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Population counts are typically referred to as discrete or quantitative data. Why is population density a continuous data type when it is typically measured for aggregate areas such as census tracts or districts/neighbourhoods (ie, it can't be measured at any point on a surface like gradient or temperature).

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Updated my answer. It is continuous if you consider the values. But if you're asking if it's spatially continuous then the answer is no. –  R.K. Nov 5 '12 at 15:00

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There are two definitions of continuous data ( at least that I could find online).

continuous data

Data such as elevation or temperature that varies without discrete steps.

enter image description here

spatially continuous data

A surface for which each location has a specified or derivable value. Typically represented by a tin or lattice (e.g., surface elevation). enter image description here

So to answer your question, it considered continuous in the sense of the first definition. Population densities are ratios and therefore, have values that vary continuously, unlike population counts which have values that vary in discrete increments. It is not spatially continuous data. It is areal data as @Radar have said.

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+1 This answer addresses both spatial and non-spatial forms of data continuity, which is important considering the forum in which the question was asked. –  Nick Ochoski Nov 5 '12 at 19:32
    
Although these are common attempts at describing "continuous" data, neither is an effective definition. The first is always incorrect: data, being finite in nature, must always vary in discrete steps. Lurking here is the distinction between what the data are and how we model them. The second is vague: what is meant by a "surface," which "locations" are involved, and what is meant by a "specified or derivable value"? It is also worth being aware that other fields--most notably mathematics--have standard definitions of "continuous" which differ markedly from these two. –  whuber Nov 5 '12 at 22:00

A population count is a point measure. Density is an areal measure and in itself implies that there exists a container variable (e.g. jar that holds water, or in this case a census tract that holds people).

For a census tract you can ask the question: How many people exist per unit area? This is a means of aggregating many point measures.

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Ah, but the caveat is the application of a kernel function on the point data. When a kernel smoothing function (i.e, Kernel Density Estimate) is applied to point data it is not considered an areal aggregation as in summary by census track. –  Jeffrey Evans Nov 2 '12 at 22:42
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Aggregating many point measures doesn't make data continuous. To produce spatially continuous data from point data, you'd have to interpolate values to produce a surface which is spatially continuous. Something areal data is not. –  R.K. Nov 5 '12 at 15:19
    
+1 for both above. For a spatial application the consideration of interpolation is important. My answer focuses on density from a non-spatial data standpoint. –  Nick Ochoski Nov 5 '12 at 19:30

The measurements themselves are discrete, but how you represent them need not be.

For example, you could represent them as a continuous density surface:

School Population Density

Or, as discrete 3D bar charts extruded from the census tracts (in this case a grid):

Kansas City, MO, 3-D View, 2010 Population

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I love the second map! –  Devdatta Tengshe Nov 5 '12 at 3:14

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