91Ó°ÊÓ

Relative uncertainty is a key concept when calculating how precise or uncertain a measurement is in comparison to the actual measurement itself. It is expressed as a fraction or percentage, representing the uncertainty of a measurement divided by the measurement's actual value. This measure helps to understand the precision of an instrument or procedure.
To compute relative uncertainty, use the formula: \[ \text{Relative Uncertainty} = \frac{\Delta x}{x} \] where \( x \) is the measured value and \( \Delta x \) is the absolute uncertainty.

• For example, if you measure something to be 36.2 ± 0.4, the relative uncertainty is \( \frac{0.4}{36.2} \approx 0.011 \).
• It indicates that the uncertainty is approximately 1.1% of the measurement.

This concept is practical because multiplying relative uncertainties simplifies the process of dealing with products or quotients of several measurements. It's particularly useful in complex calculations like those shown in problems involving multiple operations.

Absolute uncertainty quantifies the margin of error in a measurement. It is different from relative uncertainty, as it is expressed in the same units as the measurement itself, providing a specific range within which the true value is expected to lie.
Absolute uncertainty is often denoted alongside measurements as \( x \pm \Delta x \), where \( x \) is the measurement and \( \Delta x \) represents the uncertainty.
When you have the relative uncertainty, converting it back to absolute uncertainty helps in directly associating the error with the original measurement or result.

• For instance, if you calculate a value to be 0.0141 with a relative uncertainty of 0.0263, then the absolute uncertainty is \( 0.0141 \times 0.0263 = 0.00037 \).
• This tells you that the true quantity may be as low as \( 0.0141 - 0.00037 \) or as high as \( 0.0141 + 0.00037 \).

Recognizing the absolute uncertainty is crucial when presenting final results to indicate their precision.

When dealing with multiplication or division of values that have uncertainties, one does not simply multiply or divide the absolute uncertainties. Instead, the relative uncertainties are used, as the process involves percentage-based calculations.
For multiplication or division of variables \( x \pm \Delta x \) and \( y \pm \Delta y \), the relative uncertainty of the result \( z = xy \) or \( z = \frac{x}{y} \) can be calculated using:
\[ \frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y} \]

• This formula shows how the errors spread through multiplications and divisions.
• Once the relative uncertainty is known, it can be converted to absolute uncertainty by multiplying by the resultant value. For example, combining two measurements, \( 50.23 \pm 0.07 \) and \( 27.86 \pm 0.05 \), through multiplication and division, yields a relative uncertainty of 0.00576.
• The final step is to multiply this relative uncertainty with the result obtained from multiplying or dividing the quantities, such as \( \frac{(50.23 \times 27.86)}{0.1167} \approx 11987.9 \).
• Thus, finding \( 11987.9 \times 0.00576 = 69.09 \) as the absolute uncertainty, indicating the range of possible error in the calculated result.

Understanding these procedures helps ensure the integrity and reliability of scientific measurements and computations.