Mean and reported “Mean” temperatures and the consequences of the difference
Guest essay by Tom Quirk
The convention in meteorology is to report mean temperatures as the average of minimum and maximum temperatures. This assumption has been tested using temperatures recorded every 30 minutes through the 24 hour day at various locations in Australia by the Bureau of Meteorology and made available on their website. The period examined is from the middle of March 2013 to the end of April 2013. Analysis shows that distortions are introduced by the use of thermometers that measure minimum and maximum temperatures and more importantly that the averaging of minimum and maximum temperatures does not represent the mean for the period examined. Whether this is also true for the entire year should be tested.
The Bureau of Meteorology (BOM) on its website (http://www.bom.gov.au/australia/index.shtml) provides temperatures recorded every 30 minutes through the 24 hour day at various locations in Australia, an example, Canberra is at http://www.bom.gov.au/products/IDN60903/IDN60903.94926.shtml.
The convention in meteorology is to report daily, monthly or yearly mean temperatures as the average of minimum and maximum temperatures. This assumption can be tested using the BOM data.
13 locations around Australia have been selected for analysis. Figure 1 shows the average of the 30 minute intervals for 43 days in March and April for Cairns and Alice Springs. The figures show errors on the mean, not standard deviations. The data for the 13 sites divided into continental and coastal locations are shown in the Appendix.
The measurements at Alice Springs and Cairns are a perfect illustration that the mean is not always the average of minimum and maximum temperatures. For Alice Springs the average of the minimum and maximum temperatures is 0.12 +/- 0.12 above the average of all 48 30 minute readings while for Cairns the average of all 48 30 minute readings is 0.45 +/- 0.07 below the average of the minimum and maximum temperatures.
Figure 1: Temperatures measured at 30 minute intervals through a 24 hour day. The sample is for 37 days in March and April and the errors not the standard deviations are shown. The difference for Mean (30 minute Tmin & Tmax) – Mean (all 30 minute T) is -0.13 +/- 0.10 for Alice Springs and 0.46 +/- 0.07 for Cairns.
The minimum and maximum temperatures that are reported by the BOM are a result of readings from the two thermometers at 9.00 am each morning. This gives a record of the minimum temperature for the day of the reading since minima in general occur between midnight and about 7.30 am local time. This can be seen in Figure 1. The maximum temperature is a record from 9.00 am on the previous day. In general this maximum occurs before midnight.
As a test of the 30 min readings, the temperature differences of the 24 hour minimum thermometer and the 30 minute thermometer minimum value and likewise the temperature differences of the 24 hour maximum thermometer and the 30 minute thermometer maximum value were calculated. The results are shown in Figure 2. There are biases with the 24 hour readings being equal to or below the 30 minute minimum and the 24 hour maximum readings being equal to or above the 30 minute readings.
Figure 2: Maximum or minimum temperature differences for the 24 hour and 30 minute measurements as a function of the maximum or minimum 30 minute temperature measurement.
However these thermometer differences are not dependent on temperature as shown in Figure 2 where there are no significant trends. Note that there are a number of large differences. Some are due to the 24 hour record assumption that minimum temperatures occur between midnight and 9.00 am where in fact the minimum comes from the previous day after 9.00 am but before midnight. Other measurements may not to date have had a quality control check.
There is evidence of a systematic error in Figure 2 that is more obvious in Figure 3 and detailed in Table 1.
Figure 3: Maximum or minimum temperature differences for 24 hour thermometer temperatures and 30 minute temperature measurements
This bias is not unexpected as an extreme might occur in the 30 minute interval between regular measurements. A measure of this is to look at the temperature differences that occur in the 30 minute interval before and after the extreme minimum or maximum. The scatter is 0.6 0C for the maximum readings and -0.2 0C for the minimum readings, the same magnitude as the difference in Figure 3 and Table 1.
Table 1: 24 hour thermometer reading – 30 minute temperature readings
| Coastal | Continental | |||
| Temperature extremes | Minimum | Maximum | Minimum | Maximum |
| Number | 400 | 435 | 123 | 125 |
| 24 hour value – 30 minute value | -0.18 | 0.51 | -0.26 | 0.60 |
| Standard Deviation | 0.19 | 0.43 | 0.20 | 0.31 |
The effect that these systematic errors have on the mean temperature is given in Table 2 and shown in Figure 4. The average systematic error from the 24 hour thermometer readings is an increase in mean temperature of 0.14 +/- 0.04 0C.
