Measuring Physical Quantities

Physical quantities can be measured in two main ways:

• Direct Measurement
  Comparing the quantity directly against a unit of measure, such as using a one-meter ruler or checking speed with a speedometer.
• Indirect Measurement
  Calculating the value of a quantity based on other measurable quantities. For example, determining speed by using the distance traveled and time taken.

In both approaches, measurements are subject to different types of errors.

Measurement Errors in Physics

A measurement error is the difference between the observed value and the actual value of a physical quantity.

Errors can be represented in three main forms:

• Absolute Error
  The error is expressed in the same units as the measured quantity (e.g., 10 cm ± 0.1 cm) $$ x \pm \delta $$
• Relative Error
  The error is expressed as a ratio of the error to the quantity’s value, resulting in a pure, unitless number (e.g., 0.1/10 = 0.01). $$ e_r = \frac{ \delta }{x} $$

  Note: Relative error is often also presented as a percentage $$ e_p = \frac{ \delta }{x} \cdot 100 $$

A Practical Example

A scale measures weight with an error margin of ±1%.

When weighing an object of 80 kg:

$$ \frac{ \delta }{80 \ kg} \cdot 100 $$

The absolute error in this measurement is:

$$ \delta = \frac{ 80 \ kg }{100} = 0.8 \ kg $$

The relative error is:

$$ e_r = \frac{ 0.8 \ kg }{80 \ kg} = 0.01 $$

The Difference Between Precision and Accuracy

Each measurement can be evaluated based on its precision and accuracy.

• Precision
  Precision indicates the uncertainty in the measurement due to the instrument used.

  Example: One stopwatch can measure to the hundredth of a second, while another measures to the thousandth. The latter is more precise.

• Accuracy
  Accuracy reflects the uncertainty due to the measurement method or procedure.

  Example: Two stopwatches are precise to the thousandth of a second, but one is manually operated while the other is triggered automatically by a sensor. Although both are equally precise, the manually operated stopwatch is less accurate because of the human reaction time delay.

Sources of Measurement Errors

Measurement errors can stem from various sources:

• Random Errors
  Every measurement is influenced by random errors caused by changing environmental conditions. For example, wind might affect speed readings. To reduce random errors, repeat the measurement multiple times (x_i) and calculate the average.

  Note: Random error is measured using the standard deviation and decreases as the number (n) of repetitions increases. $$ \delta = \frac{ \sum_i^n (x_i - \bar{x} )^2 }{n(n-1)} $$

• Errors from Instruments and/or Procedures
  These errors arise from the precision of the instrument or the accuracy of the measurement procedure. Repeating the measurement does not reduce these errors. To minimize them, a more precise instrument or a more accurate measurement procedure should be employed.

  Example: The precision and accuracy of a manual stopwatch used to time an Olympic event can be improved by using a more precise stopwatch integrated with an automatic timing system.

• Error Propagation
  This phenomenon occurs in indirect measurements. In indirect measurements, a quantity is estimated using direct measurements of other quantities, which themselves carry errors that propagate to the estimated quantity.

  Note: All else being equal, propagation error increases with the number of quantities used in the calculation. For instance, the sum of two quantities $$ x=x_1 + x_2 $$ results in an absolute error for x that equals the square root of the sum of the squared absolute errors of x₁ and x₂: $$ \delta = \sqrt{\delta_1^2 + \delta_2^2} $$

And so forth.