1.3 Accuracy, Precision, and Significant Figures

Measurement and Uncertainty

Every measurement in physics carries some degree of uncertainty. No instrument is perfect, and no human reading is exact. The concepts of accuracy, precision, and significant figures give you a consistent way to express how confident you are in a measurement and to keep that honesty intact through calculations.

Accuracy vs. Precision in Measurements

These two words sound similar but mean very different things, and mixing them up is one of the most common early mistakes in a physics course.

Accuracy describes how close a measured value is to the true or accepted value. If a standard mass is known to be 100 g and your scale reads 99.8 g, that's high accuracy. Accuracy depends heavily on proper instrument calibration.

Precision describes how close repeated measurements are to each other, regardless of whether they're near the true value. If you measure the same object three times and get 10.1 cm, 10.2 cm, and 10.1 cm, your measurements are precise. Two related ideas fall under precision: repeatability (consistency when the same person repeats the measurement) and reproducibility (consistency when different people perform it).

You need both in physics. Accurate measurements let you verify theories (for example, confirming the predicted value of gravitational acceleration, $g$). Precise measurements ensure your results are reproducible and reliable (for example, different labs obtaining consistent values for the speed of light, $c$).

Significant Figures in Calculations

Significant figures (sig figs) tell you how many digits in a measurement are actually meaningful. They include every digit known with certainty, plus one final estimated digit.

Rules for counting significant figures:

• Non-zero digits are always significant. (245 has 3 sig figs)
• Zeros between non-zero digits are significant. (101 has 3; 1002 has 4)
• Leading zeros are not significant. They just locate the decimal point. (0.0023 has 2 sig figs)
• Trailing zeros after a decimal point are significant. (1.0 has 2; 2.300 has 4)
• Trailing zeros without a decimal point are ambiguous. (1500 could be 2, 3, or 4 sig figs, which is why scientific notation helps: $1.50 \times 10^3$ clearly has 3)

Sig fig rules for calculations:

1. Addition and subtraction — round the result to the fewest decimal places of any measurement in the calculation.

   • Example: $1.2 + 3.45 = 4.65$, rounded to $4.7$ (one decimal place, matching 1.2)

2. Multiplication and division — round the result to the fewest significant figures of any measurement in the calculation.

   • Example: $2.3 \times 1.2345 = 2.83935$, rounded to $2.8$ (two sig figs, matching 2.3)

Scientific notation is useful for expressing very large or very small numbers while making the number of significant figures unambiguous.

Percent Uncertainty in Experiments

Every measuring instrument has a limit to how finely it can measure. This limit introduces uncertainty into your result.

Absolute uncertainty is the smallest reliable increment of your instrument. A ruler with millimeter markings has an absolute uncertainty of about $\pm 1$ mm.

Relative uncertainty compares that absolute uncertainty to the value you measured:

$\text{Relative uncertainty} = \frac{\text{Absolute uncertainty}}{\text{Measured value}}$

For example, if your absolute uncertainty is 1 mm and you measured a length of 50 mm:

$\text{Relative uncertainty} = \frac{1}{50} = 0.02$

Percent uncertainty is just the relative uncertainty expressed as a percentage:

$\text{Percent uncertainty} = \text{Relative uncertainty} \times 100\%$

So 0.02 becomes 2%.

When reporting a final result, you express it with an appropriate uncertainty. A measured value of 10.5 cm with 2% uncertainty becomes $10.5 \pm 0.2$ cm (since 2% of 10.5 is about 0.2).

Notice something practical here: the longer the object you measure with the same ruler, the smaller your percent uncertainty. Measuring a 500 mm table with that same $\pm 1$ mm ruler gives only 0.2% uncertainty. This is why physicists prefer to measure larger quantities when possible.

Error Analysis and Propagation

All physical measurements carry some uncertainty, and when you use those measurements in calculations, the uncertainties combine. Error propagation is the process of tracking how individual measurement uncertainties affect your final calculated result.

For example, if you calculate speed from measured distance and time, the uncertainties in both the distance and the time contribute to the uncertainty in the speed. Understanding this process is how you determine whether your experimental conclusion is actually supported by your data or falls within the range of measurement error.