# Confidence Interval

On SUMMARY tables, the Statisics - Cells menu contains options to show the Upper Confidence Interval and Lower Confidence Interval. On a crosstab, there are no built-in options, but you can use the following rules to add confidence intervals to the table:

## Default confidence intervals

In most instances in Q, the lower and upper bounds of confidence intervals are computed using whichever is appropriate of:

${\bar {x}}\pm t_{\alpha /2,df}s_{\bar {x}}$ or ${\bar {x}}\pm z_{\alpha /2}s_{\bar {x}}$ where:

${\bar {x}}$ is the observed Average, %, Column %, Row %, Probability %, Total % or Coefficient,
$s_{\bar {x}}$ is the estimated or computed Standard Error which includes any computer and/or specified design effects,
$t_{\alpha /2,df}$ is the $\alpha /2$ th quantile of student's t-distribution,
$df$ is $n-1$ , and
$z_{\alpha /2}$ is the $\alpha /2$ th quantile of the normal distribution.

## Confidence intervals for percentages with unweighted samples

The Agresti-Coull interval is used to computed confidence intervals for categorical questions where there are no weights, except where Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests or when Extra deff is not 1. The Agresti-Coull interval is given by:

${\tilde {\bar {x}}}\pm z_{\alpha /2}{\sqrt {\frac {{\tilde {\bar {x}}}(1-{\tilde {\bar {x}}})}{\tilde {n}}}}$ where:

${\tilde {n}}=n+z_{\alpha /2}^{2}$ ,
$n$ is Base n,
${\tilde {\bar {x}}}={\frac {x+{\frac {{z}_{\alpha /2}^{2}}{2}}}{\tilde {n}}}$ , and
$x$ is n.

## Confidence intervals where Weights and significance has been set to Un-weighted sample size in tests

Where Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests, confidence intervals are computed using:

${\bar {x}}\pm t_{\alpha /2,n-1}s_{\bar {x}}$ where:

$s_{\bar {x}}={\sqrt {\frac {d_{eff}{\bar {x}}(1-{\bar {x}})}{n-b}}}$ if ${\bar {x}}$ represents a proportion,
$s_{\bar {x}}=s_{x}{\sqrt {\frac {d_{eff}}{n}}}$ otherwise,
$s_{x}$ is the Standard Deviation,
$d_{eff}$ is Extra deff,
$b$ is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise.