an introduction to ringand modules

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The standard deviation does not change its value with a change of origin, but this is because it is a second order measure concerned with the distances between variables, which, of course, are independent of origin. ) The mean is not invariant to the transformations preserving the relationships on ordinal variables. As a very simple example, a variable with values 0, I, 3 has a mean lying between I and 3. The same variable with new values 0, 6, 7 has a mean lying between 0 and 6. Thus order, which must remain invariant, is not preserved for the mean.

Secondly, we would like the variance of our estimates to be as small as The Theory of Causal Analysis 29 possible so that they tend to cluster close to the population value (assuming unbiasedness). To ensure this we need to assume the homoscedasticity of ~4 on x1, x2 and x3. IO This assumption is additional to that involved in unconstrained regression. If we give X 1 , X 2 and X 3 probability distributions, then we can express the resultant unbiasedness and minimum variance unconditionally. This depends upon the values of X 1 , X 2 and X 3 in the sample being chosen at random from the population; this is not always practicable, however.

This interpretation of P4 1 2 in terms of variance (or of P4 1 in terms of standard deviation) is claimed to establish the path coefficients as measuring the direct causal effect of X 1 on X 4 • There is, however, an important objection to this. 2 3u 2 . Thus b41 2 is independent of the variance of X 2, X 3 or U4 in the population. 1t follows that if we compare the value of b 4 1 in two populations, then the comparison is quite independent of the variance of any implicit causes in the two 40 An Introduction to Causal Analysis in Sociology populations.

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