Normal distribution

One of the simplest pivotal quantities is the z-score. Given a normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, and an observation 'x', the z-score:

${\displaystyle z={\frac {x-\mu }{\sigma }},}$

has distribution ${\displaystyle N(0,1)}$ – a normal distribution with mean 0 and variance 1. Similarly, since the 'n'-sample sample mean has sampling distribution ${\displaystyle N(\mu ,\sigma ^{2}/n)}$, the z-score of the mean

${\displaystyle z={\frac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}}$

also has distribution ${\displaystyle N(0,1).}$ Note that while these functions depend on the parameters – and thus one can only compute them if the parameters are known (they are not statistics) — the distribution is independent of the parameters.

Given ${\displaystyle n}$ independent, identically distributed (i.i.d.) observations ${\displaystyle X=(X_{1},X_{2},\ldots ,X_{n})}$ from the normal distribution with unknown mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}}$, a pivotal quantity can be obtained from the function:

${\displaystyle g(x,X)={\frac {x-{\overline {X}}}{s/{\sqrt {n}}}}}$

where

${\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}{X_{i}}}$

and

${\displaystyle s^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}{(X_{i}-{\overline {X}})^{2}}}$

are unbiased estimates of ${\displaystyle \mu }$ and ${\displaystyle \sigma ^{2}}$, respectively. The function ${\displaystyle g(x,X)}$ is the Student's t-statistic for a new value ${\displaystyle x}$, to be drawn from the same population as the already observed set of values ${\displaystyle X}$.

Using ${\displaystyle x=\mu }$ the function ${\displaystyle g(\mu ,X)}$ becomes a pivotal quantity, which is also distributed by the Student's t-distribution with ${\displaystyle \nu =n-1}$ degrees of freedom. As required, even though ${\displaystyle \mu }$ appears as an argument to the function ${\displaystyle g}$, the distribution of ${\displaystyle g(\mu ,X)}$ does not depend on the parameters ${\displaystyle \mu }$ or ${\displaystyle \sigma }$ of the normal probability distribution that governs the observations ${\displaystyle X_{1},\ldots ,X_{n}}$.

This can be used to compute a prediction interval for the next observation ${\displaystyle X_{n+1};}$ see Prediction interval: Normal distribution.

Bivariate normal distribution

In more complicated cases, it is impossible to construct exact pivots. However, having approximate pivots improves convergence to asymptotic normality.

Suppose a sample of size ${\displaystyle n}$ of vectors ${\displaystyle (X_{i},Y_{i})'}$ is taken from a bivariate bivariate normal distribution with unknown correlation ${\displaystyle \rho }$.

An estimator of ${\displaystyle \rho }$ is the sample (Pearson, moment) correlation

${\displaystyle r={\frac {{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(Y_{i}-{\overline {Y}})}{s_{X}s_{Y}}}}$

where ${\displaystyle s_{X}^{2},s_{Y}^{2}}$ are sample variances of ${\displaystyle X}$ and ${\displaystyle Y}$. The sample statistic ${\displaystyle r}$ has an asymptotically normal distribution:

${\displaystyle {\sqrt {n}}{\frac {r-\rho }{1-\rho ^{2}}}\Rightarrow N(0,1)}$.

However, a variance-stabilizing transformation

${\displaystyle z={\rm {{tanh}^{-1}r={\frac {1}{2}}\ln {\frac {1+r}{1-r}}}}}$

known as Fisher's 'z' transformation of the correlation coefficient allows creating the distribution of ${\displaystyle z}$ asymptotically independent of unknown parameters:

${\displaystyle {\sqrt {n}}(z-\zeta )\Rightarrow N(0,1)}$

where ${\displaystyle \zeta ={\rm {tanh}}^{-1}\rho }$ is the corresponding distribution parameter. For finite samples sizes ${\displaystyle n}$, the random variable ${\displaystyle z}$ will have distribution closer to normal than that of ${\displaystyle r}$. An even closer approximation to the standard normal distribution is obtained by using a better approximation for the exact variance: the usual form is

${\displaystyle \operatorname {Var} (z)\approx {\frac {1}{n-3}}}$.