Question

The mean and variance of 7 observations are 8 and $16,$ respectively. If five observations are $2,4,10,12,14,$ then the absolute difference of the remaining two observations is :

• A. 2
• B. 4
• C. 3
• D. 1

Answer 1

The Correct Option is A

To solve this problem, we start by using the given mean and variance of the 7 observations to find the unknown observations. 

1. The mean of 7 observations is given as 8. Therefore, the sum of all observations is \(7 \times 8 = 56\).
2. Five of the observations are 2, 4, 10, 12, and 14. The sum of these five observations is \(2 + 4 + 10 + 12 + 14 = 42\).
3. Let the remaining two observations be \(x\) and \(y\). Since the total sum of observations is 56, we have:
   \[ x + y = 56 - 42 = 14 \]
4. Next, we use the variance formula. The variance is given as 16, which is the square of the standard deviation. The formula for variance is:
5. Variance = \(\dfrac{\sum(x_i - \bar{x})^2}{n}\)
6. Substituting given values, the variance becomes:
   \[ \dfrac{\sum(x_i - 8)^2}{7} = 16 \]
   Therefore, \[\sum(x_i - 8)^2 = 7 \times 16 = 112\]
7. Now, calculate \(\sum(x_i - 8)^2\) for the five known values:
   • For 2: \((2 - 8)^2 = 36\)
   • For 4: \((4 - 8)^2 = 16\)
   • For 10: \((10 - 8)^2 = 4\)
   • For 12: \((12 - 8)^2 = 16\)
   • For 14: \((14 - 8)^2 = 36\)
8. The sum of these values is \(36 + 16 + 4 + 16 + 36 = 108\).
9. Let \(x\) and \(y\) be the unknown values such that:
   \((x - 8)^2 + (y - 8)^2 = 112 - 108 = 4\)
10. We also know \(x + y = 14\). Let's assume \(x = 7 + d\) and \(y = 7 - d\) where \(d\) is the difference from the mean of 14/2 = 7.
11. Then:
    • \((7 + d - 8)^2 = (d - 1)^2\)
    • \((7 - d - 8)^2 = (-d - 1)^2\)
    • Thus, \((d - 1)^2 + (-d - 1)^2 = 4\)
12. Expand and solve:
    • \((d - 1)^2 = d^2 - 2d + 1\)
    • \((-d - 1)^2 = d^2 + 2d + 1\)
    • Add: \(d^2 - 2d + 1 + d^2 + 2d + 1 = 4\)
      • \(2d^2 + 2 = 4\)
      • \(2d^2 = 2\)
      • \(d^2 = 1\)
      • \(d = \pm 1\)
13. Thus, \(d = 1\) leads to the values \(x = 8\) and \(y = 6\) or vice versa.
14. Therefore, the absolute difference of the two remaining observations is \(|8 - 6| = 2\).

Hence, the absolute difference of the remaining two observations is 2.