Measurement, Data, and Uncertainty: Qualitative/Quantitative Data, Accuracy, Precision, Errors, Significant Figures, and Uncertainty Propagation

A vocabulary set covering key concepts of data, accuracy, precision, errors, significant figures, and uncertainty propagation from the notes.

Qualitative data

Non-numerical data consisting of observations such as colour, smell, fizzing, or bubbles.

Quantitative data

Data expressed numerically, obtained from measurements.

Accuracy

How close a measured value is to the true or accepted value.

Precision

How close repeated measurements are to each other; a measure of the reliability and reproducibility of results.

Random error

Unpredictable variation in measurements that can be reduced by repeating experiments; primarily affects precision.

Systematic error

A consistent bias in measurements in one direction; cannot be reduced by repetition and can be eliminated by improved procedures.

Absolute uncertainty

The uncertainty reported with a measurement in the same unit (e.g., 12.34 ± 0.05 g).

Percentage uncertainty

Absolute uncertainty expressed as a percentage of the measured value.

Least count / smallest division

The smallest scale division on a measuring instrument used to estimate uncertainty.

Digital instrument uncertainty

Uncertainty on a digital readout, typically half of the last digit (e.g., ±0.001 for a reading to three decimal places).

Analogue instrument uncertainty

Uncertainty on an analogue instrument, typically half the smallest division.

Propagation of uncertainties

Rules for combining uncertainties when performing calculations.

Addition and subtraction (uncertainties)

When adding or subtracting values, the absolute uncertainties add.

Multiplication and division (uncertainties)

When multiplying or dividing values, the percentage (relative) uncertainties add.

Significant figures

Rules determining which digits are considered reliable in a measurement. Non-zero digits are always significant; zeros have specific rules depending on position and decimal point.

Decimal places (in calculations)

In addition/subtraction, quote results to the same number of decimal places as the measurement with the fewest decimals; in multiplication/division, use the fewest significant figures.

Quoting values with uncertainties

Uncertainty is usually rounded to one significant figure, and the final result is written so that the uncertainty is the last significant figure.

Mean value

The average of a set of measurements.

Uncertainty in the mean

An estimate of the error in the mean value; a rough rule is to take it as about two-thirds of the deviation from the mean.

Fractional (relative) uncertainty

Absolute uncertainty divided by the measured value; used for propagation, especially with exponents.

Error propagation for exponents

If a quantity is raised to a power n, the total fractional uncertainty is multiplied by n; units are also raised to the power n.

Absolute vs. percentage error (percent error)

Percent error = (|experimental − accepted| / accepted) × 100%; always expressed as a positive percentage to assess accuracy.

Quoting experimental results

Experimental result = value ± uncertainty, quoted with uncertainty in the same units and to the appropriate significant figures.

Rounding in calculations

Avoid rounding during multi-step calculations; round only when an answer to a part is required to minimize cumulative error.

Uncertainty from instruments

Uncertainty depends on the instrument type: digital instruments use half the last digit; analogue instruments use half the smallest division.