Use the summary that matches the question
The mean divides the total by the number of observations. Every observation contributes to the total. The median uses the middle of the sorted observations. For an odd count it is the central value, and for an even count it is the average of the two central values. A large observation can change the mean noticeably. Its effect on the median depends on whether the central positions change. When a question replaces a reading, sort the updated list before deciding what happened to the median.
Worked example with a changed reading
Consider readings of 3, 7, 8, 10, and 27. Their total is 55, so the mean is 11. The median is 8. Replacing 27 with 12 lowers the total to 40 and the mean to 8. The median stays at 8 because the order at the center is unchanged. If the replacement were 4 instead, the middle value would be 7 after sorting. A change at one end of the original list can affect the median if the new value lands on the other side of the center.
Read a count table from the selected group
A greenhouse checks fifteen herb plants, of which nine are healthy, and twenty flower plants, of which eight are healthy. A plant chosen uniformly from all thirty-five has a probability of 17/35 of being healthy. A plant chosen uniformly from the herbs has probability 9/15 of being healthy. If the task instead chooses from all plants and asks for a healthy herb, the numerator stays nine while the denominator becomes thirty-five. Describe the selection group first, then count the qualifying members of that group.
Handle overlapping groups before taking a complement
Some categories overlap. If six volunteers delivered flyers, five made phone calls, and two did both, adding six and five counts two people twice. Subtract the overlap once to get nine volunteers who did at least one job. If the full list has twelve volunteers, three did neither. With an equally likely selection, the chance of choosing someone who did neither is 3/12. A quick check is to add the counts for at least one job and for neither. Together they must recover the full list.
Transfer check with a different denominator
A shuttle log has sixteen weekday trips, including ten with a bike on board, and nine weekend trips, including three with a bike. Find the chance of a bike on board when choosing a weekend trip uniformly, then find the chance of a weekend trip with a bike when choosing uniformly from the full log. Pause and label each sample space. The two probabilities are 3/9 and 3/25. The numerator is the same because the three qualifying trips are the same. The denominator changes with the trips available for selection.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original OpenExam explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
College Board ACCUPLACER Math test contentOpenStax Prealgebra 2eOpenStax Introductory Statistics 2e