4 Measurement and Engineering Data

Measurement and SI units

A measurement is a numerical estimate of a quantity, expressed as a number and a unit. The International System of Units (SI) provides the standard measurement system used in science and engineering.

The SI has seven base units: the metre ($\text{m}$) for length, kilogram ($\text{kg}$) for mass, second ($\text{s}$) for time, ampere ($\text{A}$) for electric current, kelvin ($\text{K}$) for temperature, mole ($\text{mol}$) for amount of substance, and candela ($\text{cd}$) for luminous intensity. Other units are derived from these. For example, speed is measured in metres per second, and force in newtons:

Use units consistently in calculations. A conversion factor is a ratio equal to one, so it changes the unit without changing the quantity. For example:

Write a space between a number and its unit symbol, as in $25.4\ \text{mm}$. Unit symbols are not pluralized and are case-sensitive: $\text{m}$ means metre, whereas $\text{M}$ is the SI prefix mega.

Significant figures and rounding

Significant figures indicate the digits used to express a measured value. They do not, by themselves, give a complete account of its uncertainty. For example, $1.200\ \text{km}$ communicates four significant digits, while $1.2\ \text{km}$ communicates two. Scientific notation can clarify the intended digits: $1.20\times10^3\ \text{m}$ has three significant digits.

Use these practical calculation conventions:

• For multiplication and division, round the result to the number of significant figures in the least precise measured input.

• For addition and subtraction, round to the least precise decimal place among the inputs.

• Keep extra digits during intermediate calculations, then round the final result.

• Counts and defined conversion factors are exact and do not limit significant figures. For instance, a count of $12$ bolts is exact, whereas a ruler reading is a measurement.

These are practical rounding conventions. When uncertainty is known, report it directly rather than relying on significant figures alone.

Uncertainty and repeated measurements

No measurement is perfectly exact. Measurement uncertainty describes the range of values reasonably associated with a measured result. It may arise from instrument resolution, calibration, environmental conditions, the measurement procedure, or variation between repeated readings.

Random effects cause readings to vary. Systematic effects, such as a consistent calibration offset, tend to shift readings in one direction. Repeating a measurement can help estimate random variation, but it does not automatically reveal or remove systematic error.

Precision concerns the spread of repeated readings, while accuracy concerns closeness to an accepted reference value. A set of readings can be tightly grouped yet consistently biased.

For repeated measurements, report the mean and describe the observed spread or estimated uncertainty. For example, if repeated length readings average $12.46\ \text{cm}$ and the estimated standard uncertainty is $0.03\ \text{cm}$, report:

Round the uncertainty and measured value to compatible decimal places. State what the uncertainty means when needed: a standard uncertainty, an expanded uncertainty, and an interval with a stated confidence level are not interchangeable. For expanded uncertainty, report the coverage factor and, where appropriate, the probability interpretation.

Uncertainty can carry through calculations that combine uncertain measurements. For independent measurements added together, their standard uncertainties are commonly combined by taking the square root of the sum of their squares:

More complex relationships require appropriate propagation methods. Do not report a calculated result with more precision than its uncertainty supports.

Communicating engineering results

A useful engineering result lets another person understand what was measured, how it was measured, and how dependable the value is. Include the quantity, numerical result, and unit; the uncertainty or relevant measurement spread; the instrument and method, including calibration information when relevant; important conditions and assumptions; and the number of repeated trials. Include units and labels on every relevant table column, graph axis, and reported value.

For example, “The plate thickness was $2.46\ \text{mm}$” gives a value and unit, but not its uncertainty or method. A more informative report is: “Plate thickness was $(2.46\pm0.03)\ \text{mm}$, measured at five locations with a calibrated micrometer; $\pm0.03\ \text{mm}$ is the estimated standard uncertainty.” State the uncertainty type accurately, and do not imply a confidence level unless it is justified.

Use consistent units throughout calculations and reports. If another unit system is useful to the audience, give the SI value first and put the converted value in parentheses.

• Systematic effect