If the following table represents the three yearly moving averages of a data, then the value of \( \displaystyle\frac{a+b+c+d}{3} \) is-
| Year | Value | 3-yearly moving average |
| 1 | 2 | |
| 2 | \( a \) | 4 |
| 3 | 6 | \( b \) |
| 4 | 8 | 7 |
| 5 | \( c \) | 9 |
| 6 | 12 | \( d \) |
| 7 | 11 |
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 9
Each 3-yearly moving average is the mean of the value of that year and the values of the years just before and after it, and is placed against the middle year.
Year 2:
\( \displaystyle \frac{2+a+6}{3}=4\ \Rightarrow\ a=4 \)
Year 4:
\( \displaystyle \frac{6+8+c}{3}=7\ \Rightarrow\ c=7 \)
Year 3 and year 6:
\( \displaystyle b=\frac{a+6+8}{3}=\frac{18}{3}=6 \)
\( \displaystyle d=\frac{c+12+11}{3}=\frac{30}{3}=10 \)
Check with year 5: \( \frac{8+7+12}{3}=9 \), as given.
\( \displaystyle \frac{a+b+c+d}{3}=\frac{4+6+7+10}{3} \)
\( \displaystyle =\frac{27}{3}=9 \)