Represent distributions and estimate uncertainty
Quantitative chemistry • Measurements and conservation of mass
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Types of uncertainty and their causes
Random uncertainty arises from unpredictable variations in repeated measurements; causes include small fluctuations in experimental conditions and reading differences. Random uncertainty produces scatter around a central value and affects precision. Systematic uncertainty arises from consistent bias such as faulty calibration, zero error, or a misreading technique. Systematic uncertainty shifts all results and affects accuracy. Distinguishing the two clarifies what corrective action is required: reduce random uncertainty by repeating measurements and reduce systematic uncertainty by calibration or method correction.
Representing the distribution of results
A numerical summary such as mean, median, mode, and range describes the central tendency and spread of results. The mean gives the average value; the median gives the middle value when results are ordered. The range shows the total spread between highest and lowest values. Graphical methods such as histograms, box plots, and scatter plots display the shape and spread of the distribution and make outliers easy to spot. Visual representation clarifies whether the distribution is clustered, symmetric, skewed, or contains anomalous readings.
Estimating uncertainty from repeated measurements
For a small number of repeats, an uncertainty estimate commonly uses half the range (range/2) as an approximation of random uncertainty; cause: range reflects spread, so halving it approximates the maximum deviation from the mean. For larger data sets, standard deviation provides a better measure because it accounts for each result's deviation from the mean; cause: standard deviation weights all deviations and describes typical scatter. Reporting a result as mean ± uncertainty communicates both the best estimate and the expected variation around it.
Estimating uncertainty from apparatus and resolution
Instruments impose a limiting resolution that produces a minimum reading uncertainty. For digital instruments, the uncertainty often equals half the smallest division (resolution/2); cause: the true value can lie up to half a division above or below the displayed reading. For analogue scales, parallax and scale division cause similar limits. When a single measurement relies mainly on instrument resolution, report the reading ± resolution/2 as the uncertainty.
Combining uncertainties through calculations
For addition or subtraction, combine absolute uncertainties by adding them; cause: absolute uncertainties represent possible shifts in each term, so worst-case shifts add. For multiplication or division, combine percentage (relative) uncertainties by adding their percentages; cause: multiplicative operations scale uncertainties proportionally. For powers, multiply the percentage uncertainty by the power. Correct propagation produces a realistic uncertainty for derived quantities used in comparisons and further calculations.
Interpreting error bars and significance
Error bars show uncertainty on plots, typically as mean ± uncertainty. Overlap of error bars between two means suggests that differences may not be significant; cause: overlapping ranges imply possible shared true values. Non-overlapping error bars suggest a likely real difference, but strict statistical significance requires more formal tests. Error bars also reveal asymmetry or unusually large scatter that indicates poor precision or unrecognized systematic effects.
Key notes
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