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,
$\Delta O_{\text{mean }}=\sum _{i=1}^5\frac{\left|\Delta O_i\right|}{n}$
$=\frac{|80.0-\mu |+|80.5-\mu |+|81.0-\mu |+|81.5-\mu |+|82-\mu |}{5}$
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 $=\frac{\Delta O_{\text{mean }}}{O_{\text{mean }}}\times 100\%$
$=\frac{0.6}{81}\times 100\%=0.74\%$