Errors Of Measurements - Practice Questions & MCQ

It is the magnitude of the difference between the true value and the measured value of the quantity.

It may be positive in certain cases and negative in certain other cases

If $a_1, a_2, a_3 \ldots \ldots \ldots a_n$ are a measured value
then $a_m=\frac{a_1+a_2+\ldots \ldots a_n}{n}$
where $a_m=$ true value
then
1) Absolute Error for nth reading $=\Delta a_n=a_m-a_n=$ true value - measured value

So $\Delta a_1=a_m-a_1$

$
\Delta a_2=a_m-a_2
$

2) Mean absolute error

$
\Delta \bar{a}=\frac{\left|\Delta a_1\right|+\left|\Delta a_2\right|+\ldots .\left|\Delta a_n\right|}{n}
$

3) Relative error or Fractional error

The ratio of mean absolute error to the mean value of the quantity measured.
Relative error $=\frac{\Delta \bar{a}}{a_m}$
$\Delta \bar{a}-$ mean absolute error
$a_m=$ mean value

4) Percentage error

Percentage error $=\frac{\Delta \bar{a}}{a_m} \times 100 \%$

1) Error in sum ( $x=a+b)$

- Error in $\operatorname{sum}(\mathrm{x}=\mathrm{a}+\mathrm{b})$ :-
- absolute error in $\mathrm{x}=\Delta x= \pm(\Delta a+\Delta b)$
where
$\Delta a=$ absolute error in measurement of a
$\Delta b=$ absolute error in measurement of b
$\Delta x=$ absolute error in measurement of x
- The percentage error in the value of $\mathrm{x}=\frac{\Delta x}{x} \times 100=\frac{(\Delta a+\Delta b)}{a+b} \times 100$

2) Error in difference ( $x=a-b$ )
- Absolute error in $\mathrm{x}=\Delta x= \pm(\Delta a+\Delta b)$
- Percentage error in the value of $\mathrm{x}=\frac{\Delta x}{x} \times 100=\frac{(\Delta a+\Delta b)}{a-b} \times 100$

1) Error in product $x=a b$
- maximum fractional error $=\frac{\Delta x}{x}= \pm\left(\frac{\Delta a}{a}+\frac{\Delta b}{b}\right)$
where
$\Delta a=$ absolute error in measurement of a
$\Delta b=$ absolute error in measurement of b
$\Delta x=$ absolute error in measurement of x

- The percentage error in the value of $\mathrm{x}=\frac{\Delta x}{x} * 100= \pm\left(\frac{\Delta a}{a} * 100+\frac{\Delta b}{b} * 100\right)$
$=(\%$ error in value of $a+\%$ error in value of b)
2) Error in division $x=\frac{a}{b}$
- maximum fractional error in $\mathrm{x}=\frac{\Delta x}{x}= \pm\left(\frac{\Delta a}{a}+\frac{\Delta b}{b}\right)$
- The percentage error in the value of $\mathrm{x}=\frac{\Delta x}{x} * 100= \pm\left(\frac{\Delta a}{a} * 100+\frac{\Delta b}{b} * 100\right)$
$=(\%$ error in value of $a+\%$ error in value of b)

when $\left(x=\frac{a^n}{b^m}\right)$

- The maximum fractional error in x is:- $\frac{\Delta x}{x}= \pm\left(n \frac{\Delta a}{a}+m \frac{\Delta b}{b}\right)$
- Percentage error in the value of $\mathrm{x}==\frac{\Delta x}{x} * 100= \pm\left(n \frac{\Delta a}{a} * 100+m \frac{\Delta b}{b} * 100\right)$