# Met Office mean average temperatures and margin of error

#### UncleBucket

##### New member
I have been debating with a few other people on another board, regarding the correct way to calculate the mean average of a list of values, where those values are recorded temperatures.

A great many people seem to believe, for example, that if each temperature reading is accurate to +/- 0.5°, the reading should be rounded to the nearest integer. I don't have a problem with that, but when it comes to the calculation of the mean average, we have a difference of opinion. Many people seem to believe that if the list of values, for which the mean average is to be calculated, are all integers, then the mean average itself must also be an integer! This makes no sense at all to me.

Another person argues that because the list of values contains "measured approximations", it is not appropriate to express the mean average to 1 or more decimal places, and that any decimal digits must be truncated!

I welcome your views on these matters.

Just recently, I was browsing the Met Office website, and found a document which gives details of the accuracy of each instrument, and the margin of error when calculating the mean average. I will quote it:

The random error in a single thermometer reading is about 0.2°C (1 σ) [Folland et al., 2001]; the monthly average will be based on at least two readings a day throughout the month, giving 60 or more values contributing to the mean. So the error in the monthly average will be at most 0.2/√60 = 0.03°C and this will be uncorrelated with the value for any other station or the value for any other month.
I am not an expert in this field, so could someone please explain to me the validity of that formula and why it is correct, assuming it is. Clearly, the accuracy of the mean average appears to be greater than that of any of the individual values in the original list. Would this be because with a large sample of data, the errors within each instrument reading tend to cancel each other out?

#### Klaas van Aarsen

##### MHB Seeker
Staff member
Welcome to MHB, UncleBucket!

I have been debating with a few other people on another board, regarding the correct way to calculate the mean average of a list of values, where those values are recorded temperatures.

A great many people seem to believe, for example, that if each temperature reading is accurate to +/- 0.5°, the reading should be rounded to the nearest integer.
Not true. A reading should always be made to a digit more than the measurement markers can indicate. The last digit will be somewhat of a guess, but it still improves precision.

If the precision is for instance $\pm 0.7°$, it is customary to register the measurement in as many digits as this precison. So you would have for instance $20.1 \pm 0.7°$.
On the other hand, when a measurement is given as $21°$ without any precision, it is customary to assume a precision of $\pm 0.5°$. The measurement could then be written as $21.0 \pm 0.5°$.

I don't have a problem with that, but when it comes to the calculation of the mean average, we have a difference of opinion. Many people seem to believe that if the list of values, for which the mean average is to be calculated, are all integers, then the mean average itself must also be an integer! This makes no sense at all to me.
Agreed. That makes no sense at all.
An average of integers will typically be a fractional number.
Moreover, a temperature is not an integer to begin with.

Furthermore, when you take the average of, say, 100 temperatures that are all supposed to measure the same temperature, the effective precision is 10 times more accurate.
So if the original measurements have a precision of $\pm 0.5°$, then the mean measurement will have a precision of $\pm 0.05°$.
This means the result should be written down with 2 digits after the decimal point.

Another person argues that because the list of values contains "measured approximations", it is not appropriate to express the mean average to 1 or more decimal places, and that any decimal digits must be truncated!
Not true.
For starters, any intermediary results should always have a couple of digits more than the final result to avoid unnecessary rounding errors.
The final result should have as many digits as is appropriate for the final precision.
A precision is usually specified in 1 significant digit, although in cutting-edge research 2 digits might be used.

I welcome your views on these matters.

Just recently, I was browsing the Met Office website, and found a document which gives details of the accuracy of each instrument, and the margin of error when calculating the mean average. I will quote it:

I am not an expert in this field, so could someone please explain to me the validity of that formula and why it is correct, assuming it is. Clearly, the accuracy of the mean average appears to be greater than that of any of the individual values in the original list. Would this be because with a large sample of data, the errors within each instrument reading tend to cancel each other out?
Yes. The errors in the measurements will partially cancel each other out.
The precision of an average of $n$ measurements is $\sqrt n$ times more accurate than each individual measurement under a couple of assumptions.
The most important assumptions are that the precisions of all measurements are the same and that all measurements have been executed independently from each other.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
All this assumes of course that people care about the precision.

Suppose you have a digital thermometer that measures the temperature up to 3 decimal digits, but people only want to know if it is warm or cold, it makes little sense to accurately specify the temperature like a fusspot.

#### HallsofIvy

##### Well-known member
MHB Math Helper
All this assumes of course that people care about the precision.

Suppose you have a digital thermometer that measures the temperature up to 3 decimal digits, but people only want to know if it is warm or cold, it makes little sense to accurately specify the temperature like a fusspot.
We fusspots object! String that temperature out to as many decimal digits as possible!

#### UncleBucket

##### New member
And on the other board, still the arguing goes on!

It is now being said...

As SF (significant figures) says you cannot gain precision better than you least accurate instrument, any global average that includes NOAA data cannot be (following the rules of science taught at all universities in freshman science class) more accurate than 1 degree Fahrenheit.

I think that's tosh, for all the reasons we've discussed in this thread so far.

#### Klaas van Aarsen

##### MHB Seeker
Staff member
And on the other board, still the arguing goes on!

It is now being said...

I think that's tosh, for all the reasons we've discussed in this thread so far.
I don't know what this other board is, but can I assume it is not a dedicated math or physics forum? There is only so much to discuss before reaching a resolution about something that is as fundamental as this.

#### UncleBucket

##### New member
I don't know what this other board is, but can I assume it is not a dedicated math or physics forum? There is only so much to discuss before reaching a resolution about something that is as fundamental as this.
No, it isn't. It's a board dedicated to global warming denialism, where the findings of organisations like NOAA and NASA are dismissed as a "liberal agenda" or "communist conspiracy". Even the basic rules of maths are distorted to help these people believe what they want to believe.