Question

What is error? Explain the category of errors.

Answer 1

What is error? Explain the category of errors.

The aim of measurement is to find the true value $x_0$ of the measured parameter. Different circumstances, such as inaccuracy of measuring device, external interference, human carelessness etc. affect the measurement and the operator obtains an experimental value $x_{\mathrm{exp}}$ as a result. Note: $x_{\mathrm{exp}} \neq x_0$.

Category of errors:

The absolute error is a modulus of the difference between the experimental and true values:

$$\Delta = \left| x _ {\exp} - x _ {o} \right|$$

Relative error characterizes the proportion of absolute error from the true value:

$$\eta = \frac {\left| x _ {\exp} - x _ {o} \right|}{\left| x _ {o} \right|} = \frac {\Delta}{\left| x _ {o} \right|}$$

Percent error is the relative error, expressed in terms of percents:

$$\delta = \frac {\left| x _ {\exp} - x _ {o} \right|}{\left| x _ {o} \right|} \cdot 100 \% = \frac {\Delta}{\left| x _ {o} \right|} \cdot 100 \% = \eta \cdot 100 \%$$

Also, when in calculus approximations are used instead of the real data (e.g., 5.4 instead of 5.389), to find the errors we operate with the real value $x_0$ and approximated $x_{\mathrm{app}}$.

As you can see from above, error characterizes the deviation of the real data from its true value.