Intuitive Meaning of Covariance


Sometimes we can “augment knowledge” with an unusual or different approach. I would like this reply to be accessible to kindergartners and also have some fun, so everybody get out your crayons!

Given paired (x,y)(x,y) data, draw their scatterplot. (The younger students may need a teacher to produce this for them. 🙂 Each pair of points (xi,yi)(xi,yi), (xj,yj)(xj,yj) in that plot determines a rectangle: it’s the smallest rectangle, whose sides are parallel to the axes, containing those points. Thus the points are either at the upper right and lower left corners (a “positive” relationship) or they are at the upper left and lower right corners (a “negative” relationship).

Draw all possible such rectangles. Color them transparently, making the positive rectangles red (say) and the negative rectangles “anti-red” (blue). In this fashion, wherever rectangles overlap, their colors are either enhanced when they are the same (blue and blue or red and red) or cancel out when they are different.

Positive and negative rectangles

(In this illustration of a positive (red) and negative (blue) rectangle, the overlap ought to be white; unfortunately, this software does not have a true “anti-red” color. The overlap is gray, so it will darken the plot, but on the whole the net amount of red is correct.)

Now we’re ready for the explanation of covariance.

The covariance is the net amount of red in the plot (treating blue as negative values).

Here are some examples with 32 binormal points drawn from distributions with the given covariances, ordered from most negative (bluest) to most positive (reddest).

Covariance plots

Let’s deduce some properties of covariance. Understanding of these properties will be accessible to anyone who has actually drawn a few of the rectangles. 🙂

  • Bilinearity. Because the amount of red depends on the size of the plot, covariance is directly proportional to the scale on the x-axis and to the scale on the y-axis.
  • Correlation. Covariance increases as the points approximate an upward sloping line and decreases as the points approximate a downward sloping line. This is because in the former case most of the rectangles are positive and in the latter case, most are negative.
  • Relationship to linear associations. Because non-linear associations can create mixtures of positive and negative rectangles, they lead to unpredictable (and not very useful) covariances. Linear associations can be fully interpreted by means of the preceding two characterizations.
  • Sensitivity to outliers. A geometric outlier (one point standing away from the mass) will create many large rectangles in association with all the other points. It alone can create a net positive or negative amount of red in the overall picture.

Incidentally, this definition of covariance differs from the usual one only by a universal constant of proportionality (independent of the data set size). The mathematically inclined will have no trouble performing the algebraic demonstration of the equivalence.


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