Mean, median and mode: which one describes your data?
One number can summarise a list while hiding its shape. A long average wait may be driven by one exceptional case even when most waits were shorter. Decide which description you need before choosing the calculation.
Three summaries, not three synonyms
The arithmetic mean is the sum divided by the count. The median is the middle of the sorted list, or the mean of its two middle values for an even count. The mode is the most frequent value.
The mean uses every value’s magnitude, so a distant large value affects it. The median identifies the centre position; the mode highlights repetition. Name the statistic instead of saying only “average”.
References: OpenStax — Mean, median and mode
Each value must represent the same kind of observation
Define the unit, period and observation before calculating. Do not mix minutes with hours or monthly totals with weekly totals without a clear conversion or purpose.
Zero is a measured value; an unrecorded observation is not automatically zero. An accidentally duplicated record also gives that case extra weight. Keep the count and sum to make the result interpretable.
When weights are needed
Values representing groups of different sizes cannot always be combined by averaging the averages. Multiply each group mean by its count, add those amounts and divide by the combined count, assuming the measurements are comparable.
A weight should have a meaning: course credits, observation counts or purchased quantities. Do not choose weights merely to move an answer toward an expected result.
Worked example: five waits and two separate groups
Illustrative waits in minutes are 4, 5, 5, 6 and 30. They sum to 50, giving a mean of 10. The median and mode are both 5. This is not contradictory: the 30-minute case pulls the mean above the middle experience.
In a separate grouping example, 2 observations have a mean of 85 and 6 have a mean of 70. The combined mean is (85 ×2 +70 ×6) ÷8 =73.75, not the unweighted average of 85 and 70, which is 77.5.