Real-life Examples of Absolute and Relative Errors
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 Absolute Error and Relative Error in Physics
1. What are the main types of errors in Physics measurements according to the CBSE syllabus?
In Physics, measurement errors are broadly classified into two main types as per the NCERT curriculum for the 2025-26 session:
- Systematic Errors: These errors consistently affect the measurement in a single direction, meaning they are either always positive or always negative. Common causes include instrumental faults like zero error, imperfect experimental techniques, or personal bias.
- Random Errors: These errors occur irregularly and without a predictable pattern. They arise from sudden, unpredictable fluctuations in experimental conditions (like temperature or voltage) or limitations in an observer's judgment.
2. What is the fundamental difference between absolute error and relative error?
The fundamental difference lies in what they signify. Absolute error is the magnitude of the difference between an individual measured value and the true value, expressed in the same units as the quantity being measured. In contrast, relative error is the ratio of the mean absolute error to the true value. It is a dimensionless quantity that indicates the error in proportion to the size of the measurement, making it crucial for comparing precision.
3. What is the formula for calculating percentage error, and how does it relate to relative error?
The formula for percentage error is a direct extension of relative error. It is calculated by multiplying the relative error by 100 to express it as a percentage. The formula is:
Percentage Error = (Mean Absolute Error / True Value) × 100%
This conversion makes the magnitude of the error more intuitive and easier to interpret in practical applications.
4. How do you calculate the mean absolute error from a series of measurements?
To calculate the mean absolute error for a set of observations, you should follow these steps as per the CBSE guidelines:
- First, take a series of measurements (a₁, a₂, a₃, ..., aₙ) of the physical quantity.
- Second, calculate their arithmetic mean (a_mean), which is considered the best estimate of the true value.
- Third, determine the absolute error for each individual measurement by finding the positive magnitude of the difference between it and the mean (e.g., |a₁ - a_mean|).
- Finally, calculate the arithmetic mean of all these individual absolute errors to find the mean absolute error.
5. Why is relative error often more useful than absolute error for comparing the accuracy of two different measurements?
Relative error is more useful because it provides context. An absolute error of 1 cm is very significant when measuring a 10 cm pencil but almost negligible when measuring a 100 m racetrack. Relative error expresses the error as a fraction of the actual size of the measurement. This allows for a fair and meaningful comparison of the precision between measurements of vastly different scales, which absolute error alone cannot provide.
6. Can absolute error be negative? Explain with an example.
No, the absolute error itself cannot be negative because it is defined as the magnitude (or absolute value) of the deviation from the true value, which is always positive or zero. However, the *error* (without the 'absolute' term) can be negative. For example, if the true length of a rod is 15.0 cm and you measure it as 14.9 cm, the error is 14.9 - 15.0 = -0.1 cm. But the absolute error is |-0.1 cm| = 0.1 cm.
7. What is the important distinction between accuracy and precision in the context of measurement errors?
Accuracy and precision are two different indicators of measurement quality:
- Accuracy describes how close a measurement is to the actual or true value. High accuracy implies low systematic error.
- Precision describes how close multiple measurements of the same quantity are to each other. High precision implies low random error.
An experiment can be very precise (all readings are close together) but inaccurate (all readings are far from the true value), and vice-versa.
8. In a school lab experiment, what practical steps can a student take to minimise both systematic and random errors?
To improve experimental results, a student can minimise errors using these methods:
- To reduce systematic errors, one must ensure instruments are correctly calibrated, check for and correct any zero error in devices like vernier callipers, and use proper, consistent experimental techniques.
- To reduce random errors, the most effective strategy is to take a large number of readings for the same measurement and then calculate their arithmetic mean. This approach helps to cancel out the random positive and negative fluctuations, yielding a result closer to the true value.
