Describing data, z-scores, counting and uncertainty.
Statistics turns raw numbers into understanding. This guide covers the core ideas — describing data, standardizing values, counting possibilities and quantifying uncertainty — and links to the calculator for each.
The first step with any data is summarizing its center and spread. The mean is the average, the median is the middle value (more robust to outliers), and the mode is the most frequent. Spread is captured by the range and the standard deviation. The Mean, Median, Mode Calculator computes all of these at once from a pasted list of numbers.
A z-score expresses how many standard deviations a value sits from the mean, which lets you compare values from different scales and find percentiles under a normal curve. About 95 percent of values fall within two standard deviations of the mean. The Z-Score Calculator converts a value to its z-score and percentile.
Probability often starts with counting. Permutations count ordered arrangements; combinations count unordered selections. The difference matters: a podium finish (order matters) is a permutation, while a lottery draw (order doesn't) is a combination. The Permutation & Combination Calculator computes both for any n and r.
Surveys report a margin of error because a sample only approximates the whole population. It depends on sample size and the confidence level, and is largest when a result is near 50/50. The Margin of Error Calculator computes the margin and confidence interval, which is why a poll might say a candidate is at 48% plus or minus 3 points.
Numbers can mislead when context is missing. A mean hides outliers that the median would reveal; a small sample inflates the margin of error; and a 'significant' result still needs a meaningful effect size. The habit that protects you is simple: ask how the data was collected, how big the sample was, and how much uncertainty surrounds the headline number.
Mean is the average; median is the middle value, which resists outliers.
How many standard deviations a value is from the mean — useful for percentiles and comparisons.
Permutations count ordered arrangements; combinations count unordered selections.
The uncertainty around a sample estimate; it shrinks with larger samples and peaks near a 50/50 split.
Bigger samples reduce the margin of error roughly with the square root of n.