Error Measurement Formula Types and Solved Examples
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 in Mathematics Explained Clearly
1. What is error measurement in mathematics?
Error measurement in mathematics is the process of quantifying the difference between a measured or approximate value and the true value. It helps determine the accuracy and reliability of calculations and experiments.
- Error = True Value − Measured Value
- It shows how far a result deviates from the exact value.
- Used in numerical methods, experiments, and data analysis.
2. What is the formula for absolute error?
The formula for absolute error is |True Value − Measured Value|. It measures the magnitude of the error without considering its direction.
- Absolute Error = |T − A|
- Always non-negative.
- Example: If true value = 50 and measured value = 47, absolute error = |50 − 47| = 3.
3. What is relative error and how is it calculated?
Relative error is the ratio of the absolute error to the true value and is calculated as |T − A| / |T|. It shows how significant the error is compared to the true value.
- Relative Error = Absolute Error / True Value
- Usually expressed as a decimal or fraction.
- Example: If absolute error = 2 and true value = 40, relative error = 2/40 = 0.05.
4. How do you calculate percentage error?
Percentage error is calculated using the formula (|T − A| / |T|) × 100%. It expresses the error as a percentage of the true value.
- Step 1: Find absolute error.
- Step 2: Divide by true value.
- Step 3: Multiply by 100.
- Example: If true value = 80 and measured value = 76, percentage error = (4/80) × 100 = 5%.
5. What is the difference between absolute error and relative error?
The main difference is that absolute error measures the actual difference, while relative error compares that difference to the true value.
- Absolute Error: |T − A| (unit-based)
- Relative Error: |T − A| / |T| (unit-free ratio)
- Relative error helps compare accuracy across different scales.
6. What is mean absolute error (MAE)?
Mean Absolute Error (MAE) is the average of the absolute errors between predicted and actual values. It measures overall prediction accuracy.
- MAE = (1/n) Σ |Actual − Predicted|
- Common in statistics and data analysis.
- Lower MAE indicates better accuracy.
7. What is root mean square error (RMSE)?
Root Mean Square Error (RMSE) is the square root of the average of squared errors and is calculated as √[(1/n) Σ (Actual − Predicted)²]. It penalizes larger errors more heavily.
- Used in regression and forecasting.
- Always non-negative.
- Smaller RMSE means better model fit.
8. Why is percentage error important?
Percentage error is important because it shows the size of the error relative to the true value in percentage form.
- Helps compare accuracy across different measurements.
- Common in experiments and scientific calculations.
- Provides an intuitive measure of accuracy.
9. Can you give an example of calculating error measurement?
Yes, if the true value is 100 and the measured value is 92, different error measures can be calculated as follows:
- Absolute Error = |100 − 92| = 8
- Relative Error = 8/100 = 0.08
- Percentage Error = 0.08 × 100 = 8%
10. What are common mistakes when calculating error?
Common mistakes in error measurement include ignoring absolute value signs and using the wrong true value.
- Forgetting | | in absolute error.
- Dividing by measured value instead of true value for relative error.
- Not converting relative error to percentage correctly.
- Rounding too early in calculations.
