Understanding Quantities with Uncertainties in Measurements

Accuracy and precision are paramount in scientific and engineering fields. However, it is crucial to understand that all measured quantities inherently have some form of uncertainty. This article aims to clarify the concept of quantities with uncertainties and how to handle them effectively in different scenarios.

The Nature of Measurement Uncertainty

When we measure anything, we cannot hope for perfect accuracy. Even the most advanced instruments and sensors have inherent limitations that introduce errors. For example, if a digital speedometer reads 63 km/hr, it is more accurate to state that the speed is between 62 and 64 km/hr. However, the true speed could be anywhere within a broader range, typically plus or minus a certain value. This value represents the uncertainty in the measurement.

Example: Speedometer Reading

Suppose a speedometer reads 63 km/hr, and the specification states a possible range of 58 km/hr to 65 km/hr. This range gives us a clear indication that the true speed is within these bounds, but we can never know the exact value. Understanding this uncertainty is crucial for making accurate assessments in various applications.

Special Case: What are Quantities with Uncertainties?

The phrase "6.8_0.3-4.1-0.5" represents a specific case of quantities with uncertainties. This notation indicates a range where the value 6.8 could be as low as 6.5 and as high as 7.3. Similarly, the value -4.1 could be as low as -4.6 and as high as 3.6. This notation helps us understand the range of possible values for each quantity and the uncertainty involved.

Handling Subtraction of Quantities with Uncertainties

When dealing with the subtraction of quantities with uncertainties, the largest possible value for the result occurs when the first term is at its maximum and the second term is at its minimum. Conversely, the smallest possible value for the result occurs when the first term is at its minimum and the second term is at its maximum. Applying this principle to the example given:

Calculating the Range

For the first term: 7.3 - 3.6 3.7 (maximum possible value)

For the second term: 6.5 - 4.6 1.9 (minimum possible value)

Therefore, the result lies between 1.9 and 3.7, which can be written as 1.9 x 3.7 or 2.8 ± 0.9.

Asymmetric Errors and Engineering Tolerance

In some cases, errors are not symmetric, meaning the positive and negative uncertainties are different. For instance, a quantity might be specified as 7.0 ± 0.1 - 0.5. This means that on average, the true value is expected to be 7.0, but it could be as high as 7.1 or as low as 6.5. Such asymmetric uncertainties are common in engineering applications where precision is paramount.

The Statistical Aspect of Uncertainty

It is important to distinguish between statistical uncertainty and engineering tolerances. Statistical uncertainty is based on the standard deviation of the measurement. For example, if a measurement is 68.0, we expect the true value to fall within 1 standard deviation (68.0 ± 1) about 68% of the time. This is different from engineering tolerances, which are defined ranges within which a product or component must function reliably.

Mathematical Representation

For the given example, we have:

a 6.8, σa 0.3

b -4.1, σb 0.5

Using the formula for combining uncertainties:

σab √(σa2 σb2)

Substituting the values:

σab √(0.32 (-0.5)2) √(0.09 0.25) √0.34 0.583

Thus, the result is 2.7 ± 0.6.

Reporting Uncertainty

It is essential to report the uncertainty with the same number of decimal places as the measured value. This ensures that the reported value reflects the precision of the measurement and provides clear information about the level of confidence in the result.

Conclusion

Understanding quantities with uncertainties is vital for accurate and reliable measurements in science and engineering. By comprehending the nature of measurement errors and properly reporting uncertainties, we can make informed decisions based on the data we collect. Whether it is handling symmetric or asymmetric uncertainties, or dealing with statistical or engineering tolerances, the principles outlined in this article provide a solid foundation for tackling these challenges.