Table 2: Difference for mean temperature for 24 hour thermometer reading – 30 minute temperature readings
| Latitude0S | Longitude0E | 24 hour value – 30 minute value0C | +/- Error0C | |
| Continental | ||||
| Alice Springs | 24 | 134 | 0.22 | 0.03 |
| Kalgoorlie | 31 | 121 | 0.13 | 0.03 |
| Broken Hill | 32 | 142 | 0.15 | 0.03 |
| Coastal | ||||
| Darwin | 12 | 131 | 0.12 | 0.02 |
| Cairns | 17 | 146 | 0.10 | 0.02 |
| Port Hedland | 20 | 119 | 0.16 | 0.03 |
| Brisbane | 27 | 153 | 0.18 | 0.03 |
| Perth | 32 | 116 | 0.18 | 0.03 |
| Sydney | 34 | 151 | 0.25 | 0.05 |
| Canberra | 35 | 149 | 0.10 | 0.05 |
| Wangaratta | 36 | 146 | 0.06 | 0.05 |
| Melbourne | 38 | 145 | 0.09 | 0.05 |
| Hobart | 43 | 147 | 0.03 | 0.05 |
Figure 4: Location differences of mean temperature for 24 hour thermometer reading – 30 minute temperature readings. The overall difference is 0.14 +/- 0.01 0C.
The corrections to the mean temperatures are therefore increased if the BOM 24 hour thermometer measurements are used rather than the 30 minute measurements.
This systematic error is a consequence of the “one-way” temperature recording where, for example, a 10 minute 10C fluctuation in temperature would give a 0.50C increase in mean temperature rather than the properly weighted 0.010C change.
Summary of comparison
The results of the analysis of mean temperatures are presented in Table 3 and Figure 5. The comparison of the average of the minimum and maximum temperatures with a mean of 48 measurements throughout the day shows an overestimate of the mean temperature from averaging minimum and maximum temperatures. All the differences are equal or increased with the use of the BOM 24 hour thermometer measurements. The values highlighted in yellow are over 2 standard deviations from no difference of mean and “mean”. If the distribution were normal this is a probability of 98% that the difference is real.
The variations in temperature difference are a function of latitude and longitude. For this analysis the locations have been grouped as coastal and continental. The map of Australia shows the locations selected for temperature analysis.
Table 3: Temperature differences comparing the average of Tmin and Tmax with a 24 hour mean.
Figure 5: Temperature differences comparing the average of Tmin and Tmax with a 24 hour mean.
These temperature variations are complicated as shown by the correlation coefficients for locations where the correlation coefficients for Wangaratta and Canberra are significantly different to other coastal locations while the continental locations are no different to the remaining coastal locations (Table 4 and Figure 6).
Table 4: Correlation coefficients for Tmin and Tmax
| Latitude 0S | Longitude0E | Numberof days | CorrelationMin & Max T | Error | |
| Continental | |||||
| Alice Springs | 24 | 134 | 43 | 52% | 11% |
| Kalgoorlie | 31 | 121 | 42 | 55% | 11% |
| Broken Hill | 32 | 142 | 43 | 74% | 7% |
| Coastal | |||||
| Darwin | 12 | 131 | 43 | -7% | 15% |
| Cairns | 17 | 146 | 43 | -14% | 15% |
| Port Hedland | 20 | 119 | 43 | 13% | 15% |
| Brisbane | 27 | 153 | 43 | 40% | 13% |
| Perth | 32 | 116 | 43 | 35% | 13% |
| Sydney | 34 | 151 | 51 | 58% | 9% |
| Canberra | 35 | 149 | 40 | 12% | 16% |
| Wangaratta | 36 | 146 | 43 | 19% | 15% |
| Melbourne | 38 | 145 | 49 | 59% | 9% |
| Hobart | 43 | 147 | 43 | 60% | 10% |
Figure 6: Correlation coefficients for Tmin and Tmax by latitude.
Discussion
The data and analysis covers up to 45 days of 48 temperature measurements made every 30 minutes. The results indicate significant systematic distortion of the reported mean temperature. Variations in this difference should be expected as the daylight hours are longer in summer than in winter with the extremes being in January and July. This is also a function of latitude where in Melbourne the extremes are 10 to 15 hours of daylight and Darwin 11 to 13 hours of daylight. However the period covered is from mid March to the end of April and lies between the extremes. It may well represent the average result.
However a full year is needed to establish the extent of the systematic distortions.
Conclusion
There is a systematic error using minimum and maximum recording thermometers. This is a consequence of the “one-way” temperature recording where, for example, a 10 minute 10C fluctuation in temperature would give a 0.50C increase in mean temperature rather than the properly weighted 0.010C change.
