Normal theory quantile

A quantile estimator derived from a normally distributed population with unknown mean and standard deviation.

A normal theory quantile is most powerful when a random sample is from a population with a normal distribution.

Estimator Description
MVUE The uniformly minimum unbiased variance quantile estimator uses the sample mean and unbiased standard deviation as the best estimate of the population parameters (Xbar ± Z(alpha) * (s / c4(n))). The factor c4(n) is applied to the sample standard deviation to account for the bias in the estimate for small sample sizes.
t-based The t-based prediction interval quantile estimator uses the Student's t distribution for a prediction interval (Xbar ± t(alpha, n-1) * s * sqrt(1+1/n)) for a single future observation (Horn, 2005). Note that this method produces a wider interval than the MVUE estimator for small samples.

In many cases, the data are skewed to the right and do not follow a normal distribution. A Box-Cox (or logarithmic) transform can correct the skewness, allowing you to use the Normal theory quantile. If not, a distribution-free estimator may be more powerful (IFCC, 1987).