Measurement Error is the difference between the approximated value and the actual value of an object/quantity. The relative error can be computed in the percent with the help of relative error formula.
Errors in measuring instruments occur due to inescapable faults in the measuring instrument and limitations of the human eye. Errors come in all sizes, and sometimes we are required to decide if the error is so big that it makes the measurement meaningless. The smaller the error the more we are close to the actual value.
Types of Errors
There are mainly 3 types of measurement errors:
A. Absolute Error
The absolute error is the difference between the measured value and the real value. The expression for absolute error is:
Eabsolute=|xmeasured−xaccepted|
For example, if you know a length is 4.635 m + 0.007 m, then 0.007 m is an absolute error.
B. Relative Error
It represents the ratio of the absolute error of the measurement to the accepted measurement. Relative Error can be represented as;
Erelative = Eabsolute/xaccepted
Relative Error formula =|measured value−actual value| / actual value
C. Percentage Error
Percentage error is similar to relative error besides that the error here is converted to a percent value. The expression for percent error is
Percent of Error=|measured value−actual value| / actual value∗100 %
What is the Relative Error Formula?
Percent error informs us about to what extent these inescapable errors affect our experimental outcomes.
The formula used to find percent relative error is:
Percent Relative Error = ∣ Actual Value - Estimated Value/Actual Value∣×100
That being said, Absolute value can also be sometimes referred to as theoretical value or true value. Mostly, the percentage error is represented as a positive value. The value of the absolute error is divided by an accepted value and presented as a percent.
Relative Error and Percent Error
Relative Error = \[\frac{丨measured - real丨}{real}\]
Percent Error = \[\frac{丨measured - real丨}{real}\] x 100%
Define Absolute Error
In mathematics, Absolute error is the difference between the measured or the inferred value and the actual value of a quantity. The absolute error is insufficient because of the fact that it does not provide any information with respect to the significance of the error. That said, while measuring distances between cities kilometres apart, an error of some centimetres is trivial and is immaterial.
What is Absolute Error Formula
Suppose that x is the actual value of a quantity or an instrument and x0 is the measured value of the quantity, then the absolute error value could be computed with the below-stated formula
Δx = x0-x.
Here, Δx is termed as an absolute error.
If we take into account multiple measurements, then the arithmetic mean of absolute error of individual measurements must be the ultimate absolute error.
How To Find Cumulative Error
cumulative error calculation is done by finding the error of the real equation and multiplying that error by the number of times the error has been repeated. Find out the number of times the error has been made and multiply that by the actual error to calculate your cumulative error.
Solved Examples on Relative Error Formula
Example:
Jessica measured his height and found it to be 5 feet. However, after she carefully measured her height a second time, she found her actual height to be 4.5 feet. What is the percent error Jessica made when she measured the first time?
Solution:
Before solving the problem, let us identify the information:
Actual value: 4.5 ft and Estimated value: 5 ft
Now,
Step1: Subtract one value from others in order to obtain the absolute value of error. Error = |4.5−5|=0.5| = 0.5
Step-2: Then, Divide the error by the actual value, you will get; 0.5/4.5=0.1111(up to four decimal places)
Step-3: Multiply that outcome by 100 and attach a% sign to express the answer in the form of a percentage. 0.111×100=11.11
Thus: Percentage error = 11.11%
Example:
Police penalized Alex for speeding up the car. Alex got a traffic penalty notice for travelling 90 mph in a 70 mph zone. Alex insisted his speedometer was 70 mph, not 90 mph. What could Alex profess as his percent error?
Solution:
Let us arrive at % error in the below given three steps:
Absolute Error: |90−70| = 20
Relative percent error: 20/60 = 0.3333
= 0.3333 × 100
=33.33%
Thus: Alex can claim 33.33% as his percent error.
