The values of a variable for six consecutive years are 12, 18, 15, 21, 27, 24. Which of the following statements are correct?
(A) The 3-year moving averages are 15, 18, 21, 24.
(B) The first 3-year moving average is placed against the first year.
(C) The 4-year moving averages, before centring, are placed between two years.
(D) The first 4-year moving average is 17.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
Each 3-year moving average is placed against the middle year of its three years:
\( \displaystyle \frac{12 + 18 + 15}{3} = \frac{45}{3} = 15 \)
\( \displaystyle \frac{18 + 15 + 21}{3} = \frac{54}{3} = 18 \)
\( \displaystyle \frac{15 + 21 + 27}{3} = \frac{63}{3} = 21 \)
\( \displaystyle \frac{21 + 27 + 24}{3} = \frac{72}{3} = 24 \)
A 4-year period has no middle year, so each 4-year moving average falls between its second and third years. The first one is
\( \displaystyle \frac{12 + 18 + 15 + 21}{4} = \frac{66}{4} = 16.5 \)
So the 3-year moving averages are 15, 18, 21, 24, the first of them stands against the second year, and the first 4-year moving average is 16.5, placed between the second and third years.
Checking the statements
(A) The 3-year moving averages are 15, 18, 21, 24. — Correct. These are the four means of three consecutive values.
(B) The first 3-year moving average is placed against the first year. — Incorrect. It is placed against the second year, the middle of the first three.
(C) The 4-year moving averages, before centring, are placed between two years. — Correct. Four years have no middle year.
(D) The first 4-year moving average is 17. — Incorrect. It is 66 ÷ 4 = 16.5.