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Then is called a pivotal quantity (or simply a pivot). Pivotal quantities are commonly used for normalization to allow data from different data sets to be compared. It is relatively easy to construct pivots for location and scale parameters: for the former we form differences so that location cancels, for the latter ratios so that scale cancels.
The pivotal method is based on a random variable that is a function of both the observations and the parameters but whose distribution does not depend on the parameter. Such random variables are called pivotal quantities. By using these, probability statements about the observations and parameters may be made in which the probabilities do not ...
A ancillary statistic is a specific case of a pivotal quantity that is computed only from the data and not from the parameters. They can be used to construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics. [4]
In theoretical statistics, parametric normalization can often lead to pivotal quantities – functions whose sampling distribution does not depend on the parameters – and to ancillary statistics – pivotal quantities that can be computed from observations, without knowing parameters.
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals. The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these ...
For example, in Microsoft Excel one must first select the entire data in the original table and then go to the Insert tab and select "Pivot Table" (or "Pivot Chart"). The user then has the option of either inserting the pivot table into an existing sheet or creating a new sheet to house the pivot table.
Robust statistical methods, of which the trimmed mean is a simple example, seek to outperform classical statistical methods in the presence of outliers, or, more generally, when underlying parametric assumptions are not quite correct. Whilst the trimmed mean performs well relative to the mean in this example, better robust estimates are available.
A random variable that is a function of the random sample and of the unknown parameter, but whose probability distribution does not depend on the unknown parameter is called a pivotal quantity or pivot. Widely used pivots include the z-score, the chi square statistic and Student's t-value.