Absolute and Relative Error

Scientific calculations rely heavily on measurement. It’s extremely rare to get completely accurate measurement findings. Slight mistakes are common when monitoring various parameters. There are various forms of errors that result in measurement discrepancies. Mathematical equations can be used to express all of the faults. We can compute correctly and develop strategies to remedy errors if we know the errors. Absolute and relative mistakes are the two most common forms of errors. Scientific calculations depend primarily on measurement. It’s extremely rare to get completely accurate measurement findings. Slight mistakes are common when monitoring various parameters. There are various forms of errors that result in measurement discrepancies. Mathematical equations can be used to express all of the faults. We can compute correctly and develop strategies to correct errors if we know the errors. Absolute and relative mistakes are the two most common forms of errors. We will define absolute error and relative error in this essay. Along with the definition, we provide explanations, calculations, and examples of absolute and relative inaccuracy. In measuring, the concept of error calculation is critical.

Absolute Error

Absolute error is a measure of how distant a measurement is from its real value, as well as an indication of the measurement’s uncertainty. For example, if you use a ruler with millimetre marks to measure the width of a book, the best you can do is measure the width to the next millimetre.

The discrepancy between a quantity’s measured or inferred value and its actual value is known as absolute error. The absolute error is insufficient, it provides no information about the significance of the error.

Consider another situation in which a centimetre inaccuracy is a substantial error when measuring small machine parts. Both inaccuracies are of the order of centimetres, but the second is more serious than the first.

Absolute Error Formula

The absolute error value can be determined using the formula if x is the actual value of a quantity and x’ is the measured value of the quantity.

X” is equal to x’-x.

In this case, X” is referred to as an absolute error.

When many measurements are taken, the final absolute error should equal the arithmetic mean of the absolute errors of the individual measurements.

For example: Suppose that the actual value of a measurement is 10.25 cm and the measured value of that measurement is 11.05 cm.

Then, the value of absolute error is |Measured value – Absolute value| i.e. |11.05-10.25|= 0.80 cm.

Relative Error

The relative error is defined as the ratio of the measurement’s absolute error and the actual value. We can get an indication of how accurate the measurement is when compared to the actual size by calculating the relative error. 

The magnitude of absolute error can be calculated using relative error. If the exact value is unavailable, the relative error can be estimated using the quantity’s measured value. There is no unit for relative error because it is dimensionless.

The relative inaccuracy indicates how accurate the measurement is in comparison to the size of the object measured.

Relative Error Formula

The ratio of absolute error to the actual value of the quantity is used to determine the relative error. 

The relative error can be calculated using the formula below if x is the actual value of a quantity, x’ is the measured value of the quantity, and x” is the absolute error.

Then the relative error =x”/x

Where x”=x’-x.

In actual life, this is an example of relative error. suppose that an eraser’s true length is 35 mm and the measured length of the eraser is 34.13 mm. The absolute inaccuracy is now (0.87 mm) = (35-34.13) mm.

As a result, the relative error equals to absolute error/actual length= 0.87/35= 0.02485.

Relative Error Measure of Accuracy

In different cases, the relative error is the measure of precision. At the same time, it use as a measure of accuracy. Accuracy is the extent of evaluating how accurate the value is as compared to the actual or true value. 

Because the units cancel out throughout the calculation, the relative error has no units. Because the relative error is a proportion, we may convert it to a percentage by multiplying it by 100 percent.

Relative Error Vs Absolute Error

Although there is a difference between absolute error and relative error based on their computation, they are both techniques of identifying flaws in experimental measurements. Instrumental and human errors cause the majority of measurements in scientific investigations to be inaccurate. In some circumstances, there is a pre-defined constant value for absolute error for specific measurement equipment.

It’s the discrepancy between the experimental and true values. The relative error, on the other hand, fluctuates based on the experimental value and absolute error. It is calculated by dividing the absolute error by the experimental value.

 Absolute error is the size of the difference between the precise value and the approximation, whereas relative error is derived by dividing the absolute error by the magnitude of the exact value.

Conclusion

Absolute error is a measure of how distant a measurement is from its real value, as well as an indication of the measurement’s uncertainty.

The relative error can be defined as the absolute error divided by the measurement’s size. To get the absolute error, just divide the measured value by the absolute error.