2 Handling data
- Syllabus
- 2024
- Topic
- 2
- Level
- —
Learning objectives
22A Significant figuresUse an appropriate number of significant figures.22B Arithmetic meanUnderstand and find the arithmetic mean (average).22C Bar chartsConstruct and interpret bar charts.22D Frequency tables, diagrams and histogramsConstruct and interpret frequency tables, diagrams and histograms.22F Simple probabilityUnderstand simple probability.22H Scatter diagramsUse a scatter diagram to identify a pattern or trend between two variables.22I Order-of-magnitude calculationsMake order-of-magnitude calculations.
Use significant figures to match the evidence
Significant figures count digits from the first non-zero digit and communicate the precision justified by a value. Leading zeros locate the decimal point but are not significant; zeros between non-zero digits are significant, and trailing zeros after a decimal point can show measured precision.
To round to n significant figures: find the first non-zero digit → count to the nth digit → inspect the next digit → leave the nth digit unchanged if the next digit is 0–4, or increase it by one if the next digit is 5–9 → replace discarded place values with zeros where needed and retain the unit.
Example: 0.004786s to three significant figures is 0.00479s because the significant digits begin at 4 and the next digit after 8 is 6. In standard form, the same decision is clearer: 4.786×10−3s≈4.79×10−3s.
Significant figures are not decimal places: 0.00479 has three significant figures but five decimal places. Do not report a calculated result with more meaningful precision than the measurements support. Keep unrounded calculator values during intermediate steps and round the final result, so repeated early rounding does not distort it.
Find a mean from the valid readings
The arithmetic mean shares the total of a set equally among its values. For experimental data, first decide which readings are valid; the divisor must be the number of readings actually included, not the number originally collected.
mean=number of included readingssum of included readings
Inspect repeats for an anomalous value and use evidence before excluding it. Then add the valid readings with their signs, divide by their count, attach the original unit and round only the final mean appropriately. A negative reading remains negative in both the sum and the mean.
Example: count rates are 54, 58, 52, 35 and 55 counts per minute. If repeats justify treating 35 as anomalous, the included total is 54+58+52+55=219. The mean is 219/4=54.75, which is 55 counts per minute to two significant figures.
Do not remove a value simply because it changes the mean, and do not divide by five after excluding one of five readings. A mean represents the centre of the included data; it does not show their spread, and averaging cannot correct a systematic error affecting every reading.
Use a bar chart for categories
A bar chart compares a numerical value across distinct categories. The category variable is discontinuous—for example, material or planet—so each bar belongs to a named group rather than to every value along a continuous scale.
Choose a linear numerical scale that uses most of the plotting area. Label the category axis, label the numerical axis with its quantity and unit, and draw every bar to the correct height. Keep bar widths consistent; gaps usually make separate categories clear. The orientation and category order do not change the data if the labels remain unambiguous.
Read a bar by tracing its top to the numerical scale, then compare or combine only the categories named. Example: if three non-renewable categories contribute 27.5%, 35% and 19.5%, their total is 27.5+35+19.5=82%. The calculation should use the bar heights, not an estimate based only on appearance.
Do not use a bar chart merely because the dependent variable is numerical; the deciding feature is that the horizontal groups are categories. Continuous paired measurements usually need a line or scatter graph. A non-linear axis distorts comparisons, and missing units make bar heights ambiguous.
Organise frequencies before choosing the diagram
Frequency is the number of observations in a value or class interval. A frequency table performs the essential grouping first; a frequency diagram or histogram then makes that distribution visible without changing the underlying counts.
| Representation | Horizontal structure | Vertical information | Key construction rule |
|---|---|---|---|
| frequency table | values or class intervals | written counts | intervals must not overlap or leave intended values unclassified |
| frequency diagram | discrete values or classes | frequency | label values/classes and use a linear scale |
| histogram | continuous class intervals | frequency for equal widths; frequency density for unequal widths | bars touch and bar area represents frequency |
frequency density=class widthfrequency
Example: a class from 10 up to 20 has frequency 12, so its width is 10 and its frequency density is 12/10=1.2. A class from 20 up to 25 with frequency 9 has density 9/5=1.8; its narrower but taller bar still has area proportional to 9.
A histogram is not a bar chart with the gaps removed. Its horizontal axis is continuous and bar width carries numerical meaning. If all class widths are equal, frequency heights preserve the same comparison; with unequal widths, plotting raw frequency as height gives misleading areas.
Probability measures how likely an outcome is
Probability is a number from 0 to 1: 0 means impossible and 1 means certain. For equally likely outcomes, theoretical probability is the number of favourable outcomes divided by the total number of possible outcomes.
P(A)=total equally likely outcomesfavourable outcomesP(not A)=1−P(A)
When probability is estimated from observations, use relative frequency: number of times the outcome occurs divided by the number of trials. Example: if a detector records the chosen event in 30 of 120 repeated trials, the experimental probability is 30/120=0.25 and the probability of no such event is 1−0.25=0.75.
Check that the result lies between 0 and 1; multiplying by 100 converts it to a percentage. A larger number of trials usually gives a more stable experimental estimate, but it does not force the observed proportion to equal the theoretical value exactly.
The favourable-over-total shortcut requires outcomes to be equally likely. Probability describes long-run likelihood, not a guarantee about the next single trial. Do not add probabilities unless the events being combined cannot occur together, and do not confuse a count with a probability until it has been divided by the total.
Read a scatter diagram as a pattern of pairs
A scatter diagram plots paired measurements of two variables, one point per pair. Its purpose is to reveal whether the variables tend to change together; the overall cloud of points matters more than the route from one point to the next.
| Pattern as the horizontal variable increases | Interpretation |
|---|---|
| points tend upward | positive correlation |
| points tend downward | negative correlation |
| no consistent upward or downward tendency | no clear correlation |
| points lie close to a trend | stronger correlation than a widely scattered cloud |
Label both axes with quantities and units, choose linear scales and plot every pair accurately. Identify any anomalous point, then use a line or curve of best fit to represent the central trend with points reasonably balanced around it. Estimate within the measured range by reading from that trend, not by joining dots.
Correlation does not prove that one variable causes the other; a third factor or coincidence may explain the pattern. An anomaly should be checked rather than automatically deleted, and extrapolation beyond the measured range is less secure because the relationship may change there.
Use powers of ten to judge order of magnitude
An order of magnitude is the nearest power of ten describing a value's scale. Write the value as a×10n with 1≤a<10; compare a with 10≈3.16 to decide which neighbouring power of ten is closer.
a<3.16:order=10na≥3.16:order=10n+1
Example: 7.2×10−4 is closer to 10−3 than to 10−4, so its order of magnitude is 10−3. For a calculation, first estimate the standard form: (2×103)(4×10−6)=8×10−3, whose order of magnitude is 10−2.
For products, add powers of ten; for quotients, subtract them; then normalise the coefficient before choosing the nearest power. Use the result to compare scales or check a detailed calculation. Values whose orders differ by three powers of ten differ in scale by about a factor of 1000.
Order of magnitude is not the exponent copied from unexamined standard form and is not the same as rounding to one significant figure. For example, 7×104 to one significant figure remains 7×104, but its nearest order of magnitude is 105.