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)
- Times: 2.34 s, 2.29 s, 2.31 s, 2.38 s
- mean = 9.32 ÷ 4 = 2.33 s
- range = 2.38 − 2.29 = 0.09 s
- 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.
Errors and the words for data
- Random error scatters readings both ways by different amounts (reaction time, a flickering display). Reduce its effect by repeating and taking a mean.
- Systematic error shifts every reading the same way by the same amount. Repeating does not help: correct the readings or recalibrate.
- A zero error is a systematic error: the instrument reads something when it should read zero. Take the zero reading away from each reading. In a difference of two readings it cancels.
| Word | Means |
|---|---|
| Accurate | Close to the true value |
| Precise | Repeats close together (small spread) |
| Repeatable | Same person, method and equipment gets similar results |
| Reproducible | Another person, method or equipment gets similar results |
Precise is not the same as accurate: a balance with a zero error can give readings that agree perfectly and are all wrong.
Let the calculator do the averaging
Paste these five drop times into the box: 2.31, 2.35, 2.28, 2.33, 2.52. Read the mean and the range. One reading is anomalous: which? Delete it and read the mean and range again. Work out the uncertainty as half the new range. You should get a mean of about 2.32 s and an uncertainty of ± 0.035 s.
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This question is in Brainlag Plus