The notion of covariation is based on an extension of the concept of variation to the case of two variables. As an example, let us pretend that at a meeting of the Midwest Club of Poets a question arose whether people who like to read Gothic novels also like to read poetry. To resolve this, our subjects administered a short questionnaire to their friends asking them to rate their liking of the Gothic novels and poetry.
I like to read Gothic novels
I like poetry
Subjects’ responses were recorded into the data matrix shown below.
The question to be
answered is to whether the liking of the Gothic novels and poetry are related.
To quantify the degree to which the above variables, X and Y, vary together, covary. Consider that for a single variable, variance is computed as
This can also be conceptualized as
Substitution of a deviation score y for the second deviation score x results in a formula that defines covariance as
The computation of the coefficient of covariance for the example is showed in the following table. The question to be answered is to whether the liking of the Gothic novels and poetry are related.
To compute covariance, means are computed for both variables. Next, both variables are transformed into deviation scores by subtracting their mean from scores in their respective distributions. The products of the deviation scores are then computed for each subject. These products are summed and averaged, resulting in the coefficient of covariance, for the example equal to 4.00.
Covariance can also be computed directly from the obtained scores. By substituting into the formula
the formulae for transformation of X and Y into deviation scores, i.e.,
and
the covariance can be written as
Expanding the above expression as
replacing the summation notation for the means with M and recalling that sum of a constant equals n times the constant, the above expression can be simplified as
Thus, the formula for computing the coefficient of covariance directly from the obtained scores can be written as
or, concisely, as
As an illustration, the computational algorithm is shown as
The mean of the product of variables X and Y equals 28. Subtracting the product of the separate means (28 - (4)(6)) yields the coefficient of covariance equal to 4.00. This value is equal to the value of the covariance coefficient computed from the formula expressed in deviation scores, computed previously.
The coefficient of covariance has no upper or lower limits. As will be seen later, this indeterminacy is its main disadvantage as compared with the coefficient of correlation. The lack of upper and lower limits is due to the sensitivity of covariance to the units of measurement that, on the other hand, can be an advantage in some instances. To illustrate this sensitivity, imagine that the above example does not pertain to responses to two rating scales, but (rather unrealistically) represents the measurements of the length of each subjects' toes in centimeters (X = [3 1 3 9) and subjects' fingers in millimeters (Y = [40 40 80 80]). There are 10 millimeters to a centimeter, so we need to multiply variable Y by 10. The product of variables X and Y is computed for this new example as XY = [120 40 240 720] and the coefficient of covariance is computed as 1120/4-4(60) which equals 40. Consider that we did not change the measurements themselves, but only the units of the second measurement. The coefficient of covariance reflects this change in the unit of measurement. After reading the next chapter, try to re-compute the relationship between the above measurements by using the coefficient of correlation. If you do this you will observe that the coefficient of correlation remains invariant with respect to change of the measurement unit.
Variance and covariance are often reported jointly as variance-covariance matrices. The variance-covariance matrix has variances in its diagonal and covariance in its off-diagonal elements. Thus the variance-covariance matrix V can be defined as
For our example,
The variance-covariance matrix can be composed as
Covariance is defined as
The variance-covariance matrix is defined as