Question

While performing some experiments, five readings are observed as : $80.0, 80.5, 81.0, 81.5$ and $82$. Calculate the mean percentage error in these observations.

• A. $0.74 \%$
• B. $1.74 \%$
• C. $0.38 \%$
• D. $1.36 \%$

Answer 1

Answer: (A)

Mean of five observations,
$O_{\text {mean }}=\mu=\frac{80.0+80.5+81.0+81.5+82}{5}$
$=\frac{405.0}{5}=81$
Absolute errors in the observations are
$\Delta O_{1}=80.0-\mu \Delta O_{2}=80.5-\mu$
$\Delta O_{3}=81.0-\mu \Delta O_{4}=81.5-\mu $
$\Delta O_{5}=82-\mu $
$\therefore$ Arithmetic mean of all the absolute errors,
$\Delta O_{\text {mean }} =\displaystyle\sum_{i=1}^{5} \frac{\left|\Delta O_{i}\right|}{n} $
$=\frac{|80.0-\mu|+|80.5-\mu|+|81.0-\mu|+|81.5-\mu|+|82-\mu|}{5}$
Substituting the value of $\mu$, we get
$\Delta O_{\text {mean }}=\frac{|80.0-81|+|80.5-81|+|81.0-81|+|81.5-81|+|82-81|}{5}$
$=\frac{1+0.5+0+0.5+1}{5}=\frac{3}{5}=0.6$
$\therefore$ Mean percentage error $=\frac{\Delta O_{\text {mean }}}{O_{\text {mean }}} \times 100 \%$
$=\frac{0.6}{81} \times 100 \%=0.74 \%$