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Some continuous variables like Precipitation have values that can be summed. But others like Temperature do not. Adding two temperatures together simply does not make sense. Do these variables whose values can be summed have an adjective like summable or additive?

To better appreciate the difference between summable and non-summable variables, simply look at a Combination Chart (aka Bar Line Graph) used to display Precipitation and Temperature on the same chart. Precipitation is shown with solid bars, because it is summable, hence could be stacked. Temperatures are shown with a solid line, because they are not; hence do not need to be stacked (and could not).

The issue is not limited to temperatures by the way. And it's important, because of all the quantities defined for the measure of physical phenomenons, temperature is pretty unique. It is one of the very few quantities for which unit conversion cannot be done with a simple ratio. Instead, it requires a ratio and an offset. But other non-summable continuous variables can be found, especially when dealing with percentages. Some percentages are summable, while others are not. And some have their total sum capped at 100%, while others do not, especially when dealing with percentages used to measure variations of another numerical variable.

In that respect, angles are similar to percentages. In some cases, they can be summed. In others, they cannot. And when summed, their total sum might be capped to a certain level like 90 degrees, 180, or 360, or not have any cap at all. For example, a measure of rotation is not capped. But a measure of slope is capped at 360 degrees.

Therefore, it is important to realize that the summable nature of a continuous variable is not a simple factor of its unit. Instead, it is a factor of its unit in a particular context. And I must suspect that the analysis of summable variables should be different from the analysis of non-summable variables. Furthermore, the way they should be graphed are clearly different, as indicated earlier.

Note: the fact that a variable cannot be summed does not mean that it cannot be averaged. Indeed, an average should not be viewed as a sum followed by a division. Instead, it should be viewed as an atomic aggregation, which could be computed in many other ways. For example, Euclidian geometry can be used to compute an average between two values with a compass and a ruler, without having to compute their sums.

Nick Cox
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Ismael Ghalimi
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  • If adding temperatures makes no sense, then wouldn't averaging them be nonsensical too? – whuber Jun 28 '15 at 09:21
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    No, averaging them makes sense. An average should not be viewed as an addition followed by a division, it should be viewed as an atomic aggregation. – Ismael Ghalimi Jun 28 '15 at 09:24
  • Would you also view addition of absolute temperatures as nonsensical? – whuber Jun 28 '15 at 09:28
  • Yes, because the reference for a temperature does not matter. You could add or substract temperature variations, but not temperatures themselves. – Ismael Ghalimi Jun 28 '15 at 09:30
  • Thanks. I'm still trying to understand: would you consider angles to be additive? – whuber Jun 28 '15 at 09:37
  • Yes, but not always. In that respect, angles look a bit like percentages. See my updates to the post for that. – Ismael Ghalimi Jun 28 '15 at 09:39
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    You can add temperatures. Temperature is the energy per unit volume. If you have 1L of gas at 400°K and another 1L of gas at 500°K and you compress them into 1L, that gas will have temperature 900°K. – Neil G Jun 28 '15 at 09:54
  • That is quite right. You can add temperatures, sometimes, much like you can add angles or percentages sometimes. But some other times, you cannot. Sometimes, it's a matter of the unit being used, and sometimes it's a matter of the context. For example, no matter what the unit in use, you cannot add the temperatures observed in two different cities, or in the same place at two different times. – Ismael Ghalimi Jun 28 '15 at 09:56
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    See the thread http://stats.stackexchange.com/questions/154463/what-is-the-terminology-for-data-aggregated-via-summed-totals-versus-data-aggreg for the terminology intensive and extensive. – Nick Cox Jun 28 '15 at 10:05
  • Pedantically: Note that the units of absolute temperature are strictly kelvin K, not $^\circ$K. (Presumably when units were cleaned up in the middle 20th century it seemed that $^\circ$C and $^\circ$F were too entrenched to be modified.) – Nick Cox Jun 28 '15 at 10:09
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    The fact that precipitation is always a sum or integral over a time period (minute, hour, day, month, year, whatever) whereas temperature is in principle defined at any instant is important to your leading example. – Nick Cox Jun 28 '15 at 10:12
  • Nick, would you mind adding an answer with a link to the question covering intensive/extensive? This is exactly what I was looking for. – Ismael Ghalimi Jun 28 '15 at 10:48
  • I have to say that the implication of the previous thread is that this question is a duplicate. – Nick Cox Jun 28 '15 at 11:41

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