Range, IQR, and Standard Deviation

Two data sets can share the same mean and still feel totally different to live with. A class where everyone scores near 80 is a calm room; a class split between 50s and 100s with the same average of 80 is not. Mean alone can't tell these apart, so you need a number that captures spread.

Range is the crudest tool: max minus min, sensitive to one weird outlier. IQR is sturdier: it's Q3 minus Q1, the width of the middle half of the data, immune to outliers since it ignores the tails entirely. Standard deviation is the most informative: it asks, on average, how far does each point sit from the mean? You find each deviation (value minus mean), square it so negatives don't cancel positives, average those squared deviations (dividing by n for a population, n-1 for a sample), then take the square root to undo the squaring and get back to original units. That square root step is what makes standard deviation a 'typical distance from the mean' rather than some abstract squared quantity.

These three measures connect directly to the shape and outlier questions that come right before and after this topic. Range and IQR pair naturally with the five-number summary and boxplots, since quartiles are what IQR is built from. Standard deviation pairs with the mean the way IQR pairs with the median: use mean and standard deviation for roughly symmetric data without outliers, and median with IQR when the distribution is skewed or outliers are present. That choice of which pair to report is a judgment call you'll keep making all through the unit.