If the random error in the arithmetic mean of $50$ observations is $\alpha$, then the random error in the arithmetic mean of $150$ observations would be

The dimensional formula for a physical quantity $x$ is $\left[ M ^{-1} L ^{3} T ^{-2}\right]$. The errors in measuring the quantities $M , L$ and $T$ respectively are $2 \%, 3 \%$ and $4 \%$. The maximum percentage of error that occurs in measuring the quantity $x$ is

In an experiment four quantities $a, b, c$ and $d$ are measured with percentage error $1\%, 2\%, 3\%$ and $4\%$ respectively. Quantity $w$ is calculated as follows $w\, = \,\frac{{{a^4}{b^3}}}{{{c^2}\sqrt D }}$  error in the measurement of $w$ is .......... $\%$

Explain effect of multiplication or division of error on final result.

The least count of a stop watch is $\frac{1}{5}$ second. The time of $20$ oscillations of a pendulum is measured to be $25$ seconds. The maximum percentage error ig the measurement of time will be ..... $\%$

A student determined Young's Modulus of elasticity using the formula $Y=\frac{M g L^{3}}{4 b d^{3} \delta} .$ The value of $g$ is taken to be $9.8 \,{m} / {s}^{2}$, without any significant error, his observation are as following.

Then the fractional error in the measurement of ${Y}$ is