Question

The mean of 12 observations was calculated as 28. Later, it was found that one observation was wrongly recorded as 46 instead of 26, and another observation was completely omitted. If the correct mean is 27, find the value of the omitted observation.

• A. \(36\)
• B. \(18\)
• C. \(35\)
• D. \(28\)

Hint:

For corrected mean problems, always follow these three steps:
(1) Find the original (wrong) sum: Wrong Mean $\times$ Wrong $n$.
(2) Correct any wrongly recorded values by subtracting wrong entries and adding correct ones.
(3) Add the omitted value and set the new sum equal to Correct Mean $\times$ Correct $n$.
Watch carefully whether the number of observations changes (it does when an observation is omitted).

Answer 1

The Correct Option is C

Concept:
Mean \(= \dfrac{\text{Sum of observations}}{\text{Number of observations}}\), so:
When an observation is wrongly recorded, we correct it by subtracting the wrong value and adding the correct value. When an observation is omitted, we need to find it such that the final correct mean holds.

Step 1: Compute the original (incorrect) sum.
The mean was calculated as 28 for 12 observations:

Step 2: Correct the wrongly recorded observation.
One observation was recorded as 46 instead of the correct value 26.

Step 3: Account for the omitted observation.
Let the omitted observation be \(x\). When we include it, the total number of observations becomes \(12 + 1 = 13\). The correct sum (including all corrections and the missing observation) is:

Step 4: Use the correct mean to find \(x\).
The correct mean is 27 with 13 observations:



Step 5: Verify.

• Original incorrect sum: \(336\).

• After fixing wrong entry: \(336 - 46 + 26 = 316\).

• After adding omitted value: \(316 + 35 = 351\).

• New mean: \(\dfrac{351}{13} = 27\).