Statistics: From Data to Decisions

A variable is either categorical or quantitative, and its level of measurement decides which summaries are even allowed on it.

Mean, median, and mode each name a different notion of a typical value, and a skewed distribution is exactly what makes them disagree.

Range, interquartile range, and standard deviation measure how far data sits from its center, and they differ chiefly in how much an outlier can move them.

A histogram or boxplot reveals whether a distribution is symmetric, skewed, or multi-peaked, and that shape decides which measure of center you can trust.

The normal curve is a symmetric, single-peaked distribution whose 68–95–99.7 rule fixes what fraction of data falls within one, two, and three standard deviations of the mean.

A z-score restates a value as the number of standard deviations it lies from its mean, letting you compare measurements taken on completely different scales.

A sample can only speak for its population when it is chosen without bias, and random selection is the one dependable defense against that bias.

A statistic like the sample mean changes from one sample to the next, and that predictable spread of estimates is measured by the standard error.

A confidence interval reports an estimate plus a margin of error, and the confidence level states how often intervals built this way capture the true value.

A scatterplot shows two measurements per subject at once, exposing the direction, form, and strength of any relationship between the variables.

The correlation coefficient r condenses the direction and tightness of a linear relationship into a single number between minus one and one.

A regression line summarizes a scatter with one equation you can predict from, but only within the range the data actually covers.

Two variables can move together because one drives the other, because the second drives the first, or because a lurking third variable moves both — and correlation alone cannot say which.

A chart can distort honest data by truncating the axis, stretching or compressing a scale, or cherry-picking the window of time it shows.

Whether a study may claim that one thing causes another depends on whether it merely observed the world or actively assigned a treatment, and on which competing explanations it ruled out.

Before a conclusion is allowed to drive a decision, it must pass a fixed checklist covering how the data was collected, how it was summarized, and how it was displayed.