Question

The random error in the arithmetic mean of $100$ observations of a physical quantity is $x$, then random error in the arithmetic mean of $400$ observations of the same physical quantity will be

• A. $\frac{x}{4}$
• B. $\frac{x}{2}$
• C. $4x$
• D. $2x$

Answer 1

The correct option is A $\frac{x}{4}$

Random error is due to any random causes in the measurement process. It can be reduced by repetitions and averaging of the observations. If more observations are taken then less random error can be found in arithmetic mean.
$\Rightarrow$ Random error $\propto$ $1/$ Number of observations
Hence, increasing the number of observations to $4$ times will decrease the random error to $1/4$ times the initial value. So, $\frac{x}{4}$ will be the random error in $400$ observations.