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.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.
Round results to appropriate significant figures
Significant figures count meaningful digits from the first non-zero digit and communicate precision without keeping unsupported calculator digits.
| Value | Significant figures | Reason |
|---|---|---|
| 0.00450 | 3 | leading zeros locate the decimal; 4, 5 and the final zero are significant |
| 20.06 | 4 | zeros between non-zero digits are significant |
| 2300 | ambiguous without notation | write 2.3×103 for 2 s.f. or 2.300×103 for 4 s.f. |
To round to n significant figures, keep the first n meaningful digits, inspect the next digit, and increase the last kept digit by 1 when the next digit is 5 or more. Keep place-value zeros in the rounded result.
6.134;\text{to 2 s.f.}=6.1,\qquad 0.009876;\text{to 3 s.f.}=0.00988
Significant figures are not decimal places. Round once at the end of a multi-step calculation unless an intermediate value must be reported separately.
Calculate and interpret an arithmetic mean
The arithmetic mean shares the total of a set equally across the number of values.
\text{mean}=\frac{\text{sum of all included values}}{\text{number of included values}}
For starting temperatures 17.8∘C and 18.4∘C, the mean is (17.8+18.4)/2=18.1∘C. For 6.1, 6.3 and 6.0, the mean is 18.4/3=6.13…, reported as 6.1 to 2 significant figures.
State which values are included, retain units, and report at precision justified by the measurements. Exclude an anomaly only when evidence justifies that decision, then adjust both the sum and the count.
Do not divide by the number of intervals or by a memorised count. The denominator is the number of values actually included.
Construct and interpret a bar chart
A bar chart compares values for separate categories; each bar's height represents the numerical value for its category.
| Feature | Requirement |
|---|---|
| category axis | label each discrete category; keep bars separated |
| value axis | use a linear, evenly spaced scale and state the quantity and unit |
| bars | equal width, common baseline and accurately plotted heights |
| interpretation | compare heights using values or differences, not visual adjectives alone |
Read a bar by tracing its top horizontally to the value scale. Compare categories with a difference, ratio or ranked statement when that answers the scientific question.
Do not join bar tops or make bars touch for separate categories. Touching intervals belong to a histogram, which is outside this Chemistry objective.
Use simple probability
Probability measures how likely an event is on a scale from 0 (impossible) to 1 (certain).
P(\text{event})=\frac{\text{number of favourable equally likely outcomes}}{\text{total number of equally likely outcomes}}
| Form | Example for P=0.25 |
|---|---|
| fraction | 41 |
| decimal | 0.25 |
| percentage | 25% |
For an event and its opposite, P(not event)=1−P(event). Repeated-event data may estimate probability as observed frequency divided by total trials.
The favourable-over-total rule requires equally likely outcomes. A probability cannot be below 0 or above 1.
Identify patterns in a scatter diagram
A scatter diagram plots paired values for two variables so their overall association can be identified.
| Pattern | Meaning |
|---|---|
| points rise from left to right | positive correlation: larger x tends to occur with larger y |
| points fall from left to right | negative correlation: larger x tends to occur with smaller y |
| no clear direction | no evident correlation |
| point far from the pattern | possible outlier or anomaly to investigate |
The more tightly points cluster around a smooth trend, the stronger the association. Describe direction first, then strength if the diagram supports it; use a best-fit trend rather than joining point-to-point.
Correlation does not by itself show that one variable causes the other. A third factor or the experimental design may explain the pattern.
Estimate with orders of magnitude
An order of magnitude describes the power-of-ten scale of a quantity and lets you judge whether a calculation is plausibly sized.
| Quantity | Scientific notation | Scale |
|---|---|---|
| 0.0048 | 4.8×10−3 | about 10−3 |
| 6200 | 6.2×103 | about 103 |
| 1.7×106 | already standard form | about 106 |
For a quick estimate, round inputs to one significant figure, calculate using powers of ten, then compare the full answer with the estimate. Multiplication adds exponents and division subtracts them.
\frac{(3\times10^4)(2\times10^{-2})}{6\times10^1}\approx1\times10^1
Order of magnitude is a scale estimate, not a precise rounded answer. Keep coefficient reasoning separate from exponent arithmetic and use the estimate to catch misplaced decimal points.