IB Biology SLSkill Set 3 — Maths for BiologyPaper 1 & 2Core skill~11 min read
Maths Skills for Biology
Nobody picks biology for the maths, and yet a good chunk of every paper is arithmetic wearing a lab coat. The good news: it is the same dozen skills over and over — percentages, ratios, averages, spread. Learn them properly once and you stop losing marks on questions where you already understood the biology.
📘 What you need to know
The everyday toolkit: decimals, fractions, percentages, ratios, proportions, frequencies, densities and reciprocals.
Surface area to volume ratio is the ratio you will meet most — it falls as an object gets bigger.
Central tendency: mean, median and mode each describe the “typical” value in a different way.
Dispersion: standard deviation, standard error and interquartile range describe the spread.
Percentage change compares back to a starting value. Percentage difference compares two values with no baseline.
Discrete data is counted; continuous data is measured.
Statistical tests you should recognise: Simpson’s reciprocal index, Lincoln index, chi-squared and the t-test.
The everyday toolkit
These come up constantly, usually buried inside a longer question.
Skill
Where it turns up in biology
Decimals
Almost every measurement — cell diameters, rates, concentrations
Fractions
Calculator answers often appear as fractions; convert to a decimal before writing them down
Percentages
Percentage change in mass, percentage cover in a quadrat, percentage of offspring with a trait
Ratios
Surface area to volume, and expected ratios in genetic crosses such as 3 : 1
Proportions
Scaling a biological drawing up or down from a micrograph
Frequencies
Allele frequency in a population, and how it shifts over generations
Densities
Population density in ecology, stomatal density on a leaf surface
Approximations
Sensible rounding in magnification and scale calculations
Reciprocals
Rate as 1 ÷ time, when you have timed how long something took
The reciprocal one catches people out. If a reaction takes a long time, the rate is low — so time and rate are inversely related, and 1 ÷ time turns your stopwatch reading into something you can plot sensibly.
Surface area to volume ratio
This is the ratio the course keeps coming back to, because it explains why cells are small, why alveoli are folded, and why a mouse eats more per gram than an elephant.
The reason is a mismatch in how the two quantities grow. Double the length of a cube and the surface area goes up four times, but the volume goes up eight times. Volume wins, so the ratio falls.
Always simplify the ratio to something over 1. “54 : 27” is correct but “2 : 1” is the answer the examiner is looking for.
WORKED EXAMPLE
A cube-shaped block of agar has sides of 2 mm. Calculate its surface area to volume ratio.
Step 1: surface area — six faces6 × (2 × 2) = 24 mm2Step 2: volume2 × 2 × 2 = 8 mm3Step 3: divide, then simplify24 ÷ 8 = 3SA : V = 3 : 1Do not forget the six faces. Using 2 × 2 for the surface area is the most common slip in this calculation.
Averages: mean, median and mode
All three claim to give you the typical value, and they disagree whenever the data is lopsided.
Mean — add everything up and divide by how many values there are. The standard choice, but a single extreme value drags it.
Median — put the values in order and take the middle one. Extreme values barely touch it.
Mode — the value that appears most often. The only one that works on categoric data such as eye colour or blood group.
Ecological counts are often lopsided like this, which is why field data is so often summarised with a median rather than a mean.
Spread: how much do the readings vary?
A mean on its own tells you almost nothing. Two data sets can share a mean of exactly 5.0 and be completely different sets of numbers.
Quoting a mean without a measure of spread is like quoting an exam average with no idea whether the class scored 48 to 52 or 12 to 88.
The three measures you should know
Standard deviation (SD) — the spread of the data around the mean. Small SD, tight data. Use it to compare how consistent two data sets are.
Standard error (SE) — how far your sample mean is likely to be from the true population mean. It measures how well your sample represents the population, and it is always smaller than the SD.
Interquartile range (IQR) — the difference between the 75th and 25th percentiles, so the range covered by the middle half of the data. Quartiles cut the ordered data into four equal parts, and the IQR ignores extremes at both ends.
Percentage change and percentage difference
These sound like the same thing and are not, and choosing the wrong one is a reliable way to lose a mark.
Percentage change — when there is a starting value
(final − initial) ÷ initial × 100
Use this whenever something has changed from a “before” into an “after”: mass of a potato chip after soaking, population size after ten years, heart rate after exercise. A negative answer means a decrease, and that is a real answer — do not drop the minus sign.
Percentage difference — when there is no baseline
(A − B) ÷ [(A + B) ÷ 2] × 100
Use this when the two values are simply two different things with no “before”. Neither one is the starting point, so you compare the gap to the average of the two.
