3.5 - Processing Results

A calculated result should show no more precision than the measurements support. Keep unrounded calculator values through intermediate steps, then round the final value once so early rounding does not distort the result.

Suppose a calculation gives $9.574\,\mathrm{m\,s^{-2}}$ from measurements quoted to 2 significant figures. Report $9.6\,\mathrm{m\,s^{-2}}$. Writing $9.574$ implies unsupported precision, while rounding intermediate values before substitution can move the final answer unnecessarily.

Choose axes and a scale that make the plotted pattern easy to judge. Put the specified independent or transformed variable on the horizontal axis and the corresponding dependent variable on the vertical axis; label each with its quantity and unit.

A transformed graph needs transformed labels and units: for example, plot $D^2/\mathrm{m^2}$ against $H/\mathrm{m}$, or label a logarithmic axis as $\log(f/\mathrm{Hz})$. The scale must remain linear in the plotted transformed values.

A number without its correct unit does not identify the physical quantity. Convert measurements to a consistent unit system before calculating, then carry the resulting unit through table headings, graph axes, gradients, intercepts and final constants.

If a graph plots energy in joules against inverse wavelength in $\mathrm{m^{-1}}$, its gradient has unit $\mathrm{J\,m}$. Any constant derived from that gradient must then be converted only if the defining equation requires it.

Interpret a graph by describing how the dependent variable changes as the independent variable changes, then use the best-fit shape and intercept to decide what relationship the evidence supports.

Use uncertainty and scatter when judging a prediction. A line that misses the origin slightly may still be consistent with proportionality if the displacement is within experimental uncertainty; a systematic offset may instead explain a meaningful intercept.

Rewrite the model in the form of the graph: $y=mx+c$. Match the plotted quantities to $y$ and $x$, identify what physical expression equals the gradient $m$ or intercept $c$, and only then calculate the required constant.

Choose two well-separated points on the best-fit line, not necessarily measured data points. Draw a triangle covering at least about half the line, read both coordinates accurately, and calculate $m=(y_2-y_1)/(x_2-x_1)$ with the gradient unit obtained from the axes.

A realistic error-reduction modification begins with the dominant limitation revealed by the data. Name the affected measurement, identify whether the problem is resolution, judgement, timing, alignment, drift or scatter, and change the apparatus or geometry that produces it.

State any condition needed for the change to work: a longer screen distance still needs measurable intensity; sensors need correct alignment and calibration; extra readings must lie in the informative region.

An experiment can be improved by using its processed results to decide where stronger evidence is needed. The aim is not only a smaller reading uncertainty, but a design that tests the relationship, locates a feature or produces a more defensible fit.

Tie the improvement to the intended conclusion and keep control variables unchanged. A targeted cluster near a turning point can complement, but should not replace, coverage of the wider relationship.

Quantitative uncertainty gives an interval or percentage around a measured result; qualitative evaluation explains where that uncertainty comes from, how it affects the result, and whether the evidence can distinguish the proposed values or relationships.

Where the mechanism is known, state the direction: using a wavelength that is systematically too small makes $1/\lambda$ too large, shifting points horizontally. Where the sign is unpredictable, describe increased scatter rather than inventing a direction.

Select the absolute uncertainty from how the measurement was obtained, then express it relative to the measured value. For this Unit, a single reading uses half the instrument resolution; repeated readings use half the range.

A length of $8.0\,\mathrm{cm}$ read with resolution $0.1\,\mathrm{cm}$ has absolute uncertainty $0.05\,\mathrm{cm}$ and percentage uncertainty $(0.05/8.0) imes100\%=0.625\%$, reported suitably as about $0.6\%$. Repeats from $12.1$ to $12.3\,\mathrm{s}$ have half-range $0.1\,\mathrm{s}$; divide by their mean to obtain the percentage.