Because Frank E. L. asked me about related issues, I have downloaded all the monthly/regional Czech Hydrometeorological Institute's temperature data from the years 1961-2012 to Mathematica and calculated all statistical quantities I considered interesting.
First, I had to figure out the right URLs of the HTML pages that contain the tables in an importable enough format, import them with some sensible formatting options to Mathematica so that they quickly become an array, and replace decimal commas by decimal points.
When we're done, we have the temperature for each of the 12 months of each of the 52 years for each of the 1+13 regions of the Czech Republic. Just to be sure, Czechia has 14 administrative regions (the division was different in the era of communism and it's not one of the things in which I considered communism to be worse: I don't really care) but in this dataset, Prague is unified with the Central Bohemian region that surrounds the capital which means that we only have 13 "weather regions" but they're sometimes supplemented by the new 14th region (put at the beginning) which is the whole country.
Well, your humble correspondent rewrote the names of the regions with the right character set and diacritics. It still displayed incorrectly in the PDF and most readers can't read Czech, anyway. ;-)
Now, I have checked the consistency of the data – the web pages show the actual monthly temperature for each month/year/region combination, the normal temperature for this month/region combination (that should be independent of the year), and their difference, the temperature anomaly.
The "normal temperatures" were listed identically almost everywhere. The only mistake appeared in January 1961 (the first year) for the Southern Bohemian region which says +2.8 although all other years agree it should be –2.8 – all these numbers are in Celsius degrees. So I suppose +2.8 is a typo which also makes the anomaly for that single month/region combination incorrect.
Then I verified whether their anomalies are really equal to the differences of the actual temperatures and the the normal temperatures. It turned out to be exact almost everywhere. However, there were 49 month/year/region combinations where the difference was plus or minus 0.1 Celsius degree, probably due to rounding. Curiously enough, all these mismatches (a tiny minority of month/year/region combinations: such rounding errors should have occurred in 1/2 of the data i.e. in thousands of combinations if the algorithm capable of producing rounding errors were used everywhere) appeared in the years 1974, 1981, 2003, 2005, and 2012. One may see some inconsistency in the rounding schemes – it was probably done by different people or at different moments for different years.
At any rate, the actual temperatures seem trustworthy enough. I haven't verified whether the tables did the correct averaging over the years and the correct averaging over the 13 weather regions – some new rounding issues could be found and discussed here as well, I guess.
So I calculated my own normal temperatures. For each region, they're a pretty nice sine that goes from –4 through –2 °C in January to +15 through +18 °C in July. You may see that the differences between the regions are slightly larger in the spring and the summer than they are in the fall or the winter.
The maximum temperature anomaly for a month/year/region combination was 6.3 degrees. The root mean square anomaly was 1.93 °C. Yes, if you average the temperature over a month, they give you a pretty nice Gaussian centered at the "normal temperature" whose standard deviation is almost two degrees. That's true in the Czech Republic. The numbers may differ in your country or region. The fluctuations are likely to be smaller near the sea and larger deep inside the continents.
When I divided the anomalies to individual years, the histograms were much less Gaussian and more "noisy". In different years, the root mean square anomaly went from 1.2 to 3.0 °C or so. When I divided the anomalies to the 13 or 14 individual regions, the Gaussians were a bit smoother and all the root mean square values of the anomalies were between 1.85 and 2.05 °C.
Now, the trends. Note that when we calculate the trends, the year variable "drops out" because it's already been used to calculate the trend. So we only have at most 12*14 = 168 month/region combinations for which the trend may be computed. The mean trend is 2.8 °C per century – Czechia has obviously seen a faster rate than the globe – the latter had close to 1 °C in the last 50 years. The histogram is somewhat noisy and almost all the entries show a trend between 0 °C and 6°C.
When the trends are divided to the individual 14 regions, they are all between 2.45 and 3.15 °C per century – we still have the degeneracy over the months. All these differences between the regions may be described as noise.
However, a shocking and a kind of curious observation is that the temperature trend vastly depends on the month – from January to December – if all the regions are clumped. It changes in a zig-zag way. The trend for Octobers is close to 0.2 °C per century, virtually zero, while the trend for Mays and Augusts exceeds 4.3 °C per century.
Do you have an explanation for this zigzag behavior?
A possible explanation you may suggest is that due to leap years, Januaries of different years are "different parts of the year", and similarly for the other 11 months. However, this explanation seems to produce far too weak an effect. At most, the shift of a month is by half a day in one way or another. Seasonally, half a day only makes 0.1 °C of a difference and a vast majority of these effects gets averaged out, anyway, because among the 52 years, some of them will have excesses and some of them won't.
So we are forced to conclude that there's been no climate change in Octobers, almost no climate change in Septembers, and a negligible climate change in Februaries. ;-) Does a similar pattern exist at other places? Do you have an explanation of this strong dependence on the month?
One more hint: it could have something to do with other drivers such as aerosols. Maybe the no-trend months – October, September, February – are the worst smog months and smog has contributed a negative amount to the warming trend which only seems to operate when there's actually smog? Do you have a better idea? What should be the differences expected from a sensible statistical/weather model you would think of?