Question

Explain different types or errors in measurement. An object was weighed by a physical balance and following readings were obtained: $5.04g,5.06g,4.17g,5g,4.93g$
Find:
1) Mean absolute error
2) Percentage error

Answer 1

Hint: Error in measurement can be described as the difference between the measured value of a physical quantity and the true value of the physical quantity. Error in measurement is sometimes called uncertainty in measurement. There are mainly six types of errors, which can occur while measuring physical quantities.

Complete answer:
When we measure a physical quantity using a measuring device, it is practically impossible to find the true value of that physical quantity. The difference between the measured value of the physical quantity using a measuring device and the true value of the physical quantity obtained using a theoretical formula is termed as error in measurement of that physical quantity. There are mainly six types of errors in measurement. They are explained as follows:

a) Constant error: Constant error is described as the error that causes measurements to consistently deviate from their true value. In other words, constant errors cause the same amount of deviation as the measured value. If the constant error of a measuring device is known well before measuring quantities, much more accurate measurements can be taken by suitable subtraction or addition.

b) Systematic error: Systematic errors are errors which occur due to some known causes while taking measurements. Systematic error is divided into four, as follows:
1) Instrumental error: This error occurs due to poor calibration of the measuring device or the measuring apparatus.
2) Observational error: This error occurs due to poor observation by the observer, while taking measurements. Observational error is also called gross error or personal error.
3) External error: This error occurs due to external factors like temperature, pressure, presence of electric field or magnetic field, etc.
4) Experimental technique error: This error occurs due to poor experimental techniques and lack of knowledge of clear procedure of measurement.

c) Random error: Random error depends on the capabilities of the observer. This error is caused by carelessness of the observer while taking measurements. In order to reduce random error, multiple measurements of the same physical quantity are taken and the mean value of all these readings is taken, to get an accurate value of the physical quantity. If ${{a}_{1}},{{a}_{2}},{{a}_{3}},......,{{a}_{n}}$ are $n$ different readings of the same physical quantity, mean value of these $n$ readings is given by
${{a}_{mean}}=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....+{{a}_{n}}}{n}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{a}_{i}}}$

d) Absolute error: The difference between mean value of a physical quantity and individual measured values of that physical quantity is termed as absolute error. If ${{a}_{1}},{{a}_{2}},{{a}_{3}},......,{{a}_{n}}$ are $n$ different readings of the same physical quantity and ${{a}_{mean}}$ is the mean value of these readings, absolute error of ${{a}_{1}}$ is given by
$\Delta {{a}_{1}}={{a}_{mean}}-{{a}_{1}}$
Similarly, absolute errors of ${{a}_{2}},{{a}_{3}},......,{{a}_{n}}$ are given by
$\begin{align}
  & \Delta {{a}_{2}}={{a}_{mean}}-{{a}_{2}} \\
 & \Delta {{a}_{3}}={{a}_{mean}}-{{a}_{3}} \\
\end{align}$
.
.
.
$\Delta {{a}_{n}}={{a}_{mean}}-{{a}_{n}}$
The mean absolute error of all these absolute errors is taken and is considered to be the final absolute error in measurement of physical quantity. Mean absolute error is given by
$\Delta {{a}_{mean}}=\dfrac{\left| \Delta {{a}_{1}} \right|+\left| \Delta {{a}_{2}} \right|+\left| \Delta {{a}_{3}} \right|+...+\left| \Delta {{a}_{n}} \right|}{n}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{\left| \Delta {{a}_{i}} \right|}$

e) Relative error: The ratio of mean absolute error to the mean value of physical quantity is termed as relative error. It is given by
Relative error, $\delta a=\dfrac{\Delta {{a}_{mean}}}{{{a}_{mean}}}=\dfrac{\dfrac{1}{n}\sum\limits_{i=1}^{n}{\left| \Delta {{a}_{i}} \right|}}{\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{a}_{i}}}}$
f) Percentage error: Percentage error is nothing but the relative error in measurement of a physical quantity, expressed in percentage. It is given by
$P.E=\dfrac{\Delta {{a}_{mean}}}{{{a}_{mean}}}\times $100%

Now, let us move on to the next part of the question.
We are told that the weight of an object is measured using a weighing balance. Different readings of weight of the object are taken, as given below.
$5.04g,5.06g,4.17g,5g,4.93g$
We are supposed to find mean absolute error and percentage error. To calculate mean absolute error as well as percentage error in measurement of weight of the object, let us use the formulas given above, in the explanation of different types of errors in measurement.
Firstly, let us determine the mean value of weight from the given readings of weight. Mean weight is given by
${{a}_{mean}}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{{{a}_{i}}}=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+....+{{a}_{n}}}{n}=\dfrac{{{a}_{1}}+{{a}_{2}}+{{a}_{3}}+{{a}_{4}}+{{a}_{5}}}{5}=\dfrac{5.04+5.06+4.17+5+4.93}{5}=4.84$
 Now, let us calculate the absolute error of each reading as follows.
$\begin{align}
  & \Delta {{a}_{1}}={{a}_{mean}}-{{a}_{1}}=4.84-5.04=-0.2 \\
 & \Delta {{a}_{2}}={{a}_{mean}}-{{a}_{2}}=4.84-5.06=-0.22 \\
 & \Delta {{a}_{3}}={{a}_{mean}}-{{a}_{3}}=4.84-4.17=0.67 \\
 & \Delta {{a}_{4}}={{a}_{mean}}-{{a}_{4}}=4.84-5=-0.16 \\
 & \Delta {{a}_{5}}={{a}_{mean}}-{{a}_{5}}=4.84-4.93=-0.09 \\
\end{align}$

Clearly, from the formula mentioned in the above explanation, mean absolute error in measurement of weight is given by
$\Delta {{a}_{mean}}=\dfrac{1}{n}\sum\limits_{i=1}^{n}{\left| \Delta {{a}_{i}} \right|}=\dfrac{\left| \Delta {{a}_{1}} \right|+\left| \Delta {{a}_{2}} \right|+\left| \Delta {{a}_{3}} \right|+...+\left| \Delta {{a}_{n}} \right|}{n}=\dfrac{0.2+0.22+0.67+0.16+0.09}{5}=0.268$
Also, percentage error is given by
Percentage error $=\dfrac{\Delta {{a}_{mean}}}{{{a}_{mean}}}\times 100\%=\dfrac{0.268}{4.84}\times 100\%=5.53\%$

Note:
While taking the arithmetic mean of absolute errors of each reading, students should take care to utilize only the magnitude of absolute errors of each reading. It is to be understood that only the difference in absolute error and mean value is required. Therefore, only positive values of individual absolute errors are taken to calculate the arithmetic mean of absolute errors or the total absolute error in measurement of the physical quantity.