Mathematical and statistical skills

Use SI units and prefixes consistently, convert scales such as kilo, milli, micro and nano, and interpret inequality or proportional symbols.

Write the unit with the number and use standard form for very large or small biological quantities.

2.5 µm is 2.5×10⁻⁶ m, not 2.5×10⁻³ m; the prefix determines the power of ten.

A correct numerical value with a mismatched unit is still an incorrect biological result.

Estimate the order of magnitude first, then calculate and report a precision justified by the measured data.

Keep extra digits during working but round the final quantity consistently.

If the least precise input has two significant figures, a calculated rate should not be reported to six unexplained figures.

Calculator precision is not measurement precision.

Magnification = image size ÷ actual size; use the same units before calculating and apply the relevant area, perimeter or volume formula.

Check whether the question asks for a length, area or volume because scaling powers differ.

A 20 µm cell shown as 40 mm long has magnification 2000× after converting 40 mm to 40 000 µm.

Do not divide by magnification without first matching units.

Use the statistic or percentage that matches the question, and express experimental error relative to the measured quantity.

State the denominator for a percentage change and keep ratios in a comparable form.

A 2-unit change from 10 is 20%, while the same change from 100 is 2%; the absolute change is identical but the relative change differs.

Percentage error is not the same as percentage change between two biological treatments.

Translate between numerical, graphical and algebraic forms; choose an appropriate chart and calculate rate from a gradient or tangent when the relationship changes.

Put the explanatory variable on x, use a meaningful scale and state the interval for any gradient.

The initial tangent to a respiration curve gives the initial rate, while the overall endpoint difference gives an average rate.

A steeper-looking graph may result from unequal axis scales; inspect units and scale before comparing gradients.

Probability helps predict genetic ratios and quantify how chance affects a biological sample.

Separate expected probability from the result observed in one sample, and increase sample size when chance variation is large.

A 1:1 genetic ratio does not require every small family to contain equal numbers of each phenotype.

Chance variation is not evidence that the genetic model is wrong by itself.

Hardy–Weinberg estimates allele/genotype frequencies, Lincoln estimates a mobile population and Simpson’s index summarises diversity.

Define each symbol from the data, substitute consistently and interpret the result in the biological context.

A recapture estimate becomes unstable when the number marked in the second sample is very small.

A formula output is only as reliable as the assumptions behind sampling and the data collection.

Mean, median, mode, range, standard deviation, standard error and confidence intervals describe different aspects of a dataset.

Use the statistic that matches the distribution and explain what error bars represent before comparing them.

A skewed dataset may be better summarised by a median, while standard error describes uncertainty in a mean rather than spread of individual values.

Error bars are not interchangeable; SD, SE and confidence intervals answer different questions.

Chi-squared compares observed and expected categories; a t-test compares means under suitable assumptions.

Calculate degrees of freedom, compare the probability with the significance level and state the biological decision.

Rejecting the null for a chi-squared test means the observed categories differ from expectation; it does not identify the mechanism.

A significant result does not show that the effect is large or biologically useful.

Pearson and Spearman coefficients describe relationships from −1 to +1, but neither alone establishes causation.

Choose the test that matches data type and pattern, inspect confounding variables and state conditions for validity.

Plant abundance may correlate with soil moisture because both respond to altitude rather than because moisture alone causes the pattern.

A strong correlation can be spurious, and a weak correlation can hide a non-linear relationship.

Use p+q=1 and p²+2pq+q²=1 to connect allele and genotype frequencies.

Identify whether the supplied value is an allele frequency, homozygous genotype frequency or heterozygote frequency before rearranging.

If q² is known, take the square root to find q, then calculate p and 2pq.

The equations are a model with assumptions; they are not a universal description of every population.

Use N=(n₁×n₂)/m₂ for mark-release-recapture and the stated Simpson formula for diversity.

Define each sample count and check that the formula version matches the convention in the question.

If few marked animals are recaptured, the estimate can become very large and uncertain.

Never interpret Simpson’s direction without checking whether the formula is D, 1−D or a reciprocal form.

Apply the provided formulae to calculate a statistic, then interpret the size or interval rather than memorising symbols in isolation.

Keep observed and expected values paired and preserve units for biological quantities.

A confidence interval around a mean expresses uncertainty in that estimate, not the full spread of individual observations.

A correct calculation can still be misinterpreted if degrees of freedom or the null hypothesis is wrong.

Use the supplied formula for the selected test and calculate degrees of freedom where required.

The final step is a decision about significance and biological meaning, not just a number.

A Pearson coefficient near +1 indicates a strong positive linear association under valid conditions; it does not prove one variable causes the other.

Do not choose a test because its formula is familiar; choose it from the data type and question.

Statistical validity depends on data type, distribution, independence, sample size and whether the question asks for a difference or relationship.

State why a method fits before calculating, then link the outcome to the biological claim.

Chi-squared suits category counts, a t-test suits two means, and Spearman suits ranked or non-normal monotonic data.

A method can be mathematically executable but biologically invalid if its assumptions are ignored.