Measurements & Uncertainties

Every measurement has a limit of precision -- learn to quantify, combine, and interpret uncertainties like a physicist.

Spec Points Covered
  • I can distinguish between systematic and random errors and give examples of each.
  • I can explain the difference between precisionHow close repeated measurements are to each other. A precise set of results has a small spread (low random error). and accuracyHow close a measurement is to the true or accepted value. High accuracy means low systematic error..
  • I can calculate the absolute uncertaintyThe uncertainty expressed in the same units as the measurement. Written as +/- a value after the reading. in a measurement from the range of repeats.
  • I can convert between absolute and percentage uncertainties.
  • I can combine uncertainties for quantities that are added or subtracted.
  • I can combine uncertainties for quantities that are multiplied or divided.
  • I can apply the powerThe rate of energy transfer. Measured in watts (W). rule for uncertainties when a variable is raised to a powerThe rate of energy transfer. Measured in watts (W)..
  • I can determine the uncertainty in the gradient and y-intercept of a graph using worst lines.
  • I can draw error bars on a graph from absolute uncertainties.
  • I can state results to an appropriate number of significant figures based on the uncertainty.
  • I can identify whether a result is accurate by comparing it to a known value.
  • I can suggest improvements to reduce systematic and random errors in an experiment.
Notes
01 Systematic error Systematic error → 02 Precision Precision → 03 Absolute uncertainty Absolute uncertainty → 04 When quantities are added or subtracted $\text{If } y = a \pm b: \quad \Delta y = \Delta a + \Delta b$ → 05 Error bars are drawn on plotted points to show the uncertainty in each measurement $\text{Uncertainty in gradient} = \frac{\text{steepest gradient} - \text{shallowest gradient}}{2}$ → 06 The number of significant figures in your final answer should match the precision of your data → 07 Use percentage difference to compare your result to a known or theoretical value $\text{Percentage difference} = \frac{|\text{experimental value} - \text{accepted value}|}{\text{accepted value}} \times 100\%$ → 08 A student measures the mass m = 0 $E_k = \frac{1}{2}mv^2 = \frac{1}{2} \times 0.250 \times 3.0^2 = 1.125 \text{ J}$ → 09 A student plots a graph of extension x against force F and draws a best-fit line $\text{Uncertainty in gradient} = \frac{0.0138 - 0.0112}{2} = 0.0013 \text{ m N}^{-1}$ → 10 To reduce random error →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
Percentage uncertainty
$$\% \text{ uncertainty} = \frac{\Delta x}{x} \times 100\%$$
  • Where:
    • $\Delta x$ = same units as x
    • $x$ = measured value
Converts an absolute uncertainty to a percentage. Essential for combining uncertainties in products and quotients.
Combining uncertainties (addition/subtraction)
$$\begin{aligned} \text{If } y &= a \pm b: \quad \Delta y \\ &= \Delta a + \Delta b \end{aligned}$$
  • Where:
    • $\Delta y$ = same units as y
    • $\Delta a$ = same units as a
    • $\Delta b$ = same units as b
When two quantities are added or subtracted, add their ABSOLUTE uncertainties. This rule also applies to a change (e.g. temperature change = final - initial: add both uncertainties).
Combining uncertainties (multiplication/division)
$$\begin{aligned} \text{If } y &= ab \text{ or } y \\ &= \frac{a}{b}: \quad \%\Delta y \\ &= \%\Delta a + \%\Delta b \end{aligned}$$
  • Where:
    • $%\Delta y$ = % (dimensionless)
    • $%\Delta a$ = % (dimensionless)
    • $%\Delta b$ = % (dimensionless)
When quantities are multiplied or divided, add their PERCENTAGE uncertainties. Then convert back to absolute if needed.
Power rule for uncertainties
$$\begin{aligned} \text{If } y &= a^n: \quad \%\Delta y \\ &= n \times \%\Delta a \end{aligned}$$
  • Where:
    • $%\Delta y$ = % (dimensionless)
    • $n$ = the exponent (dimensionless)
    • $%\Delta a$ = % (dimensionless)
Multiply the percentage uncertainty by the power. For y = \(a^{2}\), the % uncertainty doubles. For y = \(a^{3}\), it triples. For y = sqrt(a), multiply by 0.5.
Uncertainty in gradient from worst lines
$$\Delta m = \frac{m_{\text{max}} - m_{\text{min}}}{2}$$
  • Where:
    • $\Delta m$ = same units as the gradient
    • $m_{max}$ = gradient of steepest worst line
    • $m_{min}$ = gradient of shallowest worst line
Draw two worst lines through the error bars. The steepest and shallowest gradients define the range. Half the range is the uncertainty.
Percentage difference
$$\% \text{ difference} = \frac{|x_{\text{exp}} - x_{\text{acc}}|}{x_{\text{acc}}} \times 100\%$$
  • Where:
    • $x_{exp}$ = experimental value
    • $x_{acc}$ = accepted/theoretical value
Used to validate results. If percentage difference < percentage uncertainty, the result is consistent with the accepted value.