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I read in some article that the month of the year was being considered as qualitative nominal variable, but for me the month of the year has a clearly ordered structure and should therefore be considered as qualitative ordinal. Am I right?

On the same article it was said that the year was a qualitative ordinal variable. I am OK with that, but I also wanted to ask if it was possible to consider the year as quantitative discrete.

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    $\begingroup$ Ordinarily it's none of the above. In many applications it would be considered a coarse circular variable: January precedes February precedes ... precedes December precedes January, with roughly the same distance (elapsed time) between each. You ought therefore to consider whether this is even a useful question to ask. $\endgroup$
    – whuber
    Jan 23, 2018 at 19:46

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I find it hard to believe that there are grounds for regarding year or month as qualitative. You don't give a precise reference and you don't report the argument, so further comment on that view is difficult for me.

A year variable with values such as 2018 is evidently quantitative and numeric (I don't distinguish between those) and ordered (2018 > 2017 > 2016) and also interval in so far as differences such as 2017 $-$ 1947 are well defined (as indeed we all know from childhood in working with people's ages). It's not a ratio scale in so far as the zero point is arbitrary. The test is that ratios such as 2017/1947 make no substantive sense. (Detail: There was in history, even in retrospect, no year zero; 1 BC/BCE is deemed to have been followed immediately by 1 AD/CE; no one complained at the time if only because the labelling was introduced much later. Recall that zero took some time to be accepted as a basic mathematical idea.)

(The illustration here uses the "Western" calendar; the same arguments apply to any other calendar, so far as I am aware.)

It's not even essential to regard year as discrete. Whenever in physical or environmental science (for example) something varies continuously with time, so we have (e.g.) monthly or daily measurements, so too time can be regarded as continuous: 2017.5 is well defined as half-way through 2017.

The case of month is interesting. First, we need to be clear that monthly dates such as January 2018 are quantitative, numeric, ordered and interval insofar as for the purposes of CV they will generally be handled in terms of a count of months before and after some origin used by particular software. Good software arranges that dates are shown conventionally, but calculations are based on integers on a defined scale.

Month of the year January to December or 1 to 12, say, is quantitative and numeric and ordered in so far as no person who completed elementary education reasonably well has a problem in putting the months in order. But the order is circular, in so far as a particular 12 is necessarily followed by another particular 1.

This may clash with any narrow definition of ordinal scale you encounter, but such a clash just shows lack of imagination and experience on the part of the definer, or more charitably an attempt to keep things simple by leaving out complications from some introduction or elementary treatment. Ordinal, I suggest, means "can be placed in a definite and repeatable order" and doesn't exclude that order being circular.

Evidently it's just a convention to start with January: we all know at least a little about calendars associated with particular religions or say academic, financial and hydrological years. So, to spell it out, month of year is not a ratio scale as the origin is quite arbitrary. And indeed it can be convenient, even natural, to think that years "start" in months other than January.

Circular scales are all around you.... A full and busy statistically-based career could conceivably include no need to work with circular scales, but

  • seasons: spring, summer, autumn (fall), winter

  • months: December January ... December January ...

  • day of year 1 to 365 or 366 (complicated by leap years)

  • compass direction (aspect, azimuth)

are some basic examples. Circular scales usually need care and attention, so for example the mean of directions 1 degree (just East of North) and 359 degrees (just West of North) is not sensibly 180 degrees (South).

The unequal lengths of months may or may not be important detail. In practice (at least in fields I know about) people with monthly data usually treat the months as equally spaced and as equally long, even though that's not quite right. This is a matter of convenience rather than a denial of fact.

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    $\begingroup$ +1. But I was taken aback at the intimation that the nature of a scale might depend on whether people had invented a zero. That suggests some kinds of measurement (like temperature) may be inherently of ratio type, but for historical or cultural reasons they might not ordinarily be expressed as such. As far as the ordinality of months goes, Stevens' wisdom was that the scale of a measurement is determined by its group of transformations. Because the group for months differs from the larger order-preserving group for nominal data, it's wrong to characterize months as nominal. $\endgroup$
    – whuber
    Jan 24, 2018 at 14:35
  • $\begingroup$ The history of mathematics makes clear that ratios were well understood before people had a clear concept of zero. However, it seems easiest to explain ratios not making sense by underlining whenever zeros are arbitrary (e.g. Celsius or Fahrenheit scales). Could you develop your understanding in another answer? $\endgroup$
    – Nick Cox
    Jan 24, 2018 at 14:54
  • $\begingroup$ This conversation highlights an interesting distinction. For some kinds of measurements, there is no inherent origin: historical times, points on the Equator, locations in space, etc. For other kinds of measurements there is an inherent origin and it can be discovered even by studying measurements that are far from that origin: temperature, mass, speed, charge, concentration, and other physical quantities are the exemplars. And note that not all are ratio-type measurements: concentration, for instance, cannot be. $\endgroup$
    – whuber
    Jan 24, 2018 at 15:41
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Month should be considered qualitative nominal data. With years, saying an event took place before or after a given year has meaning on its own. There is no doubt that a clear order is followed in which given two years you can say with certainty, which year precedes which.

As for months, on their own, you cannot. I agree that there is, in some sense, an order to months but the order or relative positions (without the years the months correspond to) provide no information on their own. For example, if someone deduces that car accidents are most likely to occur in May of all the months by looking at historical data, the relative position of the months have nothing to do with the pattern. I think it is fundamentally incorrect to think of months as a circular scale. In theory, January 2018 can be distinct from January 2019 in every conceivable way. There is no weight in saying that there is a circularity in months. They are labels, arbitrarily chosen by humans, to literally 'nominate' time.

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    $\begingroup$ So at the end of December does it come as a surprise to you when it becomes January? If it were nominal anything could happen including a repeat of December. $\endgroup$
    – mdewey
    Oct 10, 2020 at 14:13
  • $\begingroup$ The climate doesn't believe you, nor does anything or anybody that responds to seasonality. Months may be invented, but they are invented to recur completely predictably. $\endgroup$
    – Nick Cox
    Oct 10, 2020 at 15:04
  • $\begingroup$ Many, perhaps most, ordinal scales are invented for the purpose. Try poor, adequate, good, excellent as gradings. $\endgroup$
    – Nick Cox
    Oct 10, 2020 at 15:42
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    $\begingroup$ @mdewey It will most definitely come as a surprise to me if after December 2019 it became January 2019. Without the crucial knowledge of which year the months correspond to, months cannot be ordered. Tell me, does January come before or after December? $\endgroup$ Oct 11, 2020 at 15:32
  • $\begingroup$ @NickCox I agree with you. I do not dispute seasonality, but seasonality can only be identified reasonably with time series data (with the year). If I gave you, say sales data with only the respective month, you CANNOT tell me the effect of seasonality because you do not know which data point fell before it and hence you do not know the effect that was brought on by the month/seasonality. $\endgroup$ Oct 11, 2020 at 15:37

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