Mean, Median, Mode, and Range

Objective: Relate the arithmetic mean to the sum and count.

Prerequisite skills: ordering numbers and arithmetic means. The arithmetic mean shares the total equally among all observations. Multiply a known mean by its count to recover the total. Mean uses every value and is sensitive to large extremes. Common error: dividing by the number of distinct values instead of observations. Restrictions and interpretation: Sort before finding a median; for even counts average the two middle values; a data set may have no unique mode.

Objective: Find the middle or most frequent value using the appropriate definition.

The median is a positional statistic in sorted data. The mode is a most-frequent value and may be nonunique. A large repeated value can be the mode without being the median. For a unique mode, one value must occur more often than every other value. Common error: taking the middle of the unsorted display or confusing frequency with magnitude. Restrictions and interpretation: Sort before finding a median; for even counts average the two middle values; a data set may have no unique mode.

Objective: Use a mean or range condition to recover an unknown value.

A mean condition is an equation about the complete sum. Multiply mean by sample size and subtract known observations. A range condition instead compares only maximum and minimum: range=max−min. If the unknown is explicitly the largest value, add the range to the minimum. Common error: using the sum when a range condition only involves the endpoints. Restrictions and interpretation: Sort before finding a median; for even counts average the two middle values; a data set may have no unique mode. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.