Range about the mean: measure of uncertainty
Quantitative chemistry • Measurements and conservation of mass
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Key concepts
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Definition of range about the mean
The range about the mean measures how far individual measurements lie from the average value. The mean is calculated first. The largest positive difference between a measurement and the mean and the largest negative difference are found. The larger of the two differences is taken as the uncertainty about the mean. The range about the mean therefore expresses an uncertainty as ± a single value around the mean. This form makes the result readable and comparable when reporting repeated measurements.
Calculation steps
Step 1: Calculate the arithmetic mean of the repeated measurements. Step 2: Subtract the mean from each measurement to obtain deviations. Step 3: Identify the largest positive deviation and the largest negative deviation in magnitude. Step 4: Select the larger magnitude and write the result as mean ± uncertainty. Example cause → effect: Larger spread in the measurements causes a larger value for the range about the mean, which increases the reported uncertainty and signals lower precision.
When the method is appropriate
Use the range about the mean for small sets of repeat measurements where a quick, easily understood uncertainty estimate is required. The method suits classroom practicals and simple experiments with limited repeats. The method is less suitable when many measurements exist or when measurement errors are known to follow a distribution; in those cases, statistical measures such as standard deviation provide a more robust estimate.
Limitations and cautions
Sensitivity to outliers causes the range about the mean to overestimate uncertainty when a single erroneous measurement is present. Small sample sizes give imprecise estimates because the range depends heavily on the particular values obtained. Instrument precision and systematic errors do not appear in the range about the mean. The method reports only random scatter and therefore underestimates total uncertainty when systematic effects are present.
Comparison with other measures
Standard deviation gives an average spread of all measurements and becomes more reliable with larger data sets. The range about the mean uses only the extreme deviations and therefore responds strongly to single extreme values. Cause → effect: When many measurements exist, the standard deviation typically provides a more representative uncertainty; when only a few measurements exist, the range about the mean gives a simple upper bound on random scatter.
Key notes
Important points to keep in mind