Let in a series of 2n observations, half of them are equal to a and remaining half are equal to -a. Also by adding a constant b in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of a² + b² is equal to :

Solution

Maths sir · 4 steps

Adding b moves the mean by b but leaves the spread untouched, so b is 5 and |a| is 20.

1. Original mean. There are n values equal to a and n values equal to -a. Their sum is na - na = 0, so the mean is 0.
2. Original SD. Every observation is at a distance |a| from the mean 0. So the variance is a² and the standard deviation is |a|.

   σ² = 1/2nΣ xᵢ² - x̄² = (2n · a²)/2n - 0 = a²

3. Add b. Adding b to every observation adds b to the mean, so 0 + b = 5 and b = 5. A shift does not change the spread, so the SD is still |a| = 20.
4. So. a² + b² = 400 + 25 = 425, which is option (a).

The idea to keep. Adding a constant to every observation shifts the mean by that constant but does not change the standard deviation or the variance.

Answer from the NTA final answer key. Solution by PrepWiser.