Lesson 19: Data Analysis

Data only becomes evidence when it is organised, graphed and interrogated. Data analysis is the skill of turning tables into conclusions: choosing the right graph, calculating rates and averages, spotting patterns and anomalies, and knowing how far a conclusion can stretch. Data-based questions carry heavy marks in every exam — this lesson is pure return on investment.

Choosing and drawing graphs

• Bar chart — comparing categories (discrete data): numbers of species, favourite fuels.
• Line graph — continuous change, usually over time or against a continuous variable: cooling curves, growth, rates. Join points with straight lines or draw a smooth curve of best fit.
• Scatter graph — looking for a relationship between two variables; the line of best fit shows the trend, passing as near as possible to all points with roughly equal numbers above and below.

Graph craft that earns marks: labelled axes with units, a sensible scale using most of the grid, accurately plotted points, and — for rates — a gradient calculation: gradient = change in y / change in x, using two widely separated points on the LINE (not two raw data points), with units (e.g. °C/min).

Worked example: the cooling curve

A beaker of hot water cools in a classroom:

Cooling rate between 10 and 20 minutes = (55 − 42) / 10 = 1.3 °C per minute. Between 20 and 30 minutes it is only (42 − 35)/10 = 0.7 °C/min. The rate falls because the temperature difference between the water and the room shrinks — cooling is fastest when the difference is greatest (Newton's law of cooling, in words). Over the whole 30 minutes the temperature fell by 80 − 35 = 45 °C. As a percentage fall that is 45/80 × 100% ≈ 56%. Always compute a percentage change as (change ÷ original value) × 100% — never estimate it by eye.

Averages describe the middle: mean (sum ÷ count), median (middle value), mode (most frequent). For the set {2, 3, 3, 5, 9}: mean = 22/5 = 4.4, median = 3, mode = 3. When a set contains an outlier (9), the median often represents the data better than the mean — examiners ask exactly this.

Reading patterns and judging claims

Describe a trend in three moves: (1) overall pattern ("as X increases, Y decreases"), (2) the shape ("rapidly at first, then levelling off"), (3) quoted figures ("from 80 °C at 0 min to 35 °C at 30 min"). Then evaluate: are there anomalies? One point far off the line (a temperature of 50 °C at 15 min in the table above) should be named and a cause suggested. Beware extrapolation — extending a line beyond the measured range assumes the pattern continues; the cooling curve flattens towards room temperature (about 30 °C in Singapore) and cannot fall below it, so extrapolating a straight line to 0 °C is physically wrong.

Correlation is not causation: ice-cream sales correlate with drowning incidents, both driven by hot weather (a lurking variable). A claim "X causes Y" needs a mechanism and, ideally, a controlled experiment — not just a rising pair of lines.

Singapore focus: reading national data

Singapore publishes rich data: rainfall records, dengue case counts, PSI trends, reservoir levels. During dengue season, NEA's weekly case graphs rise and fall with temperature and rainfall from weeks earlier (mosquito breeding lags the rain). Reading such graphs — axes, lags, anomalies like festive-week reporting dips — is the same skill your exam tests with school-laboratory data.

Common mistakes

• Calculating a gradient from two plotted crosses instead of two points on the best-fit line. Raw points carry experimental scatter; the line is the trend.
• "The graph goes up" as a description. Always state variables and quote values: "as time increased from 0 to 30 min, temperature fell from 80 °C to 35 °C".
• Forgetting units in gradient answers. A gradient of 1.3 is meaningless; 1.3 °C/min is a rate.
• Averaging categorical or ranked data, or quoting a mean to more precision than the instrument gave (12.4333 s from a hand stopwatch is fiction).

Exam Experience

Question 1 (structured, data-based). A student heats a beaker of water and records its temperature:

(a) Calculate the average rate of temperature rise between 2 and 8 minutes. [2] (b) The student predicts the temperature at 12 minutes will be 99 °C by extending the pattern. Evaluate this prediction. [2] (c) At 6 minutes the reading was taken with the thermometer bulb touching the beaker's base. Explain how this affects the reliability of that point. [2]

Question 2. A scatter graph shows plant height increasing as fertiliser mass increases, up to 4 g, then decreasing above 4 g. The best description is: [1] A. height increases proportionally with fertiliser at all amounts B. there is no relationship between the variables C. height rises to a maximum at about 4 g, then falls D. the variables are directly proportional

