Every measurement has an uncertainty: the interval within which the true value is expected to lie. A length written as 25.0 ± 0.1 cm is expected to lie between 24.9 cm and 25.1 cm.The absolute uncertainty is the ± value itself (0.1 cm here). It has the same units as the measurement.The fractional uncertainty is the absolute uncertainty divided by the measured value, and the percentage uncertainty is the fractional uncertainty × 100%:percentage uncertainty = (absolute uncertainty ÷ measured value) × 100% Where the absolute uncertainty comes from:
• A single reading on a scale (for example a thermometer): ± half the resolution.
• A measurement made from two readings, such as a length on a ruler where both ends are read: ± one resolution (1 mm for a mm ruler).
• A digital meter: ± the resolution (one in the last digit), unless the manufacturer states more.
• Repeated readings: ± half the range.Example 1: For 25.0 ± 0.1 cm, fractional uncertainty = 0.1 ÷ 25.0 = 0.004, so the percentage uncertainty is 0.4%.Example 2: Three times are 2.31 s, 2.35 s and 2.29 s. The mean is 6.95 ÷ 3 = 2.32 s. The range is 2.35 − 2.29 = 0.06 s, so the absolute uncertainty is ±0.03 s and the percentage uncertainty is 0.03 ÷ 2.32 × 100% = 1.3%. The result is 2.32 ± 0.03 s.Measuring a larger quantity with the same instrument lowers the percentage uncertainty.
• A single reading on a scale (for example a thermometer): ± half the resolution.
• A measurement made from two readings, such as a length on a ruler where both ends are read: ± one resolution (1 mm for a mm ruler).
• A digital meter: ± the resolution (one in the last digit), unless the manufacturer states more.
• Repeated readings: ± half the range.Example 1: For 25.0 ± 0.1 cm, fractional uncertainty = 0.1 ÷ 25.0 = 0.004, so the percentage uncertainty is 0.4%.Example 2: Three times are 2.31 s, 2.35 s and 2.29 s. The mean is 6.95 ÷ 3 = 2.32 s. The range is 2.35 − 2.29 = 0.06 s, so the absolute uncertainty is ±0.03 s and the percentage uncertainty is 0.03 ÷ 2.32 × 100% = 1.3%. The result is 2.32 ± 0.03 s.Measuring a larger quantity with the same instrument lowers the percentage uncertainty.