Why does adding one extreme value raise the arithmetic mean above a threshold?

Conditions

• Using the arithmetic mean as the summary statistic.
• The sample consists of a finite set of numerical values.
• A decision threshold exists.

Reasoning, step by step

1. Sum all observations in the dataset.
2. Divide the total sum by the number of observations.
3. Compare the resulting mean to the threshold.
4. Observe how a single large value inflates the numerator and the final mean.

Example

For the scores 15, 20, 25, 20, 100, the mean is $(15+20+25+20+100)/5 = 180/5 = 36$. Because 36 is greater than the threshold 35, the mean-based decision changes.

Common misconceptions

• Believing that a single extreme value has little effect on the overall average.
• Assuming the mean represents the typical value of every individual in the dataset.

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