1.5 Measurement Uncertainty, Accuracy, and Precision

Measurement and Uncertainty

Every measurement in chemistry carries some degree of uncertainty. Understanding how to quantify and communicate that uncertainty is what separates meaningful data from guesswork. This section covers accuracy, precision, significant figures, and error analysis.

Accuracy vs Precision in Measurements

These two terms sound similar but describe very different things.

Accuracy measures how close a result is to the true or accepted value. Precision describes how close multiple measurements are to each other, regardless of whether they're correct.

The classic analogy is a dartboard:

• Accurate and precise: Darts tightly grouped right on the bullseye
• Accurate but imprecise: Darts scattered around the board, but their average lands on the bullseye
• Precise but inaccurate: Darts tightly grouped, but clustered away from the bullseye
• Neither accurate nor precise: Darts scattered all over, nowhere near the bullseye

This distinction matters because a precise result can still be wrong. If your scale is miscalibrated, you might get the same (incorrect) mass reading three times in a row. That's precise but not accurate.

Exact vs Uncertain Chemical Data

Not all numbers in chemistry carry uncertainty. Knowing the difference tells you when significant figure rules apply.

Exact numbers have no uncertainty:

• Counted values (12 eggs, 5 molecules)
• Defined values (1 kg = 1000 g, 1 inch = 2.54 cm)
• Subscripts in chemical formulas (the "2" and "4" in $H_2SO_4$)

Uncertain numbers come from measurements and always have some limit to their reliability:

• Measured values (5.3 g, 10.2 mL)
• Calculated values derived from measurements (like density = mass ÷ volume)

Significant figure rules only apply to uncertain numbers. You'd never round a counted or defined value.

Significant Figures for Uncertainty

Significant figures (sig figs) tell you how reliable a measurement is. They include all the digits you're certain about, plus one estimated digit.

Rules for counting significant figures:

• Non-zero digits are always significant (245 has three)
• Zeros between non-zero digits are significant (1.0023 has five)
• Leading zeros are never significant (0.0012 has two sig figs)
• Trailing zeros after a decimal point are significant (1.200 has four sig figs)
• Trailing zeros without a decimal point are ambiguous (1200 could have two, three, or four sig figs)

Rules for calculations:

Multiplication and division: Your answer gets the same number of sig figs as the measurement with the fewest sig figs.

Addition and subtraction: Your answer gets the same number of decimal places as the measurement with the fewest decimal places.

Notice the difference: multiplication/division cares about total sig figs, while addition/subtraction cares about decimal places.

Rounding Rules in Calculations

Standard rounding is straightforward:

1. If the digit to the right of your last sig fig is less than 5, round down (12.44 → 12.4)
2. If the digit to the right is greater than 5, round up (12.46 → 12.5)
3. If the digit to the right is exactly 5 (with nothing or only zeros after it), round to the nearest even number:
   • 12.350 → 12.4 (rounds up because 3 is odd)
   • 12.450 → 12.4 (rounds down because 4 is already even)

That third rule is called the "round half to even" rule (sometimes called "banker's rounding"). It prevents a systematic bias from always rounding 5s upward. Your instructor may or may not require this rule, so check your course expectations.

Sources of Error and Statistical Analysis

Errors in measurement fall into two categories:

Systematic error is a consistent deviation in the same direction every time. An uncalibrated balance that always reads 0.5 g too high is a systematic error. These affect accuracy and can be reduced through calibration, which is the process of adjusting instruments against a known standard.

Random error is unpredictable variation between measurements. Slight differences in how you read a meniscus or small temperature fluctuations are random errors. These affect precision and can be reduced by taking more measurements and averaging.

A few statistical tools help you quantify these errors:

• Standard deviation measures how spread out your data points are around the mean. A small standard deviation means high precision.
• Confidence interval gives a range of values that likely contains the true value, calculated from your mean and standard deviation.
• Error propagation describes how uncertainties in individual measurements combine to affect the uncertainty of a final calculated result. For example, if you calculate density from uncertain mass and volume measurements, both uncertainties contribute to uncertainty in the density.