Discussion
If each observation in a group is increased by 5, then the arithmetic mean will ....................... .
Answer: A
The arithmetic mean will increase by 5.
The mean is affected by a change of origin. If every observation becomes x + 5, the sum rises by 5n, so
\[ \frac{\sum (x + 5)}{n} = \frac{\sum x}{n} + 5 = \bar{x} + 5 \]
The mean cannot stay unchanged or fall when every value rises, and the change is exactly 5 whatever the observations are. (Measures of dispersion such as the standard deviation, by contrast, are unaffected by a change of origin.)
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