[Physics Class Notes] on Errors – Absolute Error, Relative Error Pdf for Exam

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.

 

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.

 

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

 

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

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%.