For a more general definition of a *Confidence Interval* see, Confidence Interval in the Data Story Guide.

## Technical details

### Default confidence intervals

In most instances in the software, 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:

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

- \(\tiny \tilde{n} = n + z_{\alpha/2}^2\),
- \(\tiny n\) is Sample Size,
- \(\tiny \tilde{\bar x} = \frac{x + \frac{{z}_{\alpha/2}^2}{2}}{\tilde{n}}\), and
- \(\tiny x\) is Count.

### 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:

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

### Notes

- 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.

- To keep this page relatively short, \(\tiny s\) is used in the formulas above where it is more conventional to use \(\tiny \sigma\).
- 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.

## Method

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: