The average of 8 numbers is 45. If each number is increased by 5, what will…

Concept:

Adding the same constant to every value of a data set moves the mean by exactly that constant. If a set of n values has total Σx, its mean is Σx/n. Adding k to each of the n values adds nk to the total, so the new mean is (Σx + nk)/n = Σx/n + k, i.e. old mean + k. The number of values n cancels out: it never changes the size of the shift, and the spread of the data is untouched — only its level moves.

Application:

1. The set has n = 8 values and a mean of 45.

2. Total of the original values = 8 × 45 = 360.

3. Every value rises by 5, so the total rises by 8 × 5 = 40, giving a new total of 360 + 40 = 400.

4. New mean = 400 ÷ 8 = 50.

Cross-check:

Using the shortcut instead of the long route, new mean = old mean + constant added = 45 + 5 = 50, which agrees with the step-by-step total. Notice that 8 appears in both the numerator and the denominator of the long route and cancels, so the same shift of 5 would apply to a set of 3 numbers or 300 numbers: the count tells you how many values share the increase, not how large the change in the mean is.