Statistics
Combined variance of two datasets
nta_pyq_2023_jan
Grade 11
Question:
Let $X = \{11, 12, 13, \ldots, 40, 41\}$ and $Y = \{61, 62, 63, \ldots, 90, 91\}$ be the two sets of observations. If $\bar{x}$ and $\bar{y}$ are their respective means and $\sigma^2$ is the variance of all the observations in $X \cup Y$, then $|\bar{x} + \bar{y} - \sigma^2|$ is equal to ______.
Step-by-Step Solution
Key Concept: Each set has 31 elements. $\bar{x} = 26$, $\bar{y} = 76$. Combined mean $\mu = 51$. Compute variance using $\sigma^2 = \frac{1}{62}\sum(x_i - \mu)^2$ with both sets forming APs.
$\bar{x} = 26$, $\bar{y} = 76$. $\mu = 51$. $\sigma^2 = 705$. $|\bar{x} + \bar{y} - \sigma^2| = |26 + 76 - 705| = 603$.
Correct Answer: 603