In the world of measurements, precision is key. But even with the best instruments, some degree of uncertainty is unavoidable. The relative uncertainty formula is a tool that helps us quantify and understand the impact of this uncertainty on our measurements. It essentially tells us how significant the uncertainty is compared to the measured value itself.
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The relative uncertainty (also known as relative error) is calculated by dividing the absolute uncertainty by the measured value. The formula is:
Relative Uncertainty = (Absolute Uncertainty / Measured Value)
If you are comparing your measurement to a known or standard value, the formula becomes:
Relative Uncertainty = (Absolute Uncertainty / Known Value)
Where:
- Absolute Uncertainty: The margin of error associated with the measurement (e.g., ± 0.5 cm). It has the same units as the measurement.
- Measured Value: The value you obtained from your measurement.
The relative uncertainty is often expressed as a percentage by multiplying the result by 100. This unitless measure allows for easy comparison of the precision of different measurements, regardless of their scale.
Why is Relative Uncertainty Important?
Relative uncertainty puts the error into perspective. For instance, an error of ± 1 cm is much more significant when measuring something that is 10 cm long, compared to measuring something that is 10 meters long. The relative uncertainty helps you understand the error within a relative context.
Examples of the Relative Uncertainty Formula
Example 1: Measuring Gram Weights
Imagine you're measuring three 1.0 gram weights, and you obtain the following measurements: 1.05 grams, 1.00 grams, and 0.95 grams.
- The absolute uncertainty is ± 0.05 grams.
- The relative uncertainty is (0.05 g / 1.00 g) = 0.05, or 5%.
Example 2: Chemical Reaction Time
A chemist measures the time for a reaction to occur and finds it to be 155 ± 0.21 hours.
- Absolute uncertainty = 0.21 hours
- Relative uncertainty = 0.21 hours / 155 hours = 0.00135, or 0.14% (rounded to two significant figures).
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