> if I have one set of people that are 50, 51, 52, and 53 inches tall, and another who are 51, 51, 52, and 53 inches tall, I can say that the second set is on average 0.25 inches taller than the first.

Thanks for making my case with a very simple example.

You can say that. Of course. However, the issue here is about whether or not what you are saying has any real meaning.

I'll use the second data set as an example of how what this paper is reporting might raise questions.

51, 51, 52, 53

minimum: 51

maximum: 53

median: 51.5

mean: 51.75

partial mean: 51 (only considering the first two samples)

standard deviation: 0.96

Let's look at how we might be able to report the differences between these four measurements.

First, the fact that the measurements were reported as "51", "52" and "53" implies the experimental data could not be recorded to an accuracy of more than one inch. If the measurements were more accurate we would have numbers like "51.1". Whatever the case may be, a measurement like "51" does not imply an infinite number of significant digits. This is important.

Here the difference between "accuracy" and "precision", which are different things, might be worth reviewing:

Accuracy: How far a value is from the actual value

Precision: How close measurements are to each other

You can have incredibly good precision and still be way off. To use a war analogy, you intended to land ten missiles within a 1 km radius of city A. Your missiles landed within a 1 km radius, however, they landed in city B. This was very precise, yet highly inaccurate. The other end of the scale would be that all missiles land in city A, yet they land within 10 km of each other, accurate, not precise.

And so, when discussing experimental data and using it to generate conclusion the nature, scale and kind of errors in the data must be accounted for. Without this information you might think your mission is successful when you actually bombed the wrong city. In other words, unless you have the full picture the conclusions are invalid, or, at the very least could be questioned and deemed to be worthy of skepticism.

Back to your data (which has no error information at all).

How can we report the differences?

Difference from minimum: 0, 0, 1, 2

Difference from maximum: -2, -2, -1, 0

Difference from median: -0.5, -0.5, 0.5, 1.5

Difference from partial mean: 0, 0, 1, 2

Difference from standard deviation: -0.78, -0.78, 0.26, 1.3

So, if I look at these results and I intend to show as dramatic a change as possible, I restrict my mean calculation to the first two samples and produce a graph showing 0, 0, 1, 2. In other words, the increase in the samples outside the partial average looks more dramatic.

This map (and possibly the entire website) uses this weird partial mean that LEAVES OUT the last twenty years and then compares the entire period to that partial mean. Huh?

To make things worse, we don't even know what kind of mean they used. For example, it could have been a weighted average calculation and, yes, this could be designed to add even more dramatism to the charts.

Looking at the standard deviation is likely to be a far better indicator of where each sample lies. Again, without error data some or all of this is meaningless. Why?

Let's say the measurement error is +/- 1 inch. Why? Hair, being in a hurry, using a tape measure and no reliable way to actually measure to the top of the skull, etc.

I don't feel like doing a bunch of statistical analysis this morning. I'll just say that, once you include such an error in the calculations you can end-up with results that anyone could consider to be "normal" and certainly within one standard deviation from the mean. In other words, no news at all.

I mentioned "significant digits" above and said this is important. This is an often misunderstood reality of calculations, particularly when it comes to real-world experimental data.

Say I machine a thousand pieces of metal to a diameter of 1.000 inches. It's a pin that is supposed to go into a hole with a diameter of 1.001 inches. After manufacturing half of them come out to 1.005 and the other half 0.995 inches. The average is 1.000 in. What does this mean? NOTHING. Why? Because fully half of the pins I machined will not go into the hole because they are too big! That's 50% waste due to the error. The average is meaningless.

If I report this as a difference with respect to the mean of a partial population the numbers that will come out of that will be utterly meaningless. Useless. The correct specification is likely to be a measure acceptable standard deviations of the pins from the standard deviation of the population of achievable hole diameters.

Back to significant digits.

You can't fabricate data where it doesn't exist.

If you have three numbers, [10.5, 9.3, 17.3] you can calculate the average to be 12.3666... However, if these numbers came from experimental data, you can't claim you have a valid result to six or more significant digits. You just can't. In this case your best hope for a meaningful average is 12. Not 12.3, 12. Two significant digits. This is limited by the value with the lowest number of significant digits.

I won't go into the details on significant digits. Here's a short document from Yale on the subject and how it applies to calculation:

http://www.astro.yale.edu/astro120/SigFig.pdf

And, BTW, this takes on a much more important meaning when you are dealing with laboratory or experimental data. Dry theoretical calculations must follow these rules. Once the values being manipulated become connected to the real world this become far more important and significant, because findings must have real meaning. Without that you might assume 100% of the pins you machined are good, when, in reality, you have to throw away (or fix) 500 of them.

Here's a good set of videos on the subject from Khan Academy:

https://www.khanacademy.org/math/arithmetic-home/arith-revie...

From the video: "to ensure"..."that the result isn't more precise than the things you actually measured".

If you are not familiar with this kind of analysis, please watch the videos. This speaks directly to the issues in the kind of reporting we see on this subject all the time. We can't fabricate data and not expect to be challenged.

As I said in a different comment. This is NOT to say I am one of these delusional anti-climate change folks. The effect is very real. What I am saying here is that if we are going to gain the hearts and minds of those who might not understand or want to accept what's going on we have to make sure we don't leave easy holes in the claims we are making. The "and then a miracle occurs" kind of math and claims you see all the time in climate change reporting provides very easy attack vectors for anyone wishing to discredit the entire discipline.