A student calculates the mean and variance of 10 observations x₁, x₂, , x₁₀ . To simplify the calculations, a constant A is subtracted from each observation. The mean and variance of the shifted values (x_i - A) are found to be 4 and 16 respectively. Later, it is discovered that one of the original observations was misread as 20 instead of 50 . If the correct mean of the original data is 15 , then the correct varianc

Step-by-step solution

Let the incorrect mean of the original observations be x . Since the mean of (x_i - A) is 4 , we have x - A = 4 x = A + 4 . The incorrect sum of the original observations is 10(A + 4) . When the misread observation 20 is replaced by 50 , the correct sum becomes: 10(A + 4) - 20 + 50 = 10A + 70 . Given that the correct mean is 15 , the correct sum is 10 15 = 150 . Equating the two sums: 10A + 70 = 150 10A = 80 A = 8 . Thus, the incorrect mean of the original data is x = 8 + 4 = 12 . Variance is invariant under a shif