JEE Main 2021 · 16 March · morning · Single correct · +4 / −1
Consider three observations a, b, and c such that b = a + c. If the standard deviation of a + 2, b + 2, c + 2 is d, then which of the following is true?
- (a) b² = 3(a² + c²) + 9d²
- (b) b² = 3(a² + c²) - 9d²
- (c) b² = 3(a² + c² + d²)
- (d) b² = a² + c² + 3d²
Answer and solution
Answer (b) b² = 3(a² + c²) - 9d²
Solution
Maths sir · 4 steps
Adding 2 to every value does not change the spread, so d is just the SD of a, b, c themselves.
- Drop the shift. Adding the same number to every observation moves the mean by that number but leaves every deviation unchanged. So the standard deviation of a, b, c is also d.
- Find the mean. Since a + c = b, the sum is a + b + c = 2b. So the mean is x̄ = 2b/3.
d² = (a²+b²+c²)/3 - (2b/3)²
- Clear fractions. Multiply both sides by 9: 9d² = 3(a²+b²+c²) - 4b² = 3(a²+c²) - b².
- So. Rearranging gives b² = 3(a²+c²) - 9d², which is option (b).
The idea to keep. Shifting all data by a constant never changes variance or SD. Then use variance = mean of squares minus square of mean, and plug in any given relation between the values.
Answer from the NTA final answer key. Solution by PrepWiser.