Mathematical Treatment of Measurement Results Summary

Summary of lessons so far

Measurement Uncertainty, Accuracy, and Precision

Quantities can be exact or measured. Measured quantities have an associated uncertainty that is represented by the number of significant figures in the measurement. The uncertainty of a calculated value depends on the uncertainties in the values used in the calculation and is reflected in how the value is rounded. Measured values can be accurate (close to the true value) and/or precise (showing little variation when measured repeatedly).

Mathematical Treatment of Measurement Results

Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities using measured quantities and appropriate mathematical relations.

Key Equations

• \(T_{^{\circ}\text{C}} = \frac{5}{9}(T_{^{\circ}\text{F}} - 32)\)
• \(T_{^{\circ}\text{F}} = \frac{9}{5} × T_{^{\circ}\text{C}} + 32\)
• \(T_{\text{K}} = T_{^{\circ}\text{C}} + 273.15\)
• \(T_{^{\circ}\text{C}} = T_{\text{K}} - 273.15\)

Glossary of Words

Accuracy

how closely a measurement aligns with a correct value

Dimensional analysis

(also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities

Exact number

number derived by counting or by definition

Fahrenheit

unit of temperature; water freezes at 32 °F and boils at 212 °F on this scale

Precision

how closely a measurement matches the same measurement when repeated

Rounding

procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation

Significant figures

(also, significant digits) all of the measured digits in a determination, including the uncertain last digit

Uncertainty

estimate of amount by which measurement differs from true value

Unit conversion factor

ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit