Home / Shared Math Details / Distribution Functions / Chi-Square Distribution
Chi-Square Distribution¶
This page defines the chi-square-based interval formulas Quantum XL shares: the confidence interval for the standard deviation, and the exact confidence interval for a Poisson rate.
Notation¶
| Term | Description |
|---|---|
| \(\chi^2_{\,df,\,q}\) | the \(q\)-quantile of the chi-square distribution with \(df\) degrees of freedom |
| \(s\) | sample standard deviation (n − 1 denominator) |
| \(n\) | sample size |
| \(\alpha\) | significance level (\(1 -\) confidence) |
| \(T\) | total number of occurrences (Poisson) |
| \(m\) | sample size or exposure (Poisson) |
Confidence interval for the standard deviation¶
\[ \left[\ \sqrt{\frac{(n-1)\,s^2}{\chi^2_{\,n-1,\,1-\alpha/2}}}\ ,\ \ \sqrt{\frac{(n-1)\,s^2}{\chi^2_{\,n-1,\,\alpha/2}}}\ \right] \]
with \(df = n - 1\). Constant data (\(s^2 = 0\)) returns a zero-width interval.
Used by: Confidence Interval and Summary Statistics.
Exact confidence interval for a Poisson rate (Garwood)¶
\[ \text{lower} = \frac{\chi^2_{\,2T,\,\alpha/2}}{2m} \quad (0 \text{ when } T = 0)
\qquad
\text{upper} = \frac{\chi^2_{\,2(T+1),\,1-\alpha/2}}{2m} \]
Used by: Confidence Interval.
See Also¶
References¶
- Garwood, F. (1936). Fiducial limits for the Poisson distribution. Biometrika, 28(3/4), 437-442.
- NIST/SEMATECH e-Handbook of Statistical Methods, section on the chi-square distribution.