Lesson 18: Data Analysis and Statistics

This lesson covers data analysis and statistics for IB Biology (SL and HL) — the quantitative toolkit used in Paper 2 data questions and in the IA. Four tools carry almost all of the marks: processing with uncertainties, choosing and drawing graphs, describing spread, and running the right statistical test.

Processing raw data honestly

Raw data go in tables with the IV first, units in the headers (never in every cell), and decimal places consistent with the instrument's precision. Processed data (means, rates, percentages) must show their working at least once, carry correct significant figures (usually match the raw data's precision), and keep units. Uncertainty: every measurement has an absolute uncertainty (± half the smallest division for analogue scales, ± the last digit for digital). When quantities are added/subtracted, add absolute uncertainties; when multiplied/divided, add percentage uncertainties. Example: a rate from volume 24.0 ± 0.5 cm³ over 60 ± 1 s: percentage uncertainties are 2.1% and 1.7%, total ≈ 3.8%, so the rate 0.40 cm³ s⁻¹ is reported as 0.40 ± 0.015 cm³ s⁻¹ (≈ ±4%). Outliers: identify them, and either justify exclusion with a stated rule (e.g. beyond 1.5 × IQR) or keep them and acknowledge the effect — silent deletion is scientific misconduct and examiners treat it as such.

Graphs that earn marks

Choose by data type: line graphs for continuous IVs (join points when the relationship is a trend you are claiming, or plot points only and add a trend line), bar charts for categorical IVs, scatter graphs for two measured variables, and histograms for distributions. Non-negotiables: axes labelled with quantity and unit, sensible scales filling the space, title stating what is plotted, and error bars where repeats allow. Error bars are usually ±1 standard deviation (spread of the data) — do not confuse them with instrument uncertainty. Reading graphs: gradients give rates (show your triangle working on the graph in an IA), intercepts can reveal thresholds (e.g. the isotonic point in osmosis work), and overlapping error bars warn against claiming a difference.

Describing data: mean, median, spread

For roughly symmetrical data use the mean and standard deviation (SD); for skewed data or small samples with outliers, the median and interquartile range (IQR) are more robust. SD measures the spread *of the data* around the mean; the standard error (SD ÷ √n) measures the precision *of the mean* — know which you are plotting. Percentage change and percentage difference are different calculations; label which you used. Worked example: five trials of oxygen volume: 12.1, 12.4, 11.8, 12.2, 14.9 cm³. Mean = 12.68; the value 14.9 sits far from a tight cluster (SD of the other four ≈ 0.25), flagging a probable anomaly (perhaps a leak-free run after four leaking ones, or a temperature drift) — investigate and report, don't quietly drop.

Choosing the statistical test

Decision logic examiners reward: (1) Comparing two sets of interval data (e.g. two habitats, before/after) → t-test (unpaired for independent samples; paired for matched measurements). (p < 0.05 means the difference is statistically significant.) (2) Frequency/count data or association in categories (observed vs expected ratios, species co-occurrence) → chi-square test, χ² = Σ(O−E)²/E, df = (rows−1)(cols−1) or categories−1 for ratios; expected frequencies should be ≥5. (3) Relationship between two continuous variables → correlation: Pearson for linear, normally distributed data; Spearman rank for monotonic/non-normal or ranked data. Report the statistic, degrees of freedom and the conclusion against the null hypothesis — and phrase it correctly: a significant result means the data are inconsistent with the null; a non-significant result means insufficient evidence of a difference, *never* proof of no difference. Worked chi-square: F₂ counts 315 purple : 101 white vs expected 3:1 (312 : 104): χ² = (3²/312) + (3²/104) = 0.029 + 0.087 = 0.116, df = 1, critical 3.84 → not significant; consistent with 3:1.

Nature of Science / TOK link

The 5% significance level is a *convention*, not a law of nature — a community agreement about acceptable risk of false positives. TOK: statistical tests convert judgement into procedure, but the judgement remains — in choosing the test, the level, and what counts as the same population. Data do not speak; they are interpreted. Knowing the machinery keeps you from being spoken for.

Common mistakes (from examiner reports)

• Units repeated inside table cells instead of the header, or missing from graph axes.
• Error bars drawn as the instrument's reading error rather than spread of repeats.
• "The t-test proved the difference" — tests never prove; and reporting p-values with no conclusion tied to the null hypothesis.
• Chi-square used on means or percentages instead of raw counts.
• Calculating means of ordinal ranks or of percentage-cover data without thought.

