I understand that latitude and longitude are interval data, but are coordinates (e.g. Point(123, -123)) also considered interval data? If so, how can the standard deviation, mean etc. be calculated?


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The classification of measurements into nominal, ordinal, interval and ratio scales is sometimes helpful, but it has wasted a great deal of individual and collective time in pedagogy and polemics, if only because (and not only because) it is far from as rigorous or as complete as it is sometimes claimed to be. As far as statistics is concerned, many developments since (and before) this scheme was first outlined by S.S. Stevens in 1946 conspire to make it seem increasingly irrelevant or unduly dogmatic. Thus "cat" and "dog" are undoubtedly no more than nominal scale data, but their statistics starts with counts of cats and dogs, which allow a great deal of mathematics. Or doctrine that you can't, or shouldn't, take means of ordinal data is denied by the fact that many methods for ordinal data do precisely that in some sense.

At its simplest, a measurement on ratio scale is one measured relative to a natural and uncontroversial zero, so that human height and weight are examples and Celsius and Fahrenheit temperatures are not. Loosely, ratios make sense for measurements on ratio scales, so that heights of 2 m and 1 m allow a ratio of 2 to 1 to be calculated and also allow statements such as Joanna is twice as tall as her sun Jim. Conversely, there is no useful meaning to a notion that 20$^\circ$C is twice as hot as 10$^\circ$C. We could if we wished say that the former is twice as far from 0$^\circ$C as the latter, but ratios don't enter into analysis of such temperature measurements. .

Similarly, coordinates of points (for simplicity, let's suppose the reference is to points in a plane, rather than on the surface of a sphere, or in any other space) allow limited statements such as comparison of different distances from the origin. But in general the origin is likely to be one of convenience at best, for all that convenience beats its opposite.

Measurement of point coordinates is more nearly interval scale than ratio -- although neither category quite applies, as the measurement has bivariate flavour too. But that doesn't rule out calculations of means, standard deviations, and much else from sets of point coordinates, with respect to one axis or to both axes. You can for example combine distances in both directions, such as combining distances from a mean, median or other centre into some overall measure of concentration or dispersion based on applying Pythagoras's problem.

  • $\begingroup$ Hi there, thanks for explaining the different types of data! For coordinate Point(123, -123) for example, how would you suggest we display the mean and standard deviation? Would it be Point(mean1, mean2) and Point(stdev1, stdev2)? $\endgroup$
    – Verum
    Aug 10 at 14:18
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    $\begingroup$ The means of x and y coordinates define a point that can be considered the bivariate mean. That is sometimes interesting or useful. The variability is usually better considered as something more like the mean distance from that location. considered Pythagoras-wise. But cases vary. If there is an underlying transport network, you will or should want something different. $\endgroup$
    – Nick Cox
    Aug 10 at 14:22
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    $\begingroup$ @Verum: The proper generalization of the variance (and standard deviation) to multivariate data is the covariance matrix. For two-dimensional data, like your coordinates, this can be naturally visualized as an error ellipse. $\endgroup$ Aug 10 at 22:54
  • $\begingroup$ As both coordindates have the sme units, a variability measure that combines values on both is defensible. An old but still useful reference is Neft's 1966 monograph. jstor.org/stable/2284047#metadata_info_tab_contents $\endgroup$
    – Nick Cox
    Aug 11 at 6:43

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