Q1.
Try to explain, using algebra, what the average is when a fixed number, e.g., 2 is subtracted from every value in the collection.
Answer
The average goes down by 2.
Let the values be x1, x2, …, xn with
(x1 + x2 + … + xn) ÷ n = a
New average = [(x1 − 2) + (x2 − 2) + … + (xn − 2)] ÷ n
= (x1 + x2 + … + xn − 2n) ÷ n
= (x1 + x2 + … + xn) ÷ n − 2n ÷ n
= a − 2
(x1 + x2 + … + xn) ÷ n = a
New average = [(x1 − 2) + (x2 − 2) + … + (xn − 2)] ÷ n
= (x1 + x2 + … + xn − 2n) ÷ n
= (x1 + x2 + … + xn) ÷ n − 2n ÷ n
= a − 2
In general, subtracting a fixed number c from every value lowers the average by exactly c.
Check it yourself: the data 8, 3, 10, 13, 4, 6, 7, 7, 8, 8, 5 has mean 7.18. Reduce every value by 1 and the mean becomes 6.18 — which is the same as the shift the book shows when the +10 data (mean 17.18) is reduced by 1 to give 16.18.