Turn measurements into evidence

Data should be recorded in a table as they are collected. Put the quantity and unit in each heading, keep repeat readings separate, and do not write only the final mean. The following illustrative results represent timing ten oscillations at each length.

The raw readings have been translated into mean times and then periods. This numerical processing is a form of mathematical and statistical analysis.

Worked example. At 0.60 m, the mean time for ten oscillations is

$(15.4 \mathrm{s} + 15.8 \mathrm{s} + 15.6 \mathrm{s}) / 3 = 15.6 \mathrm{s}$.

The mean period is therefore

$15.6 \mathrm{s} / 10 = 1.56 \mathrm{s}$.

The range of the ten-oscillation times is

$15.8 \mathrm{s} - 15.4 \mathrm{s} = 0.4 \mathrm{s}$.

Half the range is 0.2 s for ten oscillations, so the corresponding estimate for one period is $0.2 \mathrm{s} / 10 = 0.02 \mathrm{s}$. The result can be written as 1.56 s +/- 0.02 s. This means there is a margin of doubt around the measured value; it does not mean somebody made a mistake of exactly 0.02 s.

Different representations answer different questions:

• the raw table preserves individual measurements;
• a summary table makes means, ranges and uncertainties easy to compare;
• a dot plot or distribution graph can show where many repeated results cluster, how widely they spread and whether there are possible anomalies;
• a graph of mean period against length displays the relationship between the independent and dependent variables.

For that relationship graph, length belongs on the horizontal axis and mean period on the vertical axis. Both axes need quantity and unit, the scale should use the plotting area sensibly, and the points may carry vertical uncertainty bars. A smooth best-fit trend should represent the pattern; joining every point dot to dot would imply unsupported behaviour between measurements.

The table shows that mean period increases as length increases. That is an observation of a trend. The explanation that length affects the motion is an inference informed by the model and the controlled method. A reasoned conclusion is that these results support the hypothesis over the tested range; they do not prove it for every pendulum or every possible length.

Evaluate data and methods

Strong evaluation names the exact feature of the evidence. The words in this table are related, but they are not interchangeable.

Random effects change unpredictably between readings and create scatter. A student may start or stop a stopwatch slightly early on one run and slightly late on another. More repeats, a mean and timing a longer interval can reduce the effect of this variation on the final estimate.

A systematic error shifts readings in the same direction. Measuring pendulum length to the bottom of the bob instead of its centre would bias every length. Repeating that same mistake produces a precise-looking set of values but does not remove the bias. A zero check, calibration, comparison with a reference, corrected technique or genuinely different method is needed.

An anomalous result lies unexpectedly far from the pattern or other repeats. Do not remove it merely because it is inconvenient. Check the record and apparatus, repeat that condition, and exclude a value only with a scientific reason that is reported honestly.

Useful method improvements contain three linked elements: the change, how it is implemented, and why it improves the evidence.

A further investigation should follow from the evidence. It might use more closely spaced lengths where the trend changes most, extend the range safely, or ask a second group to repeat the method to test reproducibility. It should not quietly change several variables and still claim to test the original question.

Use quantities, units and symbols

Scientific vocabulary is useful only when each term has a stable meaning. In this investigation, length and period are quantities, 0.60 and 1.56 are numerical values, and metre (m) and second (s) are units. A measurement needs both a number and a unit: writing only 1.56 does not communicate a period.

A scientific quantity is defined by what is measured and how it is determined. Period means the time for one complete oscillation; here it is determined by timing ten oscillations and dividing by ten. This operational definition allows another person to reproduce the measurement.

The International System of Units, SI, gives scientists shared standards. Common Physics examples include metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for current, newton (N) for force and joule (J) for energy. Related forms such as gram (g) and prefixed units such as km, mm, mg and kJ remain useful, but the prefix changes the scale of the unit. Accepted IUPAC nomenclature is used when chemical substances need to be named; this Physics review focuses on quantities, units and symbols.

Prefix symbols are case-sensitive: M means mega while m as a prefix means milli. The powers of ten make orders of magnitude explicit. Keep the conversion factor visible and preserve the physical quantity:

$75 \mathrm{cm} = 75 \times 10^{-2} \mathrm{m} = 0.75 \mathrm{m}$

$4.2 \mathrm{kJ} = 4.2 \times 10^{3} \mathrm{J} = 4200 \mathrm{J}$

$380 \mu s = 380 \times 10^{-6} \mathrm{s} = 3.80 \times 10^{-4} \mathrm{s}$

The direction is a useful check. Converting to a smaller unit gives a larger numerical value, while converting to a larger unit gives a smaller numerical value.

Significant figures show the precision justified in a reported value. Start counting at the first non-zero digit: leading zeros locate the decimal point and are not significant, zeros between non-zero digits are significant, and trailing zeros after a decimal point are significant. Therefore 0.00480 has three significant figures.

For a calculation based on measured values, keep extra digits during working and round only the final result. Unless the question gives a different instruction, the measured input with the fewest significant figures is a useful guide for multiplication and division. Do not report many calculator digits that the measurements cannot support. In a result written with an uncertainty, give the value and uncertainty to compatible decimal places, as in 1.56 s +/- 0.02 s.

Communicating a reasoned conclusion

Scientific work continues after a graph or conclusion is produced. A paper-based or electronic report or presentation should communicate the rationale for the investigation, the hypothesis, apparatus and method, safety controls, raw and processed data, analysis, uncertainty, limitations and a conclusion linked to the evidence. Different forms serve different purposes: prose explains reasoning, a diagram can show an arrangement, a table preserves numerical readings, a graph reveals a trend, and symbols or equations express relationships compactly.

The form and detail should suit the audience. Another scientist needs enough method and data to scrutinise or reproduce the work. A public summary may use less technical language, but it must preserve units, scale, important uncertainty and limitations rather than making the conclusion sound more certain than the evidence allows.

For the pendulum data, a concise report could say: “From 0.20 m to 0.80 m, the mean period rose from 0.90 s to 1.80 s. Repeats at each length differed little, but manual timing and length measurement may be biased. The results support an increasing-period hypothesis over this range; another group should test reproducibility.” This links the numerical evidence, uncertainty and limits rather than merely announcing success.