Named methods and formulas
Use z = (x - mean) / standard deviation to standardize one value. Use combinations n! / (r!(n-r)!) when order does not matter, and permutations n! / (n-r)! when order matters. For a simple proportion margin of error, a common approximation is z x sqrt(p(1-p)/n).
Concrete example
If a score is 85, the mean is 75 and the standard deviation is 10, the z-score is 1.0. If 5 people are selected from 20 and order does not matter, the combination count is 15,504. If order matters, the permutation count is 1,860,480.
What changes the result most
- Sample size n changes margin of error through the square root, so quadrupling n roughly halves margin of error.
- A 95% confidence interval commonly uses a z critical value near 1.96 under normal-approximation assumptions.
- P-values depend on the null hypothesis and test method, not only the observed difference.
Common mistakes
- Using combinations when order matters.
- Treating a confidence interval as a guarantee.
- Using p = 0.5 for sample-size planning without explaining that it is a conservative default for proportions.
- Ignoring sampling bias, which no formula can fix after the fact.
Use the calculators
- Z-score Calculator
- Confidence Interval Calculator
- Sample Size Calculator
- Permutation and Combination Calculator
- Probability and Statistics Formulas
FAQ
When should I use combinations instead of permutations?
Use combinations when order does not matter. Use permutations when different orders count as different outcomes.
Does a 95% confidence interval mean 95% probability for one finished interval?
Not exactly. The 95% refers to the long-run behavior of the method under its assumptions.
References
- NIST/SEMATECH e-Handbook of Statistical Methods - Authoritative statistical-method reference, accessed 2026-05-15.
Last reviewed: 2026-05-15.