Last week I wrote about Paul Georgia’s review of Essex and McKitrick’s Taken by Storm. Based on their book, Georgia made multiple incorrect statements about the physics of temperature. Of course, it might have just been that Georgia misunderstood their book. Fortunately Essex and McKitrick have a briefing on their book, and while Georgia mangles the physics even worse than them, they do indeed claim that there is no physical basis to average temperature. They present two graphs of temperature trends that purport to show that you can get either a cooling trend or a warming trend depending on how you compute the average. McKitrick recently was in the news for publishing a controversial paper that claimed that an audit of the commonly accepted reconstruction of temperatures over the past 1000 years was incorrect, so it only seems fair to audit Essex and McKitrick’s graphs. As we will see, *both* of their graphs are wrong, and their results go away when the errors are corrected.

In their briefing, Essex and McKitrick claim that physics provides no basis for defining average temperature and:

“In the absence of physical guidance, any rule for averaging temperature is as good as any other. The folks who do the averaging happen to use the arithmetic mean over the field with specific sets of weights, rather than, say, the geometric mean or any other. But this is mere convention.”

Physics does, in fact, provide a basis for defining average temperature. Just connect the two systems that you want to average by a conductor. Heat will flow from the hotter system to the colder one until the temperatures are equalized. The final temperature is the average. That average will be a weighted arithmetic mean of the original temperatures. Which is why the folks doing the averaging use weighted arithmetic means rather than the geometric mean.

Of course, even if they were right and there were other equally valid ways to calulate the average temperature, they still need to show that it actually makes a difference, so Essex and McKitrick present an example the purports to show that whether you use the arithmetic or some other mean can affect whether or not you find a warming trend. They constructed the graph on the left by taking monthly observations from ten weather stations and averaging them with the arithmetic mean. They found an overall warming trend of +0.17 degree Celsius per decade.

They next present a graph where they

“treat each month as a vector of 10 observed temperatures, and define the aggregate as the norm of the vector (with temperatures in Kelvins). This is a perfectly standard way in algebra to take the magnitude of a multidimensional array. Converted to an average it implies a root mean square rule.”

Note that nobody, but nobody, averages temperatures this way. Anyway, when they calculated the trend they found an overall cooling trend of +0.17 degree Celsius per decade.

They triumphantly conclude:

“The same data can’t imply global warming and cooling can they? No they can’t. The data don’t imply global anything. That interpretation is forced on the data by a choice of statistical cookery. The data themselves only refer to an underlying temperature field that is not reducible to a single measure in a way that has physical meaning. You can invent a statistic to summarize the field in some way, but your statistic is not a physical rule and has no claim to primacy over any other rule.”

I looked at their graphs and something seemed wrong to me. Their root mean square average gives almost the same answer as the arithmetic mean. For example, it gives the mean of 0 and 20 degrees Celsius as 10.2 instead of 10 degrees. It didn’t make sense to me that it could make as big a difference to the trend as what they found.

McKitrick kindly sent me a spreadsheet containing the data they used and I almost immediately saw where they had gone wrong. You see, some stations had missing values, months where no temperature had been recorded. When calculating the root mean square they treated the missing values as if they were measurements of 0 degrees. This is incorrect, since the temperature was not actually zero degrees. Because the overall average temperature was positive this meant that the root mean square was biased downwards when there were missing observations. And since there were more missing values in the second half of the time series, this produced a spurious cooling trend.

When calculating the arithmetic mean they treated missing values differently. If only eight stations had observations in a given month, they just used the average of those stations. This isn’t as obviously wrong as the other method they used, but the stations in colder climates were more likely to have missing observations, so this biased the average upwards and produced a spurious warming trend.

I filled in the missing values by using the observation for that station from the same month in the previous year and recalculated the trends. Now both mean and root mean square averaging produced the same trend of -0.03, which is basically flat. When analysed correctly, their data shows neither warming or cooling, regardless of which average is used. The different trends they found were not because of the different averaging methods, but because of inconsistent treatment of missing data.

I also calculated the trend with their root mean square average and ignoring missing values, and with the arithmetic mean and replacing missing values with zero (spreadsheet is here). As the table below shows, the averaging method made almost no difference, but treating missing values incorrectly does.

Trend | ||
---|---|---|

Missing values | Mean | Root Mean Square |

Ignored | 0.16 |
0.15 |

Treated as 0 degrees | -0.15 | -0.17 |

Previous year used | -0.03 | -0.03 |

I emailed McKitrick to point out that arithmetic mean and root mean square did not give different results. He replied:

Thanks for pointing this out. It implies there are now 4 averages to choose from, depending on the formula used and how missing data are treated, and there are no laws of nature to guide the choice. The underlying point is that there are an infinite number of averages to choose from, quite apart from the practical problem of missing data.

Incredible isn’t it? He *still* doesn’t understand basic thermodynamics. And he seems to think that are no laws of nature to guide us in estimating the missing values so that it is just as valid to treat them as zero as any other method, even for places where the temperature never gets that low.