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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If one went to wunderground and you pull up a private station near you that updates every five min or so, and you calculate an average using hourly, 30 min, 15 min, and 5 min readings, you’ll come up with varying means. Using 9 am reading times with same protocol will obviously give different means. But you can also calculate a more precise TOB adjustment as well.
TOB adjustments that are applied currently are just an average of a range. The TOB adjust is not robust on a monthly basis, but tend to average out for the annual.
I have found in N MN, the TOB adjust for am and pm both leave negative residuals in winter and positive residuals in the summer when compared to midnight stations. Spring and fall were the transition months.
Last years monthly CRN readings versus the USHCN 2.5 readings had the same issue. CRN tended warmer in winter and cooler in summer. USHCN, since 2000, only applies TOB adjustments. Since the residuals showed up even on the national dataset of USHCN, then the TOB adjust needs to be further investigated and refined. With all the midnight stations that are out there now, a more precise TOB can be calculated on a monthly basis. In fact, when Vose 2003 investigated this, they could have used the data from the 500 stations they used to apply a more precise TOB to the record and continued with that approach. A program could be constructed to do this.
Weather patterns greatly affect the TOB temps. How have weather pattern changes on decadal scales affected the TOB temps?
Duluth, MN has hourly obs since 1941. Looks like I can pick a month and a couple obs times and investigate the matter. Anyone else game for an investigation of this sort?
Nick Stokes says @ur momisugly May 10, 2013 at 8:15 pm
“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.”
I understand that.
But a 24h record from 1951-2012 (WMO station #260, De Bilt KNMI-NL) shows a STDEV of 5,23K for the Min-Max method and a STDEV of 2,39K for the 24h mean.
Using MinMax data should give greater uncertainties when used in models.
I think this is a useful analysis but I think we knew the conclusions already. AGW is predicated on the mean/median/mid-range temperatures but these can be affected just as much by increasing min temps as max. And as I understand it, it’s the min temps that are increasing and not the max.
Of course, this is not emphasised (mentioned?) by AGW fanatics because it’s not very exciting. It’s much better to allow people to believe that it’s the max temps that are increasing. If people realised that average temp increases were being driven min temps then they’d probably be pleased that, for example, nights weren’t quite so cold.
@Peter Ward,
Actually daily max – min temps are fairly consistant, ~18 some degrees F or so. Follow the link in my name to the updated charts page.
Robert_G
I hesitate to add my two cents (and maybe this has already been discussed and I missed it), but shouldn’t all these different ways of comparing the different sampling “means,” really be compared to the measured temperatures integrated over time to get the area “under the graph”.
The unit would be a temperature (degree)-day. Short-lived extremes which are reported as representing the day’s “average,” would be appropriately muted, even though they could seriously impact the (min-max)/2 technique.
NOT A COMMENT not sure how else to contact you. [Sorry Anthony, I’ve only commented once before. i’d like “Robert_G” if that is OK]
[Well, you’ve commented, and thus, you’ve contacted him. But you have established no reason why he should spend time, money and effort contacting you. 8<) What do you need to say (to communicate) and what will he gain (the rest of us gain) or learn from that communication? Mod]
My original comment stands, although now probably quite derailed.
I thought that since it’s status was “Awaiting Moderation,” that the “Moderator” (apparently not A. Watts) would have the chance to make an editorial correction without notifying the entire world. I didn’t expect a hair-trigger response. Sorry for my misunderstanding. I apologize for any inconvenience to you and the readership.
The temp records are riddled with such errors. For example, in Perth (Metro 9225) on 14 March 2013 the real maximum was 19.7C just after 6am but the BoM official max for that day of 24.4C actually occurred just before 9am the following day, 15 March. About seven hours later around 4pm on 15 March the temperature reached 29.2C, so the same day effectively scored two different maxima. It might seem trivial but the inaccurate additional 4.7C logged for 14 March means the average max for the month of March in the capital of Perth ends up at 28C instead of 27.8C.
The Perth daily press printed very early in the morning of 15 March couldn’t possibly know that the previous day’s maximum was yet to occur more than six hours after the paper was printed (???) so the real max of 19.7C was “incorrectly” printed … http://www.waclimate.net/imgs/14-mar-2013-perth.gif. Look up BoM max for Perth (9225 – http://www.bom.gov.au/climate/data/) on 14 March and it’s 24.4C. There are similar errors at various surrounding stations including rural affected by the abnormal cold front on 14 March – the fourth coldest early March day in Perth since 1897, but that’s not what the record books will tell you.
