A.1 Handling data

Significant figures communicate the precision supported by a measurement or calculation. Zeros between non-zero digits count; leading zeros only locate the decimal point.

Keep guard digits during working, then round the final result to match the least precise relevant measurement. State a value such as 2.40 to show its precision.

A mean based on readings recorded to 0.1 s should not be reported as 2.437891 s; 2.4 s or 2.44 s may be appropriate depending on the data.

The arithmetic mean is the total of the observations divided by their count. It summarises repeated measurements when they represent the same quantity and are on the same scale.

Inspect the raw values first, calculate the mean with full precision and report spread or anomalous values rather than hiding them.

Rates 9, 10 and 11 units min⁻¹ have mean 10; if one reading was taken at a different temperature, combining it may be misleading.

Tables keep exact values visible; bar charts compare categories; histograms show how continuous measurements are distributed; diagrams clarify structure or process.

Label axes and units, choose equal scales, include a key when needed and make the display match the variable type. Keep raw data available behind any summary.

Use a histogram for a distribution of cell diameters, but a bar chart for separate treatment groups; joining category bars can falsely imply continuity.

For mutually exclusive alternatives, add probabilities. For independent events that must both occur, multiply them. Independence must come from the biological model—for example, separate meioses or independently assorting genes—not from convenience.

In an $Aa\times Aa$ cross, the probability of an $aa$ child is $1/4$. Each birth is a new independent event, so two $aa$ children in succession have probability $(1/4)(1/4)=1/16$. For independently assorting $AaBb\times AaBb$, the probability of $aabb$ is $(1/4)(1/4)=1/16$.

Observed frequency estimates probability: 7 cases among 800 000 people gives $7/800000=8.75\times10^{-6}$, about $0.000875\%$ or 1 in 114 000. Sampling variation means the observed proportion is not an exact law.

From sampled species counts, $D=\dfrac{N(N-1)}{\sum n(n-1)}$, where $N$ is total individuals and $n$ is each species count. Use the same effort, method and identification rules when comparing habitats; a larger $D$ indicates greater diversity under this index.

Increase independent sampling units, distribute them across time/space where appropriate and report uncertainty. A small sample of 23 chicks with an 8:15 sex ratio may not represent the adult population or a stable 1:1 population ratio.

The mean uses every value, the median is the middle after ordering, and the mode is the most frequent value. The best summary depends on the measurement and its distribution.

Use the median when an outlier would distort the mean; use the mode for common categories or repeated discrete values. Report the raw context and sample size.

Income-like values 2, 2, 3, 3 and 20 have mean 6 but median 3, so the median better represents a typical observation.

A scatter diagram pairs two measured variables. Direction, strength and form of the pattern describe association; they do not by themselves identify a causal mechanism.

Plot the independent variable consistently, inspect outliers and restricted ranges, and use a correlation measure only with the assumptions and data type it requires.

Light intensity and photosynthetic rate may rise together before a plateau; the pattern suggests a relationship, while temperature or CO₂ could also influence both.

A leaf section measures 50 mm in a photograph at $\times100$. Its actual thickness is $50/100=0.50$ mm $=500$ µm $=5.0\times10^2$ µm. The unit conversion happens after or before division only if it is applied consistently.

Write sizes in standard form to compare scale. $2\times10^{-5}$ m is one order of magnitude larger than $2\times10^{-6}$ m because their powers differ by 1; a power difference of 3 means a thousandfold difference. Coefficients near a power boundary must be considered when rounding to the nearest order.

A measured scale bar can recover actual size even if an image is resized, provided the object and bar were resized together. A printed magnification label may become invalid after resizing.

Calculate the statistic with unrounded data, determine degrees of freedom where required and compare its magnitude with the critical value at the stated significance level. If the result exceeds the relevant critical threshold, reject the null; otherwise do not reject it. Use the course-provided table and state the biological conclusion.

Chi-squared uses frequency counts with suitable expected values; t-tests require quantitative samples and the specified distribution/design assumptions; correlation requires paired observations and tests association, with Spearman based on ranks.

For $n$ sample values, $s=\sqrt{\dfrac{\sum(x-\bar{x})^2}{n-1}}$. Calculate the mean, subtract it from each value, square deviations, sum, divide by $n-1$, then take the square root. Retain calculator precision until the final step.

For 14, 16 and 18 g, $\bar{x}=16$ g, range $=18-14=4$ g, and $s=\sqrt{(4+0+4)/2}=2.0$ g. The SD means observations typically vary around the mean; it is not an error bar for the mean unless explicitly defined that way.

SD is usually more informative than range because it uses all values, but it can still be affected by an outlier. Compare groups using both centre and spread and consider sample size and measurement scale.

Percentage uncertainty is $\dfrac{\text{absolute uncertainty}}{\text{measured value}}\times100$. Percentage error compares a measured value with an accepted/reference value: $\dfrac{|\text{measured}-\text{accepted}|}{\text{accepted}}\times100$. They answer different questions.

A diameter $2.0\pm0.1$ mm has 5% uncertainty. For area proportional to $d^2$, the simple propagated percentage uncertainty is about $2\times5=10\%$. Report the area to precision consistent with that uncertainty.