UPSC NDA I 2022 Mathematics Solved Paper

Concept:The algebraic sum of deviations from a value $y$ is $\sum (x_i - y) = \sum x_i - n y$.If the sum of deviations is zero, $y$ equals the mean. Here the sum is 180, not zero.Using given total sum and number of observations, solve for $y$.Explanation:Let the set of values be $x_1, x_2, \dots, x_n$.Given: total number of observations $n = 20$.Given: sum of all observations $\sum_{i=1}^{20} x_i = 1980$.Given: algebraic sum of deviations from $y$ is 180, i.e., $\sum_{i=1}^{20} (x_i - y) = 180$.Expand the sum: $\sum x_i - 20y = 180$.Substitute $\sum x_i = 1980$: $1980 - 20y = 180$.Rearrange: $20y = 1980 - 180 = 1800$.Solve: $y = \frac{1800}{20} = 90$.Answer:$y = 90$