Opening
Five people earn between forty and sixty thousand dollars. Their mean and median agree. Then one of them becomes extraordinarily rich. The median sits quietly while the mean starts negotiating for a yacht.
Teaching narrative
The mean uses every numerical value as a balancing point. The median uses the ordering of observations. Neither is the one true average, despite their tendency to be advertised that way. Each describes something different, and extreme values can make the distinction very obvious.
Describe center together with spread. The range depends only on the extremes. Sample standard deviation emphasizes squared deviations from the mean. Quartiles and the interquartile range describe the central portion more robustly. Do not quietly discard the most interesting observation simply because it is inconvenient.
The same observations can produce quite different-looking histograms when the bins change. That is an important warning: graphs are representations, not the underlying data. A beautiful graph with a bad baseline or unexplained denominator can tell a misleading story without falsifying a single number.
Before running a formula, ask what the unit of measurement is, how values were collected, what may be missing, and which numerical description is useful for the question at hand.
Before the experiment
Which summary stays fixed when the largest observation increases tenfold? Can two histograms show different-looking peaks without any data changing?
Live demonstration
Use the center-and-spread outlier slider, then the histogram bin explorer. Compare a fixed median against a changing mean, and a fixed dataset against changing bins.
Exit question
Explain in two sentences why a mean and a histogram should usually appear together.