July 2017 Projected Temperature Anomalies from NCEP/NCAR Data

What might have happened if you had chosen a different sample.

And therein lies one of the significant problems with the so called time series.
The sample in 1860 is different to that in 1880, which is different to that in 1900, which is different to that in 1920, which is different to that in 1940, which is different to that in 1960, which is different to that in 1980, which is different to that in 2000, which is different to that today.
Throughout the time series almost every year involves a different sample, such that one cannot compare one year with another, or one year with the so called base period.
There is no meaningful anomaly because of the different sampling.

Nick
While I agree that the sample error is additional, it is not reasonable to dismiss the measurement uncertainty which seems to be near-universal when discussing temperatures. People are taking the temperature readings as gospel, literally. The uncertainties of each measurement are not being propagated through to the final answer. Adding a huge number of additional samples does not make the instruments more precise or more accurate. It is simply not true that measuring 10,000 times as many points (which reduces the sampling error) will reduce the uncertainty of all the individual measurements. This is basic stats too.
What I see repeatedly is people assuming the measurements are as utterly precise as their multi-decimal place calculator mantissa cares to display. There is (apparently) confusion about the difference between taking a single instrument to 1000 sites to make measurements at the same time, versus taking readings on 1000 different instruments, one at each site, at the same time. Further, taking 1 measurement with each instrument is not the same as taking 100 measurements at that moment on each of the 1000 instruments, one at each location.
Daily temperature measurements are the lowest quality of all possibilities: one measurement recorded from each instrument, a different one at each site.
Additional sampling at additional sites is a good idea of course, but each sample is of a ‘different thing’ so it cannot be treated the same as having used, for example, 10 instruments read once at each site, or 10 readings on one instrument at each site, to get a better fix on where the mean lies for each sampled location. Even estimating more exactly where the mean lies does not reduce the uncertainty about the measurements themselves which is an inherent property of the instruments.
Finally, measuring the temperature at 1000 sites using 1000 instruments that are plus-minus 0.02 degrees C does not mean the final answer, the average temperature, is plus-minus 0.02. That’s not how it works. There is a formula for error propagation.
The claim that 2015 was 0.001 degrees warmer than 2014 is not supported (or denied) by the evidence. No one knows because that is a value far smaller than the propagated uncertainties.

Nick, you persist in your avoiding the measurement uncertainty.
Emphasis added:
“Deviations cancel, and the contribution to error of the mean is reduced. Broadly, variances add, so the sum of N is N*each. But then you scale each down by N, reducing the variance of the contribution to mean by N^2. So with error 0.02 and 1000 in sample, the contribution of that source of error to the mean is .02/sqrt(1000) ~ 0.00063.”
You are referring to determining a better estimate of the true position of the mean, which is the middle of the range of uncertainty, but the measurement uncertainty remains unaffected. You get no cookie for that contribution. If you know the measurement uncertainty is ±2 degrees and that the middle of the range, the Mean, is 16.6616 degrees, you only know that the answer is between 14.6616 and 18.6616 degrees. A critical element of this is that we are not measuring the temperature of one place a large number of times, we are measuring the temperature in a large number of places once each. There is no expectation that the readings will be the same in each location. Each measurement has an uncertainty, and together the uncertainty is larger than any one measurement’s. The average cannot be better known that the contributing components.
The measurement uncertainties are irreducible when performing computations. I can suggest this very brief document for instruction:
http://ipl.physics.harvard.edu/wp-uploads/2013/03/PS3_Error_Propagation_sp13.pdf
An average is a sum of values divided by a number known exactly, so is the same as propagating the uncertainties in the usual manner:
SQRT(uncertainty1^2+uncertainty2^2+uncertainty3^2+…uncertaintyN^2)
“So with error 0.02 and 1000 in sample”, assuming all the instruments are identical, the propagated measurement uncertainty is:
SQRT(0.02^2 * 1000) for an uncertainty of ±0.63 degrees. In theory the measurement errors could be as large as 20 degrees (0.02*1000) but that is extremely unlikely. It is just as unlikely that the true average lies exactly on the mean, however well its position is known.
The true average temperature lies somewhere within a span of 1.26 degrees (68% confidence) and the center of that span is known to ±0.00063 degrees as you demonstrated. There is a 32% chance that the true average of the 1000 measurements lies outside the 1.26 degree range.
The only way to reduce the final value of the propagated uncertainty is to make more accurate and precise measurements. If they were ±0.004 which is technically possible, the range is reduced to ±0.127*2 = 0.253. Obviously when averaging 10’s of thousands of measurements that are each ±0.2 degrees, the uncertainty of the final value is large. Climate scientists have been marketing the median as the ‘average temperature known with great precision’ by pretending that each measurement was perfect which is not only untrue, but impossible.

the fact anomalies are used as opposed to outright temperatures is all people need to realise the level of obfuscation going on nick.