An inversion had been in place for a couple of weeks with the temperature 50′ above ground thirty degrees warmer than at the surface. Winds were dead calm throughout the day and kicked in near midnight to scour out the cold air.
Taking the standard (Tmax+Tmin)/2 for the mean temperature of the day gives 29 degrees. Averaging the hourly observations yields 18.96 degrees, a considerable difference!

I don’t think I made myself very clear. I’m not claiming that the digital thermometer actually MEASURES hundredths of a degree, then, for some strange reason, reports in tenths of a degree. My point is that the thermometer is a tenth-of-a-degree approximation of an actual temperature, one that could be any of an infinite number of possible temperatures between 0.05 degrees below and 0.04999999… degrees above the reported temperature. I used x.45 as an example of the lowest possible fraction of a degree that could be reported as x.5 (and then rounded, by the observer, up to x+1). But x.454 would also be rounded up, as would x.46, x.49, x.451, x.4501, x.45001, and x.45000000000000001. The point is that, due to this double-rounding effect, actual temperatures between x.45 and x+1.44999999999… will always be rounded to x+1. Across a uniform probability distribution (which is obviously what temperatures, at this range of precision, are), the true average of temperatures in this range is x.95, not x+1, but the reported temperatures will all be x+1, and therefore the average of the reported temperature will be x+1. This means that there is, in addition to everything else, a positive 0.05 degree double-rounding bias that can be blamed on the use of digital thermometers that report temperature to the nearest tenth degree and observers that round those to the nearest whole degree.

Let me try an example. Say you have 1,000 thermometers spread out over an area that varies in temperature by 10 degrees. And say that the ACTUAL temperature at each of these sites is defined as T(i) = 24+.01i, for i = 1 to 1,000, resulting in a uniform probability distribution between 24.01 and 34 degrees. The average temperature across all these stations can easily be shown to be exactly 29.005 degrees.

But what is the average REPORTED temperature? Well, the first 44 sites (between 24.01 and 24.44) would all be reported as 24 degrees. The next 100 sites (between 24.45 and 25.44) would all be reported as 25. There would also be 100 sites reported as 26, 27, 28, 29, 30, 31, 32, and 33. And finally, there would be 56 sites (between 33.45 and 34.00) reported as 34 degrees.

A weighted average of these reported temperatures would be [44(24)+100(25)+100(26)+100(27)+100(28)+100(29)+100(30)+100(31)+100(32)+100(33)+56(34)]/1000, or 29.06, which is 0.055 degrees higher than the ACTUAL average temperature over these stations. (This is actually slightly higher than the 0.05-degree bias I stated earlier, but only because I’m using a DISCRETE uniform probability distribution to approximate a CONTINUOUS uniform probability distribution. With a continuous uniform probability distribution, the resulting bias would be exactly 0.05 degrees.)

Note that, if the thermometer reported temperature to the nearest WHOLE degree, though obviously less precise, it would nullify the double-rounding bias, resulting in a reported-average temperature of 29.01, just 0.005 off from the actual average (with a continuous probability distribution, even this small bias would completely disappear)

If the thermometers have an error above and beyond the 0.05 measurement degrees that can be blamed on rounding only to the nearest tenth of a degree, that’s another issue. But the measurement error itself can never be more than 0.05 degrees for a device with a precision of 0.1.

Unless, as you hypothesize, the device is merely truncating the temperature rather than rounding it. I don’t believe that is the case. But if it is, the measurement error wouldn’t be +/- 0.1 degree. It would be only MINUS 0.1 degree, i.e., the device-reported temperature would almost always be LESS THAN the actual temperature and would NEVER be more than the actual temperature (though on rare occasions, the two would be equal). I’m not sure, but I think that would cancel out the positive bias of the observer rounding. But again, there is no evidence that digital thermometers work that way.

What you say is certainly true about analog thermometers–the short of it is, they’re rather precise, especially with a good reader.

As for digital thermometers, I am not sure you’re right. When they give you a number, e.g. 79.5, it is not necessarily the case that any rounding at all has occurred. The measurement is likely +/- 0.1 degree (marked somewhere on either the thermometer or the manual that accompanied it). There is no reason to believe that the thermometer takes readings of 79.45 and rounds them (like a human would)–why couldn’t it just truncate? Or, perhaps it doesn’t take a measurement of the last decimal place in the first place–but +/- 0.1 from 79.45 means it could report either 79.4 or 79.5 and still be accurate within its defined precision.

Say the digital thermometers have 0.1-degree precision. And say that the actual temperature is 74.45. If the obesrver knew what was there in the hundredths place, he would round this down to 74, because it’s clearly closer to 74 than to 75. But, if he’s looking at a digital thermometer that only has 0.1-degree precision, all he would see is 74.5 (74.45 rounded to the nearest 0.1 degree). The observer would round this up to 75 when he recorded it on the NCDC form. So, due to this “double-rounding” issue, not only is anything greater than or equal to x.5 degree being rounded up – anything greater than or equal to x.45 degree is being rounded up! This double-rounding issue quite clearly only works one way, because 74.54 would round up to 75 whether you rounded it directly or in two stages.

