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I have a series of daily readings (NOx emissions) measured in parts per million.

I need to aggregate the daily readings into a monthly measure.

Clearly a sum operation will be incorrect (parts per million is a ratio).

Am I correct that aggregating samples in parts per million can be achieved with a simple mean operation?

e.g (if a month had 2 days).

100ppm (day1) + 500ppm (day2) = (100+500)/2 = 300ppm

What's the correct way to get the NOx in ppm over a whole month?

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@dmckee - but then I could (theoretically) end up with a total that was 2 million parts per million? – mtmacdonald Jun 26 '13 at 16:17
But what if I want the total emissions for the month and not the average emissions? – mtmacdonald Jun 26 '13 at 16:23
up vote 2 down vote accepted

The deal here is that you have to be very careful about what quantity you are interested in and what you have.

Assumptions you have measurements $f_i$ of the fraction of stuff (doesn't really matter what) during time interval $i$. You may also have measurements of $O_i$ of the total output of the medium in which stuff makes up a fraction. Alternately you may only know $\bar{O}$ the average output or the total output over the entire time range $\mathcal{O}$. Note that for $n$ measurements at uniform spacing $\mathcal{O} = n * \bar{O}$.

  • Case 1: You want to know "how much stuff" totaled over several time periods and you have both fractional values and outputs for every period.

    This is as good as it gets.

    You need to add up the daily stuff. The daily amount of stuff is $S_i = f_i O_i$. That's just the definition of $f_i$. So assuming you have the daily outputs you get: $$ \mathcal{S} = \sum_i S_i = \sum_i f_i O_i \quad .$$

  • Case 2: You want to know "how much stuff" totaled over several time periods, but you have the total output or average output without periodic output values. The best you can do is $$\mathcal{S} \approx \sum_{i=1}^n f_i \bar{O} = \mathcal{O} \frac{1}{n}\sum_{i=1}^n f_i = \bar{O} \bar{f}$$ where $\bar{f}$ is defined by this relation as the mean fraction. In this case it makes sense to take a mean of the fractional reading, but only because you don't have enough data to get the right answer.

    This will be approximately correct if (1) the $O_i$s have small variance or (2) the $f_i$s have a small variance or (3) you have a lot of data and the $f_i$s and $O_i$s are uncorrelated.

  • Case 3: You want to show that the fraction is uniformly above or below some limit. Then you just need the $f_i$s, and computing their mean and variance is fine as long as (1) the answer is a long way from the limit and (2) the $f_i$ are uncorrelated with the $O_i$s.

  • Case 4: You want to show that the stuff is uniformly above or below some limit. Then you need the daily values to do it right. In the absence of the daily output you can fall back on the average output again, but it will only be reliable if all three conditions listed for case 2 apply.

Side note: You were concerned in the comments that 2 million parts per million doesn't make sense. And you are partially right. You can't have a fractional concentration of 2 million ppm, but there are cases when that value is meaningful. You probably already understand this. People are happy to talk about 200% in some cases, but "percent" is literally "per one hundred". Two hundred pars per hundred is the number 2 and it makes sense in cases involving changes but not in cases involve the fraction of people who qualify for something.

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Thanks. I've understood this by thinking of NOx in ppm as being like a 'concentration'. So we can either work with 'average concentration' or we will also need to know absolute quantities in some other measure if we want to consider totals. – mtmacdonald Jun 27 '13 at 7:15

I assume you are asking for the expected value of the concentration of NOx in air. If you multiplied this by the total mass of air passing through your system in a month, you would get the expected total mass of NOx passing through the system (in a month.)

If your measurements fit a Normal or Poisson distribution, then the expected value would be the arithmetic mean.

You should also calculate and report a statistic representing the spread of your measurements, for example the standard deviation, so that the uncertainty of your value for the month can be known.

Finally, you should plot your daily data - just to confirm that something unexpected was not going on (for example all the NOx was on a single day.)

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