Also worth looking at a brief BoM paper which signals thousands of questionable Melbourne temps from 1979 to 2008 – http://www.amos.org.au/documents/item/392
… and Ed Thurstan on ACORN errors … http://www.warwickhughes.com/agri/ThurstanACORN28apr13.pdf
… and section 8 (p66) of the ACORN techniques … http://cawcr.gov.au/publications/technicalreports/CTR_049.pdf … which includes:
The one-minute data indicate that the only historical observation practice which shows substantial systematic differences from the current standard is the measurement of minimum temperatures using a 0000-0000 day (i.e., midnight to midnight). Averaged across the 32 stations, this gives mean minimum temperatures 0.25°C cooler than the current standard, whilst the impact on extremes is stronger, with the mean value of the highest minimum temperature of each month being 0.58°C cooler on average. All 32 stations show cooler minimum temperatures for a 0000-0000 day than a 0900-0900 day, but the differences were smallest (typically near 0.1°C) in the tropics. They were largest (0.4-0.6°C) at some southern coastal stations (Fig. 22). As about 30% of the network was using the 0000-0000 day in some form prior to 1964, these results would suggest a potential inhomogeneity in Australian mean minimum temperatures of approximately +0.08°C in 1964.
The ACORN techniques section also deals with the unresolved issues of Automatic Weather Stations introduced since the early 1990s and the effect of 1972 metrication (http://www.waclimate.net/round/acorn/index.html).
It all seems a bit pointless comparing historic records, no matter how much raw, HQ, ACORN, AWAP “correction” is or isn’t applied by the BoM at every station for every day back to 1910.
davidmhoffer said:
“Nothing like stating for the record that you failed to understand the discussion! Let me dumb it down for you. The mean could be yielding an increasing temperature trend when the earth is actually cooling, or vice versa. That’s why.”
Ouch! Moderator – condescension alarm! You use the word “could”. I think that’s a bit wishy-washy – all sorts of things _could_ be happening. I have no objections to a new 24-point (or whatever) mean being used where it is possible, but to compare to older temperatures only a 2-point mean is available.
And I do have some sympathy with ferdberple’s nihilistic assertions about non-equivalence of temperature with respect to humidity, but how is science to progress except on the basis of analysis of objectively measured observations? Rejection of analysis of a 2-point mean must surely attract use of the dreaded D-word.
Rich.
Looked at a station nearby on wunderground. The station updated every 5 min.
The max/min mean was 44.5. The mean of the entire days readings was 43.4(-1.1). When using just the hourly readings it was 43.3. Then I included the half hour readings which gave me 43.4. So yeah, averaging out the entire days readings does net a cooler mean.
A problem with using a simple average of max and min temps as a mean temp for a day can be illustrated by areas where sudden weather changes are common. On the south coast of Western Australia for example you often get the situation on a hot day in summer where there is cooling from midnight to say 5 or 6am to about 18-20C, then rapid warming to around late morning or early afternoon, to around 40C. With the passage of a front or trough the temperature could drop back to 18C within an hour and the rest of the day would remain cool, 18C or below. The maximum would be 40c, the minimum say 16C so the mean would be 28C but in fact the true average temperature for the day would be well below 28C.
It’s all to do with the area under the curve.
Coming late to this discussion, I am baffled by the characterisation of Canberra as “coastal”. Ours is very much an inland climate, with great extremes between winter and summer (up to 50C). The daily fluctuation can be as much as 30C. It is much more like that of Alice Springs in central Australia than that of the closest coastal area some 200kms away.
This is relevant to the discussion for a couple of reasons. One is that the BOM has just invented a new metric called the National Average Temperature, which is completely bogus thanks to the inclusion of dodgy data and the inevitable unequal distribution pattern of weather stations – even allowing that such a metric has any intrinsic value or meaning, which is debatable.
The post above illustrates that local factors greatly influence the outcome of any generic statistical technique applied across diverse locations. In the tropical north, temperatures do not vary much either within a day or between seasons, compared to those in landlocked areas like Canberra (we are also in the lower parts of a mountain range here). Then, as someone pointed out above about the weather in coastal WA, some places are prone to dramatic temperature changes in a short space of time, which makes any averaging technique problematic.
As someone who has the BOM’s Canberra weather tab permanently open and checks it regularly, I can also attest that it is often just plain wrong. It is updated every 10 minutes, and has been known to show fluctuations of 5C between adjacent readings when no such event occurred.
We should all be grateful for the work of people like Anthony and his volunteers in the US, and Tom Quirk, Geoff Sherrington and others here, to try to put some rigour into the poor quality data and data analysis that our national weather agencies have been dishing out for so long.