This preliminary analysis shows that around the Australian coast the mean temperature has been overestimated by 0.6 0C. If this is the general case throughout the year then the overall Australian temperature has been over estimated.
There is clearly a need to re-examine the reported Australian temperature record in the light of this analysis rather than the seemingly endless reworking of minimum and maximum temperature by adjustments.
If the mean land temperatures are overstated from averaging minimum and maximum temperatures and the air temperatures over the oceans are measured mean values then the blending of the two data sets creates a systematic distortion.
Computer models tuned by back-casting to reported measurements will in turn overstate feedback effects. This could be particularly the case for regional modelling and consequent projections.
Appendix
Selected locations show the average of the 30 minute intervals for over 43 days from 18th March to 30th April. The figures show local times and errors on the mean, not standard deviations.
| Continental | Latitude 0S | Longitude0E | Numberof days |
| Alice Springs | 24 | 134 | 43 |
| Kalgoorlie | 31 | 121 | 42 |
| Broken Hill | 32 | 142 | 43 |
| Coastal | Latitude 0S | Longitude0E | Numberof days |
| Darwin | 12 | 131 | 43 |
| Cairns | 17 | 146 | 43 |
| Port Hedland | 20 | 119 | 43 |
| Brisbane | 27 | 153 | 43 |
| Perth | 32 | 116 | 43 |
| Sydney | 34 | 151 | 51 |
| Canberra | 35 | 149 | 40 |
| Wangaratta | 36 | 146 | 43 |
| Melbourne | 38 | 145 | 49 |
| Hobart | 43 | 147 | 43 |
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Maybe I am misunderstanding the situation, but the average (or middle) value between minimum and maximum values in a set of data is call the median, not the mean. It sounds like you are taking an average of all the median values during a time period and averaging them together. I would call that an average of median values. This not a mean, but it is still a measure of central tendency.
Do I have it wrong?
Bob
From Merriam Webster’s Dictionary:
Median
Arithmetic mean
For the purposes of trying to establish temperature anomalies, does it really matter much as long as the methodology at any one station is the same over time?
BTW, taking the average of the maximum and minimum values is neither a mean nor a median of a full set of samples… which is kind of the point to the essay.
While it is a good idea to compare the max-min average temperatures to the more reasonable average of multiple half-hour measures, the analysis is hindered by the inability to know on which day the maximum or minimum occurred. This is the time-of-observation (TOBS) problem, which has been grappled with by various investigators (Vose et al, 2003):
Russell S. Vose, Claude N. Williams Jr., Thomas C. Peterson, Thomas R. Karl, and David R. Easterling. An evaluation of the time of observation bias adjustment in the U.S. Historical Climatology Network. Geophysical Research Letters, VOL. 30, NO. 20, 2046, doi: 10.1029/2003GL018111, 2003.
I believe the Vose et al adjustment method is an OK attempt, but it is not perfect, and therefore there will be unavoidable errors in comparing the max-min average with the “true” daily average.
This problem makes it hard to achieve what Tom Quirk wishes to do here–detect the errors associated with the min-max average. The best that can be done is probably to use the Vose method or some other TOBS adjustment method to try to pick out the proper day to associate with each minimum or maximum reading and go from there.
The new Climate Reference Network of 125 stations in the US does not have a TOBs problem, because it uses automated continuous measurements of temperature to determine a minimum and maximum 5-minute average associated with each hour, and then can unambiguously determine a minimum and maximum for each day. The CRN has been in operation for about 10 years, although only at full strength for about the last 5. An analysis of the bias associated with the min-max averages over the last 4 years was made on WUWT last August. An interesting result was that the bias was largely in one direction for the coastal stations, but the other direction for the continental stations. A typical size of the bias was 0.2 C, although it could extend almost to a full degree C. A given station seemed to maintain its bias directionality through all seasons and all years. A relation with relative humidity (RH) and latitude was suggested. It happened that about as many stations had a negative as a positive bias, so that the national average temperature was about the same using both methods. However, a nation with largely coastal stations or mostly continental stations could possibly have an overall bias associated with the max-min method. It would be interesting to see if Australia, with mostly coastal stations, might have such an overall bias. However, it would first be necessary to remove the TOBS bias by some means.
Considerable discussion occurred in the WUWT entries linked below regarding the possible effect on estimated temperature trends. If the bias for a given station stays relatively constant (as found for a majority of the stations), there would be little or no effect on the trend. For that matter, if the bias varies more or less randomly, there would also be little or no long-term effect on the trend. However, some persons continued to hold a different position.
http://wattsupwiththat.com/2012/08/30/errors-in-estimating-temperatures-using-the-average-of-tmax-and-tmin-analysis-of-the-uscrn-temperature-stations/
http://wattsupwiththat.com/2012/09/12/errors-in-estimating-mean-temperature-part-ii/
Will this be yet another reason to push the older temperature record down even further, thus CREATING an even greater temperature trend for the warmist agenda.
