Home / Statistical Tools / Analysis Tools / Confidence Interval / Math Details
Math Details¶
This page gives the exact formulas Quantum XL uses to compute confidence intervals. Each equation lists what it computes and where it appears in the output.
Notation¶
| Term | Description |
|---|---|
| \(x_i, w_i\) | data value and its frequency weight (\(w_i = 1\) when no frequency column is used) |
| \(n\) | effective sample size, \(n = \sum_i w_i\) |
| \(\bar{x}\) | weighted sample mean |
| \(s\) | sample standard deviation, \(s = \sqrt{\dfrac{\sum_i w_i (x_i - \bar{x})^2}{n - 1}}\) |
| \(\alpha\) | significance level; the confidence level is \(1 - \alpha\) |
Confidence interval for the mean (t-interval)¶
Quantum XL reports a Student t confidence interval for the mean, using \(n - 1\) degrees of freedom (requires \(n \ge 2\)). See Student t Distribution for the exact interval. Used by: the mean CI result.
Confidence interval for the standard deviation (chi-square)¶
Quantum XL reports a chi-square confidence interval for the standard deviation. The larger chi-square quantile (upper tail) produces the lower bound, and the smaller quantile produces the upper bound. Constant data (\(s = 0\)) returns \([0, 0]\); requires \(n \ge 2\). See Chi-Square Distribution for the exact interval.
Confidence interval for a binomial proportion (Clopper-Pearson exact)¶
For a category observed \(x\) times out of \(n\), Quantum XL computes the exact Clopper-Pearson interval from the inverse Beta CDF, where \(x\) is the integer count for the category (frequencies are rounded to whole counts). The boundary cases are \(p_{\text{lower}} = 0\) when \(x = 0\) and \(p_{\text{upper}} = 1\) when \(x = n\), and the interval requires exactly two categories. See Beta Distribution for the exact interval.
Confidence interval for a Poisson rate (chi-square exact)¶
Let \(T = \sum_i w_i\,x_i\) be the total occurrences and \(n = \sum_i w_i\) the number of observation periods (both as integer counts). The point estimate is \(\hat{\lambda} = T / n\), and Quantum XL computes the exact Garwood interval from the chi-square distribution, with \(\lambda_{\text{lower}} = 0\) when \(T = 0\). See Chi-Square Distribution for the exact interval.
Shared Math Details used here¶
This tool uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.
| Shared concept | Used here for | Reference |
|---|---|---|
| Student t distribution | the confidence interval for the mean | Student t Distribution |
| Chi-square distribution | the confidence intervals for the standard deviation and the Poisson rate | Chi-Square Distribution |
| Beta distribution | the Clopper-Pearson interval for a proportion | Beta Distribution |
See Also¶
References¶
- Hahn, G. J., & Meeker, W. Q. (1991). Statistical Intervals: A Guide for Practitioners. New York: John Wiley & Sons.
- Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404–413.
- Garwood, F. (1936). Fiducial limits for the Poisson distribution. Biometrika, 28(3/4), 437–442.
- Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Pacific Grove, CA: Duxbury.
- Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods (8th ed.). Ames, IA: Iowa State University Press.