Errors Of Measurements MCQ - Practice Questions with Answers

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 $a_1,a_2,a_3\dots \dots \dots a_n$ are a measured value then

$a_m=\frac{a_1+a_2+\dots \dots a_n}{n}$
 

where am  = true value

then  

1)Absolute Error for nth reading $=\mathrm{\Delta }{a}_n=a_m-a_n=$ true value - measured value

$$\begin{array}{rl}{S}_0\mathrm{\Delta }{a}_1& =a_m-a_1\\ \mathrm{\Delta }{a}_2& =a_m-a_2\end{array}$$

2) Mean absolute error

$$\mathrm{\Delta }\overline{a}=\frac{\left|\mathrm{\Delta }{a}_1\right|+\left|\mathrm{\Delta }{a}_2\right|+\dots .\left|\mathrm{\Delta }{a}_n\right|}{n}$$
 

3) Relative error or Fractional error

The ratio of mean absolute error to the mean value of the quantity measured.

Relative error $=\frac{\mathrm{\Delta }\overline{a}}{a_m}$
$\mathrm{\Delta }\overline{a}-$ mean absolute error
$a_m=$ mean value
4) Percentage error

$$\text{ Percentage error }=\frac{\mathrm{\Delta }\overline{a}}{a_m}\times 100\mathrm{\%}$$
 

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

• Error in sum (x=a+b):-

• absolute error in $\mathrm{x}=\mathrm{\Delta }x=\pm (\mathrm{\Delta }a+\mathrm{\Delta }b)$
  where
  $\mathrm{\Delta }a=$ absolute error in measurement of a
  $\mathrm{\Delta }x=$ absolute error in measurement of $\times$

•          The percentage error in the value of x

               = $\frac{\mathrm{\Delta }x}{x}=\frac{(\mathrm{\Delta }a+\mathrm{\Delta }b)}{a+b}\times 100$

      2) Error in difference (x=a-b)

• absolute error in $\mathrm{x}=\mathrm{\Delta }x=\pm (\mathrm{\Delta }a+\mathrm{\Delta }b)$

• Percentage error in the value of $\mathrm{x}=\frac{\mathrm{\Delta }x}{x}=\frac{(\mathrm{\Delta }a+\mathrm{\Delta }b)}{a-b}\times 100\mathrm{\%}$

1) Error in product x=a.b

•  maximum fractional error $=\frac{\mathrm{\Delta }x}{x}=\pm (\frac{\mathrm{\Delta }a}{a}+\frac{\mathrm{\Delta }b}{b})$
  where
  $\mathrm{\Delta }a=$ absolute error in measurement of a
  $\mathrm{\Delta }b=$ absolute error in measurement of b
  $\mathrm{\Delta }x=$ absolute error in measurement of x

•   The percentage error in the value of x=

                  $=\frac{\mathrm{\Delta }x}{x}\ast 100=\pm (\frac{\mathrm{\Delta }a}{a}\ast 100+\frac{\mathrm{\Delta }b}{b}\ast 100)$

                  =(% error in value of a + % error in value of b)

2) Error in division x = a/b

• maximum fractional error in  

$=\frac{\mathrm{\Delta }x}{x}=\pm (\frac{\mathrm{\Delta }a}{a}+\frac{\mathrm{\Delta }b}{b})$

The percentage error in the value of $x=$

$=\frac{\mathrm{\Delta }x}{x}\ast 100=\pm (\frac{\mathrm{\Delta }a}{a}\ast 100+\frac{\mathrm{\Delta }b}{b}\ast 100)$

                  =(% error in value of a + % error in value of b)

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