Question

In an experiment with $15$ observations on $x$, the following results were available $\sum x^2 = 2830,\, \sum x = 170$ One observation that was $20$, was found to be wrong and was replaced by the correct value $30$. Then the corrected variance is

• A. 9.3
• B. 8.3
• C. 188.6
• D. 177.3
• E. 78

Answer 1

Answer: (E)

We have
$N=15$
Incorrect $\Sigma x_{1}^{2} i=2830$,
Incorrect $\Sigma x_{i}=170$
$\therefore $ Correct $\Sigma x_{i}=$ Incorrect $\Sigma x_{i}-20+30$
$=170+10$
$=180$
$\therefore $ Correct mean $=\bar{X}=\frac{\Sigma X_{i}}{N}($ correct $)$
$=\frac{180}{15}$
$=12$
Also, correct $\Sigma x_{i}^{2}=$ Incorrect $\Sigma x_{i}^{2}-(20)^{2}+(30)^{2}$
$=2830-400+900$
$=2830+500$
$=3330$
$\therefore $ Correct variance $=\frac{\Sigma x_{i}^{2}(\text { correct })}{N}-(\bar{X})^{2}$
$=\frac{3330}{15}-(12)^{2}$
$=222-144$
$=78$