# Confidence Interval

## 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.

## Notes

1. In most situations, the statistical tests computed by Q will not correspond to conclusions drawn if attempting to construct tests from the confidence intervals. There are many reasons for this, including:
• Multiple Comparison Corrections.
• Use of non-parametric tests in Q.
• The confidence intervals having statistical properties that make them sub-optimal from a testing perspective.
2. To keep this page relatively short, $s$ is used in the formulas above where it is more conventional to use $\sigma$.
3. Whereas the The Agresti-Coull interval is an improvement on the default formula for computing the confidence intervals, the formula used when Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests, is generally inferior and is only included for the purposes of aiding comparison with results computed using this formula in other programs.