiatus”. Spatial coverage was one contributory factor to the apparent change in the rate of warming (along with a change in the way sea surface temperatures were measured). We successfully drew attention to this issue back in 2013, however we were less successful in explaining the solution, and as a result there are still misconceptions about the motivation and justification for infilled temperature datasets. The video attempts to more clearly explain the issues.

We couldn’t find a good accessible review of the most relevant parts of the theory, so Peter Jacobs (George Mason University), Peter Thorne (Maynooth), Richard Wilkinson (Sheffield) and myself have written a paper on the subject, published in the journal Dynamics and Statistics of the Climate System. Methods and data are available on the project website.

Having watched the video, you may be interested in how the HadCRUT4 gridded observations are weighted by infilling or optimal averaging methods to estimate global temperatures. The weights are shown in the figure below. Note that while the Arctic is the principal source of bias (because rapid Arctic warming means δD_{unobs} is large, equation 13 of the paper), it is the Antarctic which is the principal source of uncertainty because large weights are attached to grid cells containing single stations. As a result infilled reconstructions show more disagreement concerning Antarctic than Arctic temperatures.

williamat 06:20 AM on 16 August, 2018A basic requirement for a good statistical analysis is that every member of the population being examined has an equal chance of being chosen in the sample you take. In this case, each point of longitude/latitude would be the population. Since sampling at such points around the poles is less likely, the various estimates have to be made with a single point near the poles being taken as the result for the points not sampled. Not perfect but a pretty good first approximation.

nigeljat 06:36 AM on 16 August, 2018Kevin Cowtan is very good at explaining things and making it accessible. I only know part of this math but could get the general idea of where he was going.

John McKeonat 08:03 AM on 16 August, 2018I really enjoyed Kevin's introductory statistics in the Climate Science Denial 101 course. Very stimulating.