Absolute Error, Relative Error, Percentage Error and Combination of Errors Physics Notes

Absolute Error, Relative Error, Percentage Error and Combination of Errors Physics Notes

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 value be a₁, a₂, a₃, ………….a_n the arithmetic mean of these values is:
a_m = \(\frac{a_{1}+a_{2}+a_{3} \ldots a_{n}}{n}\)
Usually am 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

Absolute errors may be positive in certain cases and negative in other cases.

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 Aa. Thus,

Hence the final result of the measurement may be written as a = am ± Δ\(\bar{a}\).
This implies that any measurement of the quantity is likely to lie between (am+Δ\(\bar{a}\)) and (am – Δ\(\bar{a}\)).

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,
Relative error = \(\frac{\text { Mean absolute error }}{\text { Mean value }}=\frac{\Delta \bar{a}}{a_{m}}\)

Percentage error: When the relative/fractional error is expressed in term of percentage, we call it percentage error. Thus,
Percentage error = \(\frac{\Delta \bar{a}}{a_{m}}\) × 100%

Combination of Errors:
Now, we will study how errors get combined while performing mathematical operations such as addition, subtraction, multiplication and division.
1. Error in Addition of Quantities
Let x = a + b
Let Δa = absolute error in measurements of a
Δb = absolute error in measurement of b
Δx = absolute error in addition of quantities
So, x ± Δx = (a ± Δa) + (b ± Δb)
or x ± Δx = (a + b) ± Δa ± Δb
or x ± Δx = x ± Δa ± Δb
or ±Δx = ± Δa ± Δb
Here, Δx can have four possible values:
(+Δa + Δb), (+Δa – Δb), (-Δa + Δb), (-Δa – Δb)
Maximum possible error in x Δx = ±(Δa + Δb)

2. Error in Difference of Quantities
Let x = a – b
Let Δa = absolute error in a
Δb = absolute error in b
Δx = absolute error in x
x ± Δx = (a ± Δ a) – (d ± Δ b)
or, x ± Δx = (a – d) ± Δa + Δd
or, x±Δx = x±Δa + Δb
or ±Δx = ±Δa + Δb
Here, Δx can have four possible values:
(+Δa – Δb), (-Δa + Δb), (-Δa – Δb), (+Δa + Δb)

Maximum absolute error indifference of quantities
Δx = ± (Δa + Δb)

3. Error in Relation of Multiplication of Physical Quantities
Let x = a x b
Let Δa = absolute error in measurements of a
Δb = absolute error in measurements of b
Δx = absolute error in the product of a and b
x ± Δx = (a ± Δa) x (b ± Δb)

The values of both \(\frac{\Delta a}{a}\) and \(\frac{\Delta b}{b}\) is very small. So, their product will be very very small. Hence, it can be neglected.

4. Error in Relation of Physical Quantities
Let x = \(\frac{a}{b}\)
Let Δa = absolute error in a
Δb = absolute error in b

Δx = absolute error in \(\frac{a}{b}\)

The values of both \(\frac{\Delta a}{a}\) and \(\frac{\Delta b}{b}\) is very small. So, the value of their product will be very small and it can be neglected.

5. Error Due to the Power of Physical Quantities

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