Accuracy and Errors in Measurement The measuring process is essentially a process of comparison. To measure any physical quantity, we compare it with a standard (unit) of that quantity. No measurement is perfect as the errors involved in the process cannot be removed completely. Hence, in spite of our best effort, the measured value of a quantity is always somewhat different from its actual value, or true value. Measurement error: The measurement error is defined as the difference between the true or the actual value and the measured value i.e., error is quantity = (True value — measurement value) of the quantity. Absolute Error, Relative Error and Percentage Error

1. Absolute error: Absolute error in the measurement of a physical quantity is the magnitude of the difference between the true value and the measured value of the quantity.

Let a physical quantity can be measured n times. Let the measured vahie be a₁,a₂,a₃ a the arithmetic mean of these values is: ${\mathrm{a}}_{\mathrm{m}}=\frac{{\mathrm{a}}_1+{\mathrm{a}}_2+{\mathrm{a}}_3\dots .{\mathrm{a}}_{\mathrm{n}}}{\mathrm{n}}$ 

Usually, a_m is taken as the true value of the quantity, if the same is unknown otherwise by definition, absolute errors in the measured value of the quantity are

Δa₁ = a_m − a₁

Δa₂ = a_m − a₂

Δa_n = a_m − a_n

Absolute errors may be positive in certain cases and negative in the other cases. 2. Mean absolute error: It is the arithmetic mean of the magnitude of absolute errors in all the measurements of the quantity it is represented by ta. Thus,

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

∣n Hence the final result of the measurement may be written as $\mathrm{a}={\mathrm{a}}_{\mathrm{m}}\pm \mathrm{\Delta }\overline{\mathrm{a}}$ This implies that any measurement of the quantity is likely to lie between $\left({\mathrm{\alpha }}_{\mathrm{m}}+\mathrm{\Delta }\overline{\mathrm{a}}\right)$ and $\left({\mathrm{a}}_{\mathrm{m}}-\mathrm{\Delta }\overline{\mathrm{a}}\right)$

3. Relative error: The relative error or fractional error of measurement is defined as the ratio of mean absolute error to the mean or value of the quantity measured. Thus,

$\text{ Relative error }=\frac{\text{ Mean absolute error }}{\text{ Mean value }}=\frac{\mathrm{\Delta }\overline{\mathrm{a}}}{{\mathrm{a}}_{\mathrm{m}}}$ 

4. Percentage error: When the relative/fractional error is expressed in term of percentage, we call it a percentage error. Thus,

$\text{ percentage error }=\frac{\mathrm{\Delta }\overline{\mathrm{a}}}{{\mathrm{a}}_{\mathrm{m}}}\times 100$