# Confidence Interval

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On SUMMARY tables, the Statisics - Cells menu contains options to show the Upper Confidence Interval and Lower Confidence Interval. The level used for the measurement is determined by the overall level which is set under Edit > Project Options > Customize > Statistical Assumptions (default is 95%).

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.