“There is clearly a need to re-examine the reported Australian temperature record ”
“Computer models tuned by back-casting to reported measurements will in turn overstate feedback effects”
Or are BOM preparing for the coming cooler period, and will use this as an excuse for their wild warmist projections?
Time will tell.
Sorry, Tom. I am still having a difficult time understanding exactly what you are doing. I believe that on each site’s chart, each dot represents 43 separate medians, taken from 43 sets of max-min temp readings over 43 days at that exact time.
The 24 hour mean you are talking about is the daily maximums over 43 days, with the mean drawn for that value, and the mean of 43 days of minimum daily readings also represented on the chart.
Your question is why do you see the magnitude of the differences that you see.
As you noted in the text, you should not be surprised with differences in the 24 hour and 30 minute metrics. I don’t how the magnitudes of the inevitable differences should look, but that may be a function of season, etc.
Interesting article.
Thinking about this a bit more, since you have the 30-minute measurements, you should be able to pick out with near-perfect accuracy the days when either the minimum or maximum occurred on the “wrong” day. So you could use the daily 30-minute measurements to determine the minimum and maximum of the day and not be limited to the “official” minimum and maximum reported by the BOM. Of course, the min-max thermometer might pick up slightly lower instantaneous minima and slightly higher maxima, due to responding to all the values in a day, but this is probably a small effect.
If you could do this for a few years for each station, it should result in a useful analysis.
“Note that there are a number of large differences. Some are due to the 24 hour record assumption that minimum temperatures occur between midnight and 9.00 am where in fact the minimum comes from the previous day after 9.00 am but before midnight.”
Does that assumption affect the “TOBS ‘adjustment?'”
I can’t see the point of this. The “mean” is defined as the mean of max and min, not the mean of 24 hr. There is no reason why it should be adjusted to the latter value.
There is a very good reason why it is defined as it is. We have a long record of min/max mean from older technology. We can continue it. We have only a short, recent record of 24 hr temperatures.
@David L. Hagen
David: It was not my intent to invite basic definitions of mean and median. Sorry if I conveyed that impression.
As you know, depending in the data set, the median can be radically different from the mean. both of which are expressions of central tendency of the data. I am trying to figure out what the data is, and how it was processed.
1. There are 43 days of 24 hour maximum and minimum readings for each site. Tom averaged the max and min readings over the 43 days and showed these values on the chart.
2. There are 43 days of other temperature readings, with with 48 readings taken each day, one every 30 minutes. Tom takes the max and min readings of each day.
3. The max-min 30 readings for each day added, then divided by two. This gives an mean which is identical to a median if you have only two data points. This is NOT equivalent to a mean of the 48 readings taken every 30 minutes. He then finds the mean of the 43 medians (not the 30 minute mean for the day).
4. The 43 daily max-min readings (medians) are then averaged. This is the value compared to the 24 hour reading max-min means.
It is obvious that there will be differences in the two values, since one was based on an average of medians, and the other was taken at different times, and was a process of taking means, only.
I am in danger of over-analyzing this thing, and my question is, did I get Tom’s method correct?
@Nick Stokes “The “mean” is defined as the mean of max and min…”
You just defined the median. The mean is the average of all values in the data set.
“In statistics and probability theory, the median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.” In other words, the median is the middle value, not the average value.
I didn’t intend to get into a discussion of elementary statistical parameters. Best intentions…
“Analysis shows that distortions are introduced by the use of thermometers that measure minimum and maximum temperatures and more importantly that the averaging of minimum and maximum temperatures does not represent the mean for the period examined. Whether this is also true for the entire year should be tested.”
Easy to understand. And this gives me a déjàvu with John Daly more than 10 years ago?
Another point is that UHI in urban places affects the minimum temperature reading most.
@Bob
May 10, 2013 at 8:42 pm
@Nick Stokes “The “mean” is defined as the mean of max and min…”
I think that’s CRU’s definition, not Nick’s.
I’d like to know how to calculate the mean temperature of my house, including hotplates, oven and refrigerator … and what is the mean supposed to mean ?
Dr Burns: I think the best way to calculate the mean temperature in your house it to get a thermometer, at least one, or several to get the mean temperature. In my house I mean to get one of those automatic, lockable thermostats for the hvac to keep my bride’s hands from dictating that mean.
