Uncertainty and Significant Figures

What is Uncertainty?

Uncertainty represents the range within which a measured or calculated value is expected to lie. All measurements have some degree of uncertainty due to:

Limitations of measuring instruments.
Environmental factors.
Human errors.

For example, if the length of a table is measured as $1.23 \pm 0.01 \, \text{m}$ , the true length lies between $1.22 \, \text{m}$ and $1.24 \, \text{m}$ .

Types of Uncertainty

Uncertainty can be expressed in two main ways:

Absolute Uncertainty: The actual range of possible error, e.g., $\pm 0.01 \, \text{m}$ .
Relative Uncertainty: The ratio of absolute uncertainty to the measured value, expressed as a percentage:
$\text{Relative Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100$

Significant Figures

Significant figures (or digits) indicate the precision of a measurement. They include all known digits plus one estimated digit. For example:

$123.45$ has five significant figures.
$0.00320$ has three significant figures (the leading zeros are not significant).
$4.00$ has three significant figures (the trailing zeros after the decimal point are significant).

Rules for Significant Figures in Calculations

For Addition and Subtraction:

The result should have the same number of decimal places as the quantity with the least decimal places.

Example: $12.34 + 0.1 = 12.4$ (rounded to one decimal place).

For Multiplication and Division:

The result should have the same number of significant figures as the quantity with the least significant figures.

Example: $2.34 \times 1.2 = 2.8$ (rounded to two significant figures).

Combining Uncertainties

When combining measurements with uncertainties, the total uncertainty depends on the type of calculation:

Addition/Subtraction: Add absolute uncertainties:
$\Delta x_{\text{total}} = \Delta x_1 + \Delta x_2$
Multiplication/Division: Add relative uncertainties:
$\frac{\Delta x_{\text{total}}}{x_{\text{total}}} = \frac{\Delta x_1}{x_1} + \frac{\Delta x_2}{x_2}$

Practical Tips

Use appropriate significant figures to reflect the precision of measurements and calculations.
Avoid over-representing precision by including more significant figures than warranted.
Express uncertainties clearly to ensure results are meaningful and transparent.

Key Takeaways

Uncertainty and significant figures are essential for representing the precision and accuracy of measurements in physics. Understanding and applying these concepts ensures clarity and consistency in scientific work.