The average of 8 numbers is 45. If each number is increased by 3, what will be…

Concept: The mean of a set is its total divided by how many members it has. If the same constant is added to every member, the total rises by that constant multiplied by the number of members, so the mean rises by exactly that constant. A uniform shift therefore moves the average by the size of the shift, and the number of members never enters the result.

Application: Work the rule through on these eight numbers.

1. The set has 8 numbers with a mean of 45, so the total of the numbers is 45 × 8 = 360.

2. Increasing each of the 8 numbers by 3 adds 3 × 8 = 24 to that total, which becomes 360 + 24 = 384.

3. The new mean is the new total divided by the same 8 members: 384 ÷ 8 = 48.

Cross-check: Using the rule directly, new mean = old mean + shift = 45 + 3 = 48, which agrees with the step-by-step total. The 8 cancels out, so a shift of 3 would move the mean to 48 for a set of any size.

Common slips

• Counting the shift twice gives 51.

• Leaving the mean untouched gives 45.

• Reporting how many numbers there are gives 8.

None of these follows from the rule that a uniform shift moves the mean by exactly the shift.