6.5 - Analysis

Report a calculated result only to the precision justified by the measured inputs. Keep extra calculator digits during the working, then round once at the end so intermediate rounding does not distort the final value.

For $v=1.25\,\mathrm{m}/0.42\,\mathrm{s}$, the calculator gives $2.976\ldots\,\mathrm{m\,s^{-1}}$. The time has two significant figures, so report $v=3.0\,\mathrm{m\,s^{-1}}$, not every displayed digit.

Leading zeros are placeholders, while trailing zeros after a decimal can show precision: $0.00450$ has three significant figures. Scientific notation makes the intended precision unambiguous.

Choose axes that test the intended relationship, label each quantity and unit, and use a simple scale that spreads the data across most of the available grid. Plot points accurately before drawing a best-fit line or curve.

For a power law $y=kx^n$, form dimensionless processed variables $Y=\log(y/y_0)$ and $X=\log(x/x_0)$ using stated reference units. Then $Y=nX+C$, where $C$ depends on $k$ and the chosen reference units. Plotting $Y$ against $X$ on linear graph paper can reveal a straight line.

A log-log graph labelled $\log(T/\mathrm{s})$ vertically and $\log(M/\mathrm{kg})$ horizontally has dimensionless plotted values. Use consistent decimal places for the processed logarithms.

Carry units through substitutions, transformations, graph labels, gradients and final constants. Unit algebra is an independent check that the calculation and chosen relationship are physically consistent.

Convert prefixes before combining quantities. If $d=2.4\,\mathrm{mm}=2.4\times10^{-3}\,\mathrm{m}$, then $d^2=5.76\times10^{-6}\,\mathrm{m^2}$; both the number and the unit conversion are squared.

When a gradient represents a physical constant, derive its unit from the plotted axes and compare it with the unit expected from the model. A mismatch often exposes a reversed axis, missed power or unconverted prefix.

A trend comment states what the data show: direction, form, important region and scatter. Separate this observation from a physical explanation or causal claim unless the evidence and controlled design support that conclusion.

Transform the proposed relationship when useful. To test $T\propto\sqrt D$, check whether $T/\sqrt D$ is approximately constant or whether $T^2$ plotted against $D$ is linear through the origin.

Use data to support the comment: quote a range, ratio or change and acknowledge scatter. 'Increases' is weaker than 'approximately linear with a small positive intercept'.

Rearrange the proposed model into $Y=mX+c$, choose plotted quantities $Y$ and $X$, then interpret the gradient $m$ and intercept $c$ in terms of the required relationship or constant.

Choose two well-separated points on the best-fit line, not necessarily measured data points. Draw a large triangle spanning at least about half the plotted line, read coordinates carefully and keep the subtraction signs. A large triangle reduces the percentage effect of coordinate-reading uncertainty.

Give the gradient an appropriate sign, significant figures and unit derived from vertical-axis unit divided by horizontal-axis unit. A log-log gradient is dimensionless.

Two sets can have the same mean but different ranges: the set with the smaller range is more precise. Without an accepted value, their accuracy cannot be decided from agreement alone.

Greater sensitivity can make small input changes easier to distinguish, but sensitivity is not the same as resolution, precision or accuracy. An instrument can respond strongly yet retain a calibration offset.

Use uncertainty when judging accuracy: if $g=10.0\pm0.6\,\mathrm{m\,s^{-2}}$, the accepted $9.81\,\mathrm{m\,s^{-2}}$ lies in the interval, so the result is consistent with it.

A realistic error-reduction proposal names the error source, changes one practical feature and explains which uncertainty or bias becomes smaller. Match the modification to random or systematic behaviour.

Prefer a change whose effect can be predicted. Timing ten cycles makes the measured interval about ten times larger while similar start-stop uncertainty remains, so percentage uncertainty in the period decreases.

The proposal must suit the existing experiment, available range and safety constraints. Explain any new limitation introduced by additional apparatus, such as sensor alignment or sampling rate.

An experiment is improved when its method can support the intended conclusion more securely. Diagnose the weakest link—validity, data coverage, control, measurement quality or reproducibility—then propose a feasible change and explain the evidential gain.

State how the modification changes the decision made from the data. If it adds time, complexity or a new uncertainty, weigh that cost against the expected improvement.

Use details from the actual setup and results. A good proposal names the variable, apparatus setting or data region to change rather than giving a generic list of laboratory virtues.

Measurement uncertainty expresses a justified interval around a reported value. Write $x\pm\Delta x$, where $\Delta x$ is the absolute uncertainty in the same unit as $x$; it reflects resolution, scatter and method limitations rather than a known error from the true value.

Discuss the dominant source and direction when relevant: limited resolution creates a reading interval, repeats reveal random spread, and an uncorrected zero error could shift all values. State whether a proposed change reduces the absolute or percentage contribution.

Compare an accepted value or another result with the uncertainty interval. If $E=14.3\pm0.7\,\mathrm{GPa}$, values from $13.6$ to $15.0\,\mathrm{GPa}$ are consistent with the measurement; a candidate value outside that interval is not.

Propagate uncertainties by following how measured quantities are combined. The standard school-level rules give a maximum estimated uncertainty for independent contributions.

For $R=V/I$, if $V$ has $2.0\%$ uncertainty and $I$ has $3.0\%$, then $R$ has approximately $5.0\%$ uncertainty. If $A=\pi d^2/4$ and $d$ has $1.5\%$, then $A$ has $3.0\%$ uncertainty.

After finding the combined percentage uncertainty, convert to an absolute uncertainty when an interval is needed: $\Delta Q=(\%U_Q/100)Q$. Round the uncertainty sensibly and match the value's decimal place.

For one reading, use half the instrument resolution as the absolute uncertainty. If a scale's smallest division is $1\,\mathrm{mm}$, a single length reading has $\Delta x=0.5\,\mathrm{mm}$ under this syllabus convention.

For repeated readings, calculate the mean and use half the range as the absolute uncertainty. The range is maximum minus minimum.

Repeated times for five cycles are $4.84$, $4.92$, $4.88$ and $4.96\,\mathrm{s}$. Their mean is $4.90\,\mathrm{s}$ and half range is $(4.96-4.84)/2=0.06\,\mathrm{s}$. For one period, divide both by five: $T=0.980\pm0.012\,\mathrm{s}$, with the same percentage uncertainty, about $1.2\%$.