Averages from a table, and comparing distributions

Mean from a frequency table

Do not add the values in the first column and divide by how many rows there are. Multiply each value by its frequency, add those products, then divide by the total frequency. Adding an $f\times x$ column to the table makes this almost automatic.

Median and mode from a table

The mode is the value with the highest frequency — read it straight off. The median is the $\tfrac{n+1}{2}$-th value once you count through the frequencies in order; a running total column finds it quickly.

Which average is fair

The mean uses every value, so one extreme value distorts it. The median ignores extremes. The mode is the only average that works for non-numerical data such as favourite colour. Checkpoint often asks which is most appropriate — and the answer must have a reason.