FAQ: Bivariate Choropleth Maps

What does “bivariate” mean?

Bivariate means two variate values, where a variate represents a particular outcome of a random variable. In statistics, the term bivariate analysis generally refers to an analysis based on two variables.

What is a “choropleth” map?

A “choropleth” map uses color to represent changes in variate values for observational units (usually some form of areal unit based on political boundaries, such as a county). More generally, a choropleth map is a type of “thematic” map that focuses on a particular topic/subject/theme.

What are tertiles?

Tertiles (or “terciles”) represent particular quantiles that break a distribution into three groups of (approximately) equal size. Quartiles (4 groups) and percentiles (100 groups) are also commonly used quantiles.

Technically, in a sorted series of numbers, quantiles correspond to values at specific index positions that divide the distribution into equal parts, but the term is often used synonymously to describe the rank-ordered groups that are formed after the data are split. For example in the USDSS Analysis Module, each value (associated with a county) for a particular diabetes indicator falls into one of the three groups based on the class boundaries defined by the tertiles, labeled: “L” (lowest), “M” (middle), and “H” (highest).

How are the values associated with tertiles found?

There are various algorithms for identifying values associated with particular \(q\)-quantiles, such as tertiles (= 3-quantile). The USDSS Analysis Module uses the “quantile” function from the JavaScript Simple Statistics module, which is an implementation of a standard Quantiles of Population algorithm. The formula to find the index position (\(I_p\)) associated with the \(k\)th \(q\)-quantile is:

\[I_p = N (k/q)\]

where \(N\) is the population size.

As an example, let’s imagine a simple array with 10 values sorted in ascending order:

\[[1, 2, 2, 3, 4, 6, 8, 10, 10, 10]\] Using the formula, above, the index position associated with the 1st 3-quantile is:

\[I_p = 10(1/3) = 3.\overline{3}\]If the number is decimal, you round up to the next integer. Hence, the 1st 3-quantile value can be found at the 4th index position in the array and is equal to 3. If the resulting number is an integer, then you take the average of the values at that index position and the next index position. For example, to find the 2nd 4-quantile (a.k.a. the “median”) in the above array:
\[I_p = 10(2/4) = 5\] Taking the average of values at the 5th and 6th index positions in the array gives us a median value of \((4 + 6) / 2 = 5\).