A dataset of 20 observations has a mean of 15 and a variance of 10 . It was discovered that two observations were both incorrectly recorded as 15 . When these two incorrect observations are replaced by their correct values a and b , the mean of the dataset remains unchanged, but the new variance becomes 20 . The absolute difference between the two correct observations, |a - b| , is equal to

Step-by-step solution

Given n = 20 , initial mean x = 15 , and initial variance ^2 = 10 . The sum of the initial observations is x_i = 20 15 = 300 . Since the new mean is also 15 , the new sum of observations remains 300 . The sum of the two incorrect observations is 15 + 15 = 30 . Therefore, the sum of the two correct observations must also be 30 , which gives a + b = 30 . The initial sum of squares is: x_i^2 = 20 ( ^2 + x ^2) = 20 (10 + 15^2) = 20 235 = 4700 The new variance is 20 , so the new sum of squares is: x_ i( new ) ^2 = 20 (2