Measurements, uncertainty and errors

Means, ranges and uncertainty

Repeat readings let you spot anomalies (results far from the rest) and take a mean.

Leave anomalies out before you average.

mean = sum of the readings ÷ number of readings
uncertainty ≈ ± half the range (range = largest − smallest)

1. Times: 2.34 s, 2.29 s, 2.31 s, 2.38 s
2. mean = 9.32 ÷ 4 = 2.33 s
3. range = 2.38 − 2.29 = 0.09 s
4. uncertainty = ± 0.045 s, so 2.33 ± 0.05 s

Resolution is the smallest change an instrument can show: the gap between marks on a scale, or one in the last digit of a display (0.01 s for 12.46 s).

Significant figures start at the first non-zero digit. Give a calculated answer to the same number of significant figures as the least precise data you used (usually 2 or 3).

0.0450 has 3 significant figures: the zeros at the front never count, the zero at the end after the point does.

Is a difference real? Write each result as a range: 54 ± 4 cm runs from 50 to 58 cm. If the ranges of two results overlap, the difference between them could be caused by random error, so you cannot be confident it is real. If they do not overlap, the difference is bigger than the uncertainty.

Timing many swings or waves and dividing helps because the reaction-time error is about the same each time: over a longer time it is a smaller fraction of what you measure.