1.3 Working with errors  (Page 2/3)

Relative error

We can report range of error as the ratio of the mean absolute error to the mean value of the quantity. This ratio is known as relative error. Mathematically,

$$\Delta x_r=\frac{\Delta x_m}{x_m}$$

As we use a ratio, this expression of error is also known as “fractional error”. Applying this concept to earlier example, we have :

$$\Rightarrow \Delta x_r=\frac{\Delta x_m}{x_m}=\frac{0.1}{47.6}=0.0021$$

This is the amount of error which is possible for every “centimeter of length measured”. This is what is the meaning of a ratio. Hence, if there are 47.6 cm of total length, then the amount of error possible is 47.6 X 0.0021 = 0.1 cm. Two error range in the absolute form and relative form, therefore, are equivalent and specify the same range of errors involved with the measurement of a quantity.

Percentage error

Percentage error is equal to relative error expressed in percentage. It is given as :

$$\Delta x_p=\frac{\Delta x_m}{x_m}X100=\Delta x_{r}X100$$

Applying this concept to earlier example, we have :

$$\Delta x_p=\Delta x_{r}X100=0.0021X100=0.21$$

Combination of errors

Measurement of a quantity is used in a formula in various combinations to calculate other physical quantities. The mathematical operations in the working of a formula involve arithmetic operations like addition, subtraction, multiplication and division. We need to evaluate the implication of such operations on the error estimates and what is the resulting error in the quantities derived from mathematical operations. For example, let us consider simple example of density. This involves measurement of basic quantities like mass and volume.

Clearly, we need to estimate error in density which is based on the measurements of mass and volume with certain errors themselves. Similarly, there are more complex cases, which may involve different mathematical operations. We shall consider following basic mathematical operations in this section :

• sum or difference
• product or division
• quantity raised to a power

Errors in a sum or difference

We consider two quantities whose values are measured with certain range of errors as :

$$a=a\pm \Delta a$$

$$b=b\pm \Delta b$$

Sum of the two quantities is :

$$\Rightarrow a+b=a\pm \Delta a+b\pm \Delta b=a+b+\left(\pm \Delta a\pm \Delta b\right)$$

Two absolute errors can combine in four possible ways. The corresponding possible errors in “a+b” are (Δa + Δb), -(Δa + Δb), (Δa - Δb) and (-Δa + Δb). The maximum absolute error in “a-b”, therefore, is “Δa+ Δb”.

Difference of the two quantities is :

$$\Rightarrow a-b=a\pm \Delta a-b-\left(\pm \Delta b\right)=a-b+\{\pm \Delta a-\left(\pm \Delta b\right)\}$$

Two absolute errors can combine in four possible ways. The corresponding possible errors in “a+b” are again (Δa + Δb), -(Δa + Δb), (Δa - Δb) and (-Δa + Δb). The maximum absolute error in “a-b”, therefore, is “Δa+ Δb”.

Let “Δc “ be the absolute error of the arithmetic operation of addition or subtraction. Then, in either case, the maximum value of absolute error in the sum or difference is :

$$\Delta c=\Delta a+\Delta b$$

We see here that the absolute error in the sum or difference of two quantities is equal to the sum of the absolute values of errors in the individual quantities. We can write the resulting value as :

For addition as :

$$\Rightarrow c=\left(a+b\right)\pm \left(\Delta a+\Delta b\right)$$

For subtraction as :

$$\Rightarrow c=\left(a-b\right)\pm \left(\Delta a+\Delta b\right)$$

Example

Problem 1: The values of two capacitors are measured as :