Statistics
Variance of First Four Observations
nta_pyq_2024_jan
Grade 11
Question:
If the mean and variance of five observations are $\dfrac{24}{5}$ and $\dfrac{194}{25}$ respectively and the mean of first four observations is $\dfrac{7}{2}$, then the variance of the first four observations is equal to
$\dfrac{4}{5}$
$\dfrac{77}{12}$
$\dfrac{5}{4}$
$\dfrac{105}{4}$
Step-by-Step Solution
Key Concept: From mean of 5: $x_1+x_2+x_3+x_4+x_5=24$. Mean of first 4 is $7/2$: $x_1+\cdots+x_4=14\Rightarrow x_5=10$. From variance of 5: $\sum_{i=1}^5 x_i^2/5-(24/5)^2=194/25\Rightarrow\sum x_i^2=5\times(194/25+576/25)=154$. $x_1^2+\cdots+x_4^2=154-100=54$. Var of first 4 $=54/4-(7/2)^2=13.5-12.25=5/4$.
Variance of first 4 $=54/4-49/4=5/4$.
Correct Answer: 3