This section provides a methodology of how to calculate standard errors for a difference in two random variables. From basic statistical theory, the variance (Var) of a difference is:

Var(x-y) = Var(x) +Var(y) -2Cov(x,y) (B4)

Ignoring the co-variance (Cov) term, or rather assuming it is positive and bounding it below by 0, yields formula (B5):

While this formula is theoretically valid under SRS only (i.e. assuming independence), it is widely used in publications based on complex household sample designs, such as the Labour Force Status. To test its validity under these circumstances a quick empirical study was undertaken using Indigenous data from the 1994 Australian Housing Survey (see ABS 2001 for details).

An alternative, more conservative approach, is to bound the co-variance using a Cauchy-Schwarz inequality (i.e. Cov(x,y) =< se(x)se(y)), which leads to the formula:

se(x-y) =< se(x) + se(y) (B6)

Equation B5 worked quite well in ABS (2001), while equation B6 was found to overestimate standard errors, but not drastically. While the co-variance cannot always be neglected, the ABS study provides no evidence against the use of equation B5 in surveys with Indigenous components. Also, equation B6 may not be a good alternative as, being an upper bound, it will always overestimate the true variation. Given the ABS previous experience, the confidence intervals and inferences in this paper use the formula in equation B5.