FAQs on Error Measurement
1. What is an error in measurement?
An error in measurement is the difference between the actual or true value of a quantity and the value obtained during its measurement. It represents the degree of inaccuracy in any measurement, as no measurement can be perfectly precise. Errors are an unavoidable aspect of any scientific experiment or observation.
2. What are the main types of errors in measurement as per the CBSE syllabus?
According to the NCERT/CBSE curriculum for 2025-26, errors in measurement are primarily classified into three types:
- Systematic Errors: These are errors that tend to be in one direction, either consistently positive or consistently negative. They can arise from instrumental defects, incorrect experimental techniques, or personal bias.
- Random Errors: These errors occur irregularly and randomly in both magnitude and direction. They are caused by unpredictable fluctuations in experimental conditions like temperature or voltage, and slight variations in personal judgment.
- Gross Errors: These are large errors caused by mistakes or carelessness on the part of the observer, such as misreading an instrument, recording a wrong value, or using an incorrect formula for calculation.
3. What is the difference between accuracy and precision in measurement?
Accuracy and precision are two fundamental concepts in measurement that are often confused. Accuracy refers to how close a measured value is to the true or accepted value. Precision refers to how close multiple measurements of the same quantity are to each other. For example, if you measure a 10 cm line multiple times and get readings of 10.1 cm, 10.1 cm, and 10.1 cm, your measurement is precise but not necessarily accurate. If you get readings of 9.9 cm, 10.0 cm, and 10.1 cm, your measurements are more accurate on average, but less precise.
4. Can you provide a real-world example of both systematic and random errors?
Certainly. Imagine you are measuring the boiling point of water. The true value is 100°C at standard pressure.
- A systematic error would occur if your thermometer is poorly calibrated and consistently reads 2°C higher. Every measurement you take would be around 102°C. The error is consistent and in one direction.
- A random error would occur due to slight fluctuations in the flame heating the water or you looking at the thermometer scale from slightly different angles each time. Your readings might be 99.8°C, 100.3°C, 99.9°C, etc. These errors are unpredictable and average out over many measurements.
5. What are absolute, relative, and percentage errors?
These are three different ways to quantify the magnitude of an error:
- Absolute Error: This is the magnitude of the difference between the true value and an individual measured value. It has the same units as the quantity being measured.
- Relative Error: This is the ratio of the mean absolute error to the true or mean value of the quantity. It is a dimensionless quantity and indicates the error relative to the size of the measurement.
- Percentage Error: This is simply the relative error expressed as a percentage. It is calculated by multiplying the relative error by 100 and is useful for easily comparing the significance of errors in different measurements.
6. How do errors propagate when you add or multiply measured quantities?
When you perform mathematical operations on quantities that have errors, those errors combine or 'propagate'. The rules for propagation are different for various operations:
- For Addition or Subtraction: The absolute error in the final result is the sum of the absolute errors in the individual quantities. For example, if Z = A + B, then the error in Z (ΔZ) is ΔA + ΔB.
- For Multiplication or Division: The relative error in the final result is the sum of the relative errors in the individual quantities. For example, if Z = A * B, then the relative error (ΔZ/Z) is (ΔA/A) + (ΔB/B).
7. Why is it important to distinguish between systematic and random errors during an experiment?
Distinguishing between systematic and random errors is crucial because they are handled differently to improve experimental results. Systematic errors are predictable and directional, meaning once identified, their cause can be determined and they can be corrected for, for example, by calibrating an instrument. In contrast, random errors are unpredictable and cannot be eliminated. Their effect can only be minimised by taking multiple readings and calculating an average, as the random positive and negative errors tend to cancel each other out.
8. What is the 'least count' of an instrument and how does it contribute to measurement error?
The least count of a measuring instrument is the smallest and most accurate value that can be measured by that instrument. For instance, a standard ruler might have a least count of 1 millimetre. The least count introduces an error known as 'least count error,' which is a type of random error. This error arises because the instrument cannot measure values smaller than its least count, forcing the observer to estimate the reading, leading to uncertainty in the measurement.
