Enter a list of numbers separated by commas, spaces, or new lines.
nPr = n! / (n−r)!, combinations nCr = n! / (r!(n−r)!). z is (x − mean) / SD; confidence intervals use the usual two-tailed z values (1.96 for 95%).
How many ways can you arrange or choose r items from a set of n?
Find how many standard deviations a value is from the mean.
Estimate the range that likely contains the true population mean.
Work out how many respondents you need for a survey or study.
Population standard deviation divides by n, used when your data is the entire group you care about. Sample standard deviation divides by n−1, used when your data is a sample meant to estimate a larger population — the n−1 version is slightly larger, correcting for the extra uncertainty of estimating from a sample.
Permutations (nPr) count arrangements where order matters, like 1st/2nd/3rd place. Combinations (nCr) count selections where order doesn't matter, like picking a 3-person committee. For the same n and r, nCr is always smaller than or equal to nPr.
From your sample mean and standard deviation, using the standard two-tailed z-value for your chosen confidence level — 1.96 for 95%, for example. A higher confidence level or a smaller sample size both widen the interval.
The mean can be pulled a long way by a few extreme values. The median (the middle value) and mode (the most frequent value) describe the "typical" data point differently, and are more reliable than the mean when your data has outliers or a skewed shape.