In his comment Lance Wallace linked to a really good article he had written on pretty much the same subject. It is interesting and like this article has a lot of number crunching. I suppose the object is to come up with a number that is the best representative for the mean temperature of a given day, and that’s why they are using the Min/Max method to estimate that mean. There are obvious differences to be expected.
I suppose my basic question boils down to, “What exactly do we need to know about temperatures during any given time period?” As Lance said in his article, there may not be enough data to get the “true” mean.
Bob;
I suppose my basic question boils down to, “What exactly do we need to know about temperatures during any given time period?” As Lance said in his article, there may not be enough data to get the “true” mean.
>>>>>>>>>>>>>>>>
It is my position that the issue is more complex still. CO2 supposedly changes the energy flux at earth surface. We don’t measure energy flux in degrees, we measure it in watts per square meter (w/m2). Since P(w/m2) varies with T raised to the power of 4 (Stefan-Boltzmann Law of physics) even if we had incredibly detailed temperature data, there is no way to average it and get a meaningful number.
As an example, at 0 degrees C it takes 4.6 w/m2 to raise the temperature 1 degree. At 30 degrees C, it takes 6.3 w/m2 to raise the temperature 1 degree. In other words, averaging w/m2 will give you a different trend average over the course of a day than will averaging degrees. It isn’t average temp that tells us if CO2 is changing the energy balance at earth surface, it is average w/m2 that provides this information. Different methods of finding a mean temperature or average temperature (etc) are just different methods of getting the wrong answer.
Tom Quirk’s method is “less wrong” however.
I suppose my basic question boils down to, “What exactly do we need to know about temperatures during any given time period?”
Wind speed, air pressure, and humidity???
The temperature record is a PROXY for the heat retention in the system caused by increasing GHG’s. Not including the above you still are not computing the energy in the system to a reasonable level. Yes it is probably small, but then so is .9 W/M2!! (snicker)
davidmhoffer,
Good point about the difference in energy flux (w/m^2) required to increase temp (degrees Celsius) at different temperatures. I have been following this topic for many years and had never thought about that point.
It just reinforces my opinion that AGW is so much nonsense about so little, with very little actual science to back it up.
Wow, Canberra and Wangaratta are coastal! That’s news to the people who live there!
@Nick Stokes
Nick, we’ve crossed swords (pens? keyboards?) on this before, but for a new small audience, I just want to say that for what you are interested in (trends) I agree that it doesn’t make a difference which measure we use. But if one is interested in the physical processes of the climate, it would be the 24-h mean that counts, not the artificial, slightly biased min-max average, which has the further problem of being sometimes seriously influenced by the time of observation.
Suppose the coastal-continental effect has some validity. Then a mostly coastal country next to a landlocked one may have temperature estimates 0.2 C below the true mean, and the neighboring country 0.2 C above the true mean. Then the Global Climate Model goes crazy trying to fit the wrong data.
I’m reminded of Newton trying to fit the Moon into his new theory of gravitation. It didn’t work, so he sat on his calculations for some years. Then people found out their estimate of the distance of the Moon was wrong. Newton plugged the new distance into his old equation and voila! the new estimate agreed “pretty nearly,” and the rest is history.
Air temperature by itself is a poor indicator of the thermal state of the air. It is only a loosely coupled proxy for the temperature of the surface. The surface is by far; the most significant radiator of heat into space. Using air temperature as a proxy for surface temperature in a radiation “balance” seems to me to be so bad as to not even be wrong.
Before plunging into computations to divine meaning from mountains of data, one should step back and think if it makes physical sense. I’m a lazy Engineer. Taking the time to gather perspective saves doing a lot of meaningless work.
Australian Mean for the weekend.
In excel it say an average =Returns the average of its arguments.
And a Median =Returns the median of the given numbers.
I tried this on my weather Data for May so far I got..
average-8.6c.
Median=9.6c.
Christchurch NZ.
Sorry=Median=9.4c.
Nick Stokes;
There is a very good reason why it is defined as it is. We have a long record of min/max mean from older technology.
>>>>>>>>>>>>>>>>>
Ah yes, we have lots of it, so we must use it, even if it has been demonstrated that the result is meaningless.
The important issue is to what extent the temperature trend over the last 50 or so years results from using min/max temperatures. That is, to what extent is the warming trend an artifact of using min/max temperatures.
I wrote about this using the work of statistician Jonathan Lowe, and more than 40% of the warming over the last 60 years is an artifact of using min/max temperatures and isn’t real.
http://www.bishop-hill.net/blog/2011/11/4/australian-temperatures.html