But still, as long as the same equipment and rules were in place 30 years ago, the double rounding issue couldn’t account for any of the observed rise in temperature. However, 30 years ago, I suspect most thermometers were analog, mercury-bulb thermometers. Assuming they had marks at .5 degrees, even if the mercury level was at 74.49 degrees, a keen-eyed observer could still tell that the temperature was slightly below 74.5, and when he recorded the temperature on the form, he would round it down to 74.

If, on the other hand, the analog thermometers did not have marks at .5 degrees, then there would be some range (its size depending on the eyesight of the observer, but lets say from x.45 to x.55) where the observer couldn’t tell whether it was closer to x or x+1. Though an individual observer might have a personal bias that made him consistently round such “close calls” one way (thus introducing bias for an individual station), you can make the assumption that, for every such observer, there’s another observer that consistently rounds the other way, and on average, there is no bias.

Based on this analysis, I conclude that though individual analog thermometers probably have more bias (not to mention outright error) than individual digital thermometers, an average of a statistically-large-enough sample of analog thermometers clearly has less bias than that of digital thermometers. And that IS what we are looking at – an average of a huge sample of thermometers – when we say that “global average temperatures have increased by 0.6 degrees Celcius over the last 100 years”.

NCDC takes the B91 form and transcribes into a computer database. From there (as I understand it) a daily average is created, and from those, yearly average which is what me most often see in climate studies. – Anthony,

So this isn’t even a weighted average to give a statisticly significant representation of the area? A weighted average using the hourly readings would give a more realistic representation of temperature. What you described here using a simple average to determine the climatic conditions is pretty much worthless data, IMO. Any conclusion based on this information will be as equally flawed as the data it is based upon. Just because you run it through a computer doesn’t enhance the data, in the business world we call this money laundering. Start with a false premise, apply flawless logic (computer), you end up with a flawlessly false conclusion. I am just disgusted with how far science has fallen, this wouldn’t have been acceptable even in the high school math classes of my day.

REPLY: True, but the NWS lets the observers do the rounding before recording the data, and it is unlcear if they use this method.

I don’t trust you to pick a day at random. I think you will show stuff that helps your preconception preferentially.

REPLY: TCO, its all coming, just not on the schedule you demand. I can’t do everything at once, and I don’t really care if you think you can’t trust me. That’s funny coming from you ….people who have interest and character, collaborate with me and reveal themselves. You don’t. You are still just another phantom with no name other than “TCO”. As far as I can tell your email and website are bogus. So don’t lecture me about trust please. Trust is a two way street, and you’ve provided nothing.

Like I’ve said, I’m one man, with a business, a family, and many things to do. Doing this and the surfacestations.org project simultaneously on my own dime. Its a balancing act.

When I get time there will be more, in the meantime please be patient. No wonder CA folks voted you “most irritating”.

But it seems a bit crude to round to the nearest degree when one is spitballing over amounts less than the MoE.

OTOH, I suppose one can get by with oversampling. But still …

Here’s what I mean:

Example: let’s say that the temp rises through 50.5. That would be raised to 51, at, say, 10am. If the temp never goes above 51.5, there will never be anything recorded over 51.

If the temp reaches Tmax at noon, then drops below 50.4, then the recorded temp would be drop to 50 (at, say 2pm).

That means, that no matter what the Tmax really was, you would have recorded a “window” of temps, not a “snapstot” of Tmax.

The reading would be showing the temp of 51 from 10 am to 2 pm, and totally missing the Tmax.

Maybe I’m just reading this too deep.

Was there a time when only T-Min/Max was used? If so, adding TOBS would create a spurious +0.1C upswing in the record. (Or almost 15% of last century’s warming.)

Yes. But since they aren’t measuring continually, it doesn’t show up.

Asi it is, measuring at three points gives three 10% opportunities for an upward bump.

But as St. Mosh says, it’s an offset, not a trend.

Same on the low. A temp of 51 wouldn’t appear to drop until the temp reached 50.4, then a drop to 50.

It would appear to “flat-top” the curves.

Hope I’m making sense.

“Ah’m a headin’ fer the last roundup!”

Because we are not adding and then rounding, we are rounding, then adding.

“Come to think of it, if it’s rounded to whole numbers, you get a +1C at T-max and flat at t-min. So the plus-half-a -degree dif. actually comes in at a +1C.

Hmmm.”

I see his point. It’s possible that there is a “rounding bias” that may not be accounted for.

It may not make a difference, but I’m seeing from .1-.4, round down, and from .5-.9, round up. That makes a +.1 rounding bias (4 readings would be rounded down, while 5 readings would be rounded up).

Also, why have digital thermometers with an accuracy in tenths or hundredths, if you’re going to the nearest whole number?

Hmmm.