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converting from "narrower" (float, for example) to wider (double).

In contrast, covariance is converting from "wider" (double) to

narrower (float). In statistics there are several flavors of

covariance but the most commonly used one refers to the general

case of variance: instead of using two instances of the same

measure, two different measures are used in the calculation so one

is determining how two variables change relative to each other.

For our purposes one can imagine the data as occupying a cube

whose axes are time, country and measure. We would typically

like to first calculate averages by measure and year (so exports,

unemployment, cell phones in use ... for 2010) as well as measures

by country (internet users, infant mortality, external debts ... for

Andorra, Argentina ... Zambia and Zimbabwe for 2000 through

2009). With some adroit error handling one might easily gloss

over Mozambique not reporting how many doctors were working in

2007 and still produce an average. Likewise, one can still calculate

a variance when a modest amount of data is missing. Covariance is

not so forgiving. Measure data has to occur in matched pairs. No

figure for doctors working in Mozambique in 2007 can force one to

discard or distort the country of Mozambique, the year of 2007 or

the measure of doctors working. Typically, when we first start

calculating averages and data is determined to be missing or

invalid, it sets off a bit of a scramble to check our sources.