Question 3. A class claims "students who sleep longer score higher in tests" from a scatter graph of 12 students. Give two reasons why this claim, as stated, is not yet justified, and one improvement to the investigation. [3]

Marking scheme. Q1(a) Rate = (76 − 40) / (8 − 2) = 36/6 = 6.0 °C/min [1 method, 1 answer with unit]. Q1(b) The trend is roughly +6 °C/min, giving about 87 + 12 = 99 °C — arithmetically consistent [1]; but unreliable: water approaches boiling (100 °C) and heating slows/levels near boiling as evaporation increases; also the apparatus may not sustain a constant rate — extrapolation beyond the measured range is risky [1]. Q1(c) The thermometer measured the hot beaker base rather than the bulk water [1]; so that point reads higher than the true water temperature — it is likely anomalous / reduces the accuracy of that reading [1]. Q2 C [1]. Q3 Any two limitations: correlation does not prove causation (other factors — study time, prior attainment) [1]; small sample of 12 students [1]; self-reported/measured variables unreliable [1]; improvement: larger sample with controlled variables, or a controlled study recording actual sleep and comparable tests [1]. (Max 3.)

Going deeper: spread, uncertainty and honest graphs

Two classes can share a mean yet differ completely: Class A's test scores cluster tightly (range 62–68), Class B's sprawl (range 35–95). Data analysis therefore reports spread alongside averages: the range (max − min) is the quickest measure. For a measured value like 12.4 cm³ ± 0.1 cm³, the percentage uncertainty = 0.1/12.4 × 100% ≈ 0.8% — small uncertainties certify careful work; large ones demand better instruments or more repeats.

Error bars and best-fit discipline. On scatter graphs, a line of best fit should pass through the cloud's balance point with points scattered evenly around it — not necessarily through ANY plotted cross, and never forced through the origin unless the physics demands proportionality (zero input must give zero output). A point far from the pattern is an anomaly to be flagged, circled and discussed — deleting it silently is scientific misconduct in miniature, and examiners phrase questions to catch it ("one result does not fit the pattern — identify it and suggest a cause").

Choosing the summary. For the gas volumes {24, 26, 25, 40, 26} cm³: the mean with the anomaly is 141/5 = 28.2 cm³; excluding 40 it is 101/4 = 25.25 cm³. The median, 26 cm³, resists the outlier without any deletion decision. Skilled analysts state which summary they used and why — the "why" is where the marks are.

Exam-style extra

Question. Reaction times (s) for five trials: 0.32, 0.30, 0.31, 0.58, 0.29. (a) Identify the anomalous result and suggest one cause. [2] (b) Calculate the mean of the remaining results. [2] (c) State the range of the remaining results. [1]

Marking points. (a) 0.58 s [1]; cause: distraction / delayed start / anticipation error — any plausible human-timing cause [1]. (b) (0.32+0.30+0.31+0.29)/4 = 1.22/4 = 0.305 s [1 method, 1 answer]. (c) Range = 0.32 − 0.29 = 0.03 s [1].

Workshop: choosing the chart and spotting the cheat

Match the chart to the data type: bar charts for categories (favourite drinks, species counts, scores by month), line graphs for continuous change over time or against a continuous variable (cooling, growth, rate versus concentration), pie charts for parts of a whole that sum to 100%. Choosing wrongly is not a style error — a line graph across unordered categories implies a trend that does not exist.

Then read defensively. A truncated y-axis (starting at 90 instead of 0) makes a rise from 95 to 100 look like a doubling; unequal time intervals on the x-axis fake accelerations; two y-axes can manufacture agreement between unrelated series. Media graphs use all three. The scientist's counter-routine: check where each axis starts, check the intervals, restate the actual numbers before believing the shape. In your own graphs, the same discipline reversed — full scales where honest, breaks clearly marked where used — is what "present data clearly" means in the marking scheme.

One more angle: correlation is not causation

Plot monthly ice-cream sales against monthly drowning incidents and the points march up together — a strong positive correlation with NO causal link in either direction. A third variable, temperature, drives both: hot months sell ice cream AND fill pools. This confounder pattern is the most important idea in data literacy: correlation invites a causal story, and science's job is to test rival stories before believing one. The controlled experiment (Lesson 18) is the machine built for exactly this — hold the confounder still and vary only the suspect cause. When an exam scatter graph shows a beautiful trend, the distinction-level answer still asks: what third variable could produce both?