IB Exam Experience

Paper 1 style. 1. Which test compares observed and expected frequencies in a genetic cross? A. t-test B. Chi-square test C. Spearman rank correlation D. Standard deviation 2. Error bars showing ±1 standard deviation describe… A. the precision of the measuring instrument B. the spread of repeated measurements around the mean C. the range between max and min only D. the uncertainty in the IV 3. A result is significant at p < 0.05. This means… A. the null hypothesis is certainly false B. the data would be unlikely (less than 5% probability) if the null hypothesis were true C. the effect is large D. the experiment was reliable Answer key. 1 — B. 2 — B. 3 — B (significance is about probability under the null, not size or certainty). Paper 2 style. Two fields were sampled for earthworm density (per m²): Field A: 14, 16, 15, 13, 17; Field B: 9, 11, 10, 12, 8. (a) Calculate the mean density for each field. [1] (b) A t-test gives p = 0.004. State the conclusion in terms of the null hypothesis. [2] (c) Suggest why standard deviation should be reported alongside the means. [1] *Marking notes.* (a) A = 15, B = 10 worms m⁻² [1]. (b) p < 0.05, so reject the null hypothesis of no difference [1]; there is a statistically significant difference in density between the fields [1]. (c) SD shows the variability within each field, without which the reliability of the mean difference cannot be judged [1]. IA idea. Design an investigation whose data *require* a specific test and justify the choice in advance — e.g. a two-habitat comparison with a t-test, or a species-association study with chi-square — mirroring the decision logic above.

Extended drill: the t-test, computed once by hand

Two samples of leaf lengths (mm) — shaded hedge: 42, 45, 39, 47, 44 (mean 43.4, SD ≈ 3.0); sunny hedge: 51, 48, 53, 49, 50 (mean 50.2, SD ≈ 1.9). The unpaired t statistic compares the difference between means (6.8) with the standard error of that difference: SE = √(SD₁²/n₁ + SD₂²/n₂) = √(9.0/5 + 3.6/5) = √2.52 ≈ 1.59, so t ≈ 6.8 ÷ 1.59 ≈ 4.28 with df = 8. The critical value at p = 0.05 for df 8 is 2.31; 4.28 > 2.31, so the difference is significant — reject the null hypothesis of no difference in mean leaf length. You will use software in the IA, but a hand-worked miniature teaches what the number *is*: a signal-to-noise ratio. If the SDs had been three times larger, the same means would give t ≈ 1.4 — not significant — which is why reporting spread is not decoration. Two further disciplines: the t-test assumes approximately normal data and (in its standard form) similar variances — state that you considered them; and choose paired vs unpaired by design, not by which gives significance: paired is for the *same* subjects measured twice (before/after), unpaired for independent groups.

Reading someone else's graph critically

Paper 2 graphs reward three checks before any description. First, axes: what is plotted, in what units, and is the y-axis zero-based (a truncated axis exaggerates differences — a classic misleading-graph device in media reports of the same data)? Second, error bars and sample sizes: are the plotted points means of 3 or of 300? Do intervals overlap? Third, the claim being invited: does the trend hold across the whole range or only in part (note thresholds, plateaus and reversals with their x-values)? Apply the checks to a common trap: a dual-axis graph where two quantities are scaled to *look* correlated. The IB's own graphs are honest, but your evaluation of secondary sources in the IA and in TOK-flavoured questions should name scaling and selection as ways presentation shapes conclusions — data are objective; their display is rhetoric, and scientists are expected to notice the difference.

One more application: when the test says "not significant"

A non-significant result is a result. If your t-test returns p = 0.21, the honest conclusions are: the data do not provide sufficient evidence of a difference at the 5% level; possible reasons include a genuinely absent effect, an effect smaller than your design could detect, or variability swamping the signal (inspect the SDs relative to the means). What you may not write is "the variables have no effect" or "the hypothesis is disproven" — absence of evidence is not evidence of absence, and examiners penalise the overclaim precisely because working scientists are ruined by it. The constructive move is a power reflection: given the observed SD, how many replicates would a difference of the observed size have needed to reach significance? Even a rough answer converts a flat result into demonstrated statistical maturity — and it is exactly what the Evaluation criterion means by "the effect of the weakness on the data".

Examiner's lens: final drills (data & statistics)

• Test selection in ten seconds: means of two groups → t-test; counts against an expected ratio or association → chi-square; relationship between two measured variables → correlation (Pearson if linear/normal, Spearman if ranked/monotonic).
• Report a statistical result as a full sentence: the statistic, degrees of freedom, the comparison value (critical value or p), and the conclusion in terms of the null hypothesis — four components, every time.
• Significant figures discipline: quote processed values to the precision the data support (means of cm³ readings to 0.1 cm³, not 12.3842) — excess decimals signal that processing was not understood.
• Self-test: two habitats have mean densities 12.4 and 11.9 with SDs of 6. Is the difference likely to be significant, and what would you check? (No — the spread dwarfs the difference; check the t-test/SE and the error-bar overlap before claiming anything.)