Errors in Measurement
Fully accurate results are extremely rare in the cases of Mathematics and measurement and the values are just an approximation of the actual result. If someone measures an object many times, the readings are likely to be slightly different. This is popularly known as the variation in measurements. This is the unlikely cause of uncertainty in measurements, which cannot be avoided. These uncertainties are known as errors in measurement. Errors are dissimilar to mistakes and do not imply one has obtained a wrong answer.
It is just that they are not accurate enough for the actual result. The difference between the actual value of the measurement and the obtained result of the measurement is termed an error, which is a mathematical expression of the uncertainties in our measurements. There are different types of measurement errors, and they are broadly classified into two categories- Absolute Error and Relative Error.
Methods of Obtaining Errors in Measurement
There are various types of errors in measurement due to various reasons, be it faulty apparatus or human error. It is essential to reduce these errors to increase the precision of the results. Thus, the following modes of measuring errors have been suggested so that one can be aware of the amount of error that is present in the result.
Error Tolerance
Often margin of error or error tolerance is considered as a measure of error. Manufacturing industries often set up tolerance intervals where product measurements are continuously checked, and if they fall outside the interval, they are considered flawed.
The tolerance of an instrument is often regarded as the maximum allowable variation and a portion of its precision. Thus the tolerance range is obtained by adding or subtracting half of the precision of the instrument.
Suppose if a scale has a length of 5.2 cm and has a degree of precision of 0.2 cm. Then the tolerance interval of this scale is 5.2 + 0.1 cm and 5.2 - 0.1 cm, i.e. 5.1 cm to 5.3 cm. If the results are obtained within this specified range, then they are accepted.
Absolute Error
The actual amount of error during a measurement is referred to as absolute error. It helps us to understand the amount of error that is being measured.
The mathematical expression of absolute error is Absolute E = |Measured value - True value|
Example: The length of an instrument is 4.576 m \[\pm\] \[\pm\] 0.007 m. Then the absolute error is 0.007 m.
Did You Know?
Absolute error fails to convey the importance of the error and is thus often considered inadequate.
Suppose, if someone wishes to measure the distance between two cities. In this case, an error of a few centimetres would not have a massive effect on the actual value of the measurement. However, an error of a few centimetres while measuring the length of a small machine part is quite significant. Hence the severity of an error in the second situation is much more than in the first one.
There is an absolute error which is the magnitude of the difference between the mean value and each measured value.
As an Example:
We measured the diameter of three wires using screw gauges and found that they were 1.002 cm, 1.004 cm, and 1.006 cm. We will have an absolute error of:
A solution is:
The mean is 1.02 + 1.004, plus 1.06, equalling 1.04 cm
As a result, absolute errors are
Δa1=|1.002−1.004| =0.002cm
Δa2=|1.004−1.004 |=0.000cm
Δa3=|1.006−1.004 |=0.002cm
Relative Error
The ratio of the absolute error to the accepted measurement is termed the relative error of measurement. The relative error gives us the magnitude of the absolute error in terms of the actual measurement of the object. The measured value is often used when the actual measurement of the object is not known for computing relative error. This way we can determine the magnitude of the absolute error in terms of the actual size of the measurement. If the true measurement of the object is not known then the relative error is found using the measured value.
The mathematical expression of relative error is:
Relative Error = Absolute Error/Actual value
Thus Relative Error = | measured value - actual value | / actual value
Percentage Error
Measurement of errors is also often represented in percentage terms. This way of representation is quite similar to the relative error, and the error is converted to a percentage value. The relative error is multiplied by 100% to obtain the percentage error, which is the percentage uncertainty of the error.
The mathematical expression for percentage error is:
Percentage E = ( | measured value - actual value | / actual value ) \[\times\] 100%.
FAQs on Errors - Absolute Error, Relative Error
1. What are the types of measurement errors in research, and what are their sources?
Error is the difference between the measurements of the actual value and the obtained value of any physical quantity. In Physics, there are mainly two kinds of errors- random errors and systematic errors.
Measurement of all physical quantities is inaccurate and uncertain. The leading causes of errors in measurement are:
Inexperience or less knowledge of the person.
Fault in the apparatus.
Use of wrong methods or techniques.
Random errors are those errors of measurement of quantities that provide different values in repeated measurements under the same conditions. The source of random error is not known. One way of removing random error is by taking several measurements and then taking their mean.
Systematic errors are those when all the measurements of an object are equally affected, leading to a consistent difference in the readings. The systematic error may occur due to the presence of zero error in the measuring instrument or poor and incorrect calibration on the instruments. They can be considerably reduced by comparison to a more accurate instrument. Then, all the readings can be modified by adding a correction factor to them to increase accuracy.
2. What are the differences between absolute error and relative error?
Absolute and relative errors are two types of errors in the measurement of some physical quantity. They are a bit different concerning the calculations and their mathematical expressions. Most scientific experiments consist of errors either due to faulty apparatus or negligence or incompetence of personnel involved.
Absolute error is the difference between the actual value and the calculated value while the relative error is the ratio of the absolute error and the experimental value. This is the primary difference between these two types of errors. An absolute error has the same unit as the unit of measurement. On the other hand, relative error, being a ratio of two quantities, has no unit.
3. What is relative error as measure of precision?
Relative error (RE) is a measure of precision that is based on the ratio of an absolute error against the measurement being taken. In other words, relative error (RE) is a type of error based on the size of the item being measured. RE is expressed as a percentage and has no units.
RE= Absolute error / measurement being taken
Example- A rug is being measured with a meter stick by two people; one will measure the length and the other will measure the width, which means the absolute error is 0.001 m.
Its length is 3.215 metres, which is equivalent to 0.0003%. The width is 4.075 metres, which is equivalent to 0.0002%.
4. Explain what is Absolute Error.
It is a way of evaluating regression models using a metric known as the mean absolute error. It is the mean of the absolute value of all the predicted errors in a model concerning a given test set. Mean Absolute Error is equal to the absolute value of the model's error over all the instances in the test set.
Data absolute errors are averaged together to form the mean absolute error. The mean absolute error (MAE) is often abbreviated as MAE. A statistical measure of error is calculated by dividing the total number of errors by the sum of the absolute errors. The formula for MAE is:
\[MAE =\frac{1}{n} \displaystyle\sum\limits_{i=1}^n |x_{i} - x|\]
Here,
\[|x_{i} – x|\] = absolute errors
n = number of errors