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Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
The same applies for the meta-analysed measure of effect: if the points of the diamond overlap the line of no effect the overall meta-analysed result cannot be said to differ from no effect at the given level of confidence. Forest plots date back to at least the 1970s. One plot is shown in a 1985 book about meta-analysis.
When the data fit exceeds the standard 5 sigma threshold, the pulsar timing array can declare detection of gravitational waves. [2] [3] [4] More precisely, the Hellings-Downs curve is the expected correlations of the timing residuals from pairs of pulsars as a function of their angular separation on the sky as seen from Earth.
Huber-White standard errors assume is diagonal but that the diagonal value varies, while other types of standard errors (e.g. Newey–West, Moulton SEs, Conley spatial SEs) make other restrictions on the form of this matrix to reduce the number of parameters that the practitioner needs to estimate.
For two univariate distributions and with the same standard deviation, it is denoted by ′ ('dee-prime'): ′ = | |. In higher dimensions, i.e. with two multivariate distributions with the same variance-covariance matrix , (whose symmetric square-root, the standard deviation matrix, is ), this generalizes to the Mahalanobis distance between the two distributions:
For example, if the mean height in a population of 21-year-old men is 1.75 meters, and one randomly chosen man is 1.80 meters tall, then the "error" is 0.05 meters; if the randomly chosen man is 1.70 meters tall, then the "error" is −0.05 meters.