2.1 Significant figures and measurement uncertainty

Significant Figures and Measurement Uncertainty

Every measurement in chemistry carries some degree of doubt. Significant figures and measurement uncertainty are the tools you use to quantify that doubt, communicate the precision of your data, and ensure that your results are reproducible and honestly reported.

Significant Figures in Measurement

Importance of Significant Figures

Significant figures are the meaningful digits in a measured or calculated quantity. They tell the reader how precise your measurement actually is.

• More significant figures imply higher precision and less uncertainty.
• Fewer significant figures suggest lower precision and greater uncertainty.

Reporting the right number of significant figures matters. Overreporting (e.g., writing 5.2384 g when your balance only reads to 0.01 g) implies a false sense of precision. Underreporting (e.g., writing 5 g from that same balance) throws away information you actually have. Either way, you're misrepresenting your data.

Significant figures also let other researchers assess the quality of your work. If you report a concentration as 0.1032 M, that tells someone you used a method precise to four figures. Reporting it as 0.1 M tells a very different story.

Rules for Determining Significant Figures

1. All non-zero digits (1–9) are always significant.

   • In 123.45, all five digits are significant.

2. Zeros between non-zero digits (captive zeros) are always significant.

   • In 1002.05, all six digits are significant.

3. Leading zeros are never significant. They only mark the decimal point's position.

   • In 0.0123, only the 1, 2, and 3 are significant (three sig figs).

4. Trailing zeros are significant only if a decimal point is shown.

   • In 12.3000, all six digits are significant because the decimal point is present.
   • In 12300 (no decimal point), only the 1, 2, and 3 are significant (three sig figs). If you intend five sig figs, write 12300. or use scientific notation: $1.2300 \times 10^4$.

Determining Significant Figures in Calculations

Addition and Subtraction

The result gets the same number of decimal places as the measurement with the fewest decimal places.

• $12.3 + 1.456 = 13.756 \rightarrow 13.8$ (rounded to one decimal place, matching 12.3)
• $10.1 - 9.78 = 0.32 \rightarrow 0.3$ (rounded to one decimal place, matching 10.1)

If your measurements have different units, convert to the same unit first, then apply the rule:

• $5.2 \text{ cm} + 12.34 \text{ mm} = 5.2 \text{ cm} + 1.234 \text{ cm} = 6.434 \rightarrow 6.4 \text{ cm}$

Multiplication and Division

The result gets the same number of significant figures as the measurement with the fewest significant figures.

• $2.3 \times 1.456 = 3.3488 \rightarrow 3.3$ (two sig figs, matching 2.3)
• $12.34 \div 2.1 = 5.876... \rightarrow 5.9$ (two sig figs, matching 2.1)

Units follow normal algebraic rules:

• $5.2 \text{ cm} \times 3.1 \text{ cm} = 16.12 \rightarrow 16 \text{ cm}^2$ (two sig figs)

Measurement Uncertainty and Its Sources

Concept of Measurement Uncertainty

Measurement uncertainty is the estimated range of values within which the true value is expected to lie. No measurement is perfect. Even with the best equipment and technique, some doubt always remains.

Quantifying that doubt is what lets you compare results across labs, decide whether two values actually agree, and judge whether a result is meaningful.

Sources of Measurement Uncertainty

Uncertainty comes from three main categories:

Instrument limitations contribute to systematic errors (errors that push results consistently in one direction).

• A ruler with 1 mm graduations can't reliably measure anything smaller than about 0.5 mm.
• A balance that hasn't been calibrated recently may read 0.003 g too high on every single weighing.

Environmental factors can shift measurements unpredictably.

• Temperature changes cause thermal expansion, affecting length and volume measurements. A 50 mL volumetric flask calibrated at 20 °C holds a slightly different volume at 25 °C.
• Humidity affects the mass of hygroscopic substances (compounds that absorb water from the air), causing mass readings to drift upward over time.

Operator errors introduce random errors (errors that scatter results in both directions).

• Parallax error occurs when your eye isn't level with the meniscus or scale marking, shifting the reading up or down depending on your angle.
• Inconsistent technique, like varying how firmly you seat a pipette tip, adds scatter to replicate measurements.

Estimating and Reporting Uncertainty

Expressing Uncertainty

Uncertainty can be reported in two ways:

Absolute uncertainty uses the same units as the measurement and states the range directly:

$5.2 \pm 0.1 \text{ cm}$

This means the true value is expected to fall between 5.1 cm and 5.3 cm.

Relative uncertainty (also called percent uncertainty) expresses the uncertainty as a fraction or percentage of the measured value:

$\text{Relative uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100\%$

For the example above: $(0.1 / 5.2) \times 100\% \approx 1.9\%$

Relative uncertainty is especially useful for comparing the precision of measurements of different magnitudes. A ±0.1 cm uncertainty on a 5 cm measurement (about 2%) is much worse, relatively speaking, than ±0.1 cm on a 500 cm measurement (0.02%).

When reporting, the uncertainty value should be rounded to one significant figure (sometimes two, if the leading digit is a 1), and the measured value should be rounded to match the same decimal place as the uncertainty.

Propagation of Uncertainty

When you use uncertain measurements in calculations, the uncertainty carries through. The rules depend on the operation:

For addition and subtraction, add the absolute uncertainties in quadrature:

$\delta C = \sqrt{(\delta A)^2 + (\delta B)^2}$

Example: If $A = 5.2 \pm 0.1$ cm and $B = 3.4 \pm 0.2$ cm, then $C = A + B = 8.6$ cm, and:

$\delta C = \sqrt{(0.1)^2 + (0.2)^2} = \sqrt{0.01 + 0.04} = \sqrt{0.05} \approx 0.2 \text{ cm}$

So $C = 8.6 \pm 0.2$ cm.

For multiplication and division, add the relative uncertainties in quadrature:

$\frac{\delta D}{D} = \sqrt{\left(\frac{\delta A}{A}\right)^2 + \left(\frac{\delta B}{B}\right)^2}$

Example: Using the same $A$ and $B$, $D = A \times B = 17.68$ cm²:

$\frac{\delta D}{D} = \sqrt{\left(\frac{0.1}{5.2}\right)^2 + \left(\frac{0.2}{3.4}\right)^2} = \sqrt{(0.019)^2 + (0.059)^2} \approx 0.062$

That's about 6.2% relative uncertainty, giving $\delta D \approx 1.1$ cm², so $D = 17.7 \pm 1.1$ cm² (or, rounding the uncertainty to one sig fig, $18 \pm 1$ cm²).