Outliers and Changes to Data Sets

Objective: Update a mean after a single observation changes.

Prerequisite skills: means, medians, and ordered data. Replacing a value changes the sum by new minus old while leaving the count unchanged. Therefore the mean changes by that difference divided by the original count. Common error: adding a new observation count when the prompt says replace. Restrictions and interpretation: A change to one observation can affect the mean even when the median stays fixed; distinguish replacing from adding an observation.

Objective: Update both total and count when adding a value.

Appending an observation increases the count by one. Add its value to the old total, then divide by the new count. A new value equal to the old mean leaves the mean unchanged. Common error: dividing by the old count after adding a new observation. Restrictions and interpretation: A change to one observation can affect the mean even when the median stays fixed; distinguish replacing from adding an observation.

Objective: Compare how a changed extreme value affects mean and median.

The mean incorporates the magnitude of every value, while the median depends on order and middle positions. Changing only a largest value can change the mean substantially without moving the median. This is why median is often useful for skewed distributions. Common error: assuming an unchanged median implies an unchanged mean. Restrictions and interpretation: A change to one observation can affect the mean even when the median stays fixed; distinguish replacing from adding an observation. 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.