To find the relative average error, you'll first need to calculate the relative error for each individual measurement and then find the average of those relative errors. This approach accounts for the error in relation to each specific measured value, providing a more nuanced understanding of the overall accuracy. The key is that relative error is based on an individual error compared to the individual measurement.
Here's a breakdown of the process:
Steps to Calculate Relative Average Error
- Calculate the absolute error for each measurement: The absolute error is the difference between the measured value and the actual or accepted value.
- Absolute Error = |Measured Value - Actual Value|
- Example: If you measured 10.2 cm and the actual value was 10 cm, the absolute error is |10.2 - 10| = 0.2 cm.
- Calculate the relative error for each measurement: As stated in our reference, you find the relative error by dividing the absolute error by the measured value. This expresses the error as a fraction of the measured quantity.
- Relative Error = Absolute Error / Measured Value
- Using the example above, if the measured value is 10.2 cm, the relative error is 0.2 / 10.2 = 0.0196 (approximately) or 1.96%
- Calculate the average of all relative errors: Once you have the relative error for each individual measurement, sum them and then divide by the total number of measurements.
- Relative Average Error = (Relative Error 1 + Relative Error 2 + ... + Relative Error N) / N where N represents the total number of measurements.
Example
Let's say we have three measurements:
| Measurement | Measured Value (cm) | Actual Value (cm) | Absolute Error (cm) | Relative Error |
|---|---|---|---|---|
| 1 | 5.2 | 5.0 | 0.2 | 0.2 / 5.2 = 0.0385 |
| 2 | 10.3 | 10.0 | 0.3 | 0.3 / 10.3 = 0.0291 |
| 3 | 7.8 | 8.0 | 0.2 | 0.2 / 7.8 = 0.0256 |
- Calculate the relative error for each measurement (as shown in the table above).
- Sum of relative errors: 0.0385 + 0.0291 + 0.0256 = 0.0932
- Divide by the number of measurements (3): 0.0932 / 3 = 0.0311 (approximately)
Therefore, the relative average error for this data set is approximately 0.0311 or 3.11%.
Key Considerations:
- Consistency of Units: Ensure that your measured and actual values are in the same units before calculating the absolute error.
- Percentage Representation: Relative errors are often expressed as percentages by multiplying by 100. This helps in understanding the error's impact on the measurement.
- Purpose: The relative average error is valuable for comparing the accuracy of different measurement sets or different measurement techniques, particularly when the values being measured are significantly different from each other in magnitude.
In summary, calculating the relative average error involves finding the relative error for each measurement and then averaging them. This approach offers a comprehensive look at the overall accuracy of your measurements.
