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Measurement & Uncertainty Lab

Every measurement carries uncertainty. Learn to count significant figures, convert between units, propagate errors through calculations, and perform dimensional analysis with step-by-step conversion chains.

Guided Experiment: Significant Figures and Measurement Precision

How do trailing zeros, leading zeros, and decimal points affect the number of significant figures? Does the way you write a number change its implied precision?

Write your hypothesis in the Lab Report panel, then click Next.

Controls

Type the number exactly as written, including trailing zeros and decimal points

Results

Significant Figures
3
Digit Analysis
0
.
0
0
3
4
0
SignificantNot significant
Rules Applied
  • Leading zeros are never significant
  • Trailing zeros after the decimal point are significant

Data Table

(0 rows)
#TrialTabInputOutputSig FigsUncertaintyNotes
0 / 500
0 / 500
0 / 500

Reference Guide

Significant Figures

Significant figures communicate the precision of a measurement. Leading zeros are never significant, while trailing zeros after a decimal point always are.

0.003403 significant figures0.00\mathbf{340} \rightarrow 3 \text{ significant figures}

A trailing decimal point (like 100.) makes all digits significant. Captive zeros between non-zero digits are always significant.

Unit Conversion

Multiply by conversion factors that equal 1 to change units without changing the value. The number of significant figures in the result matches the input.

5.2 km×0.6214 mi1 km=3.2 mi5.2 \text{ km} \times \frac{0.6214 \text{ mi}}{1 \text{ km}} = 3.2 \text{ mi}

Temperature conversions are special because they use offset formulas, not simple ratios.

Error Propagation

Uncertainties combine in quadrature. For addition and subtraction, add absolute uncertainties. For multiplication and division, add relative uncertainties.

δR=(δA)2+(δB)2\delta R = \sqrt{(\delta A)^2 + (\delta B)^2}

For powers, the relative uncertainty is multiplied by the absolute value of the exponent.

Dimensional Analysis

Chain conversion factors so that unwanted units cancel, leaving only the desired unit. Each step multiplies by a fraction equal to 1.

65 mi×1.609 km1 mi×1000 m1 km=104,607 m65 \text{ mi} \times \frac{1.609 \text{ km}}{1 \text{ mi}} \times \frac{1000 \text{ m}}{1 \text{ km}} = 104{,}607 \text{ m}

If the units do not cancel correctly, the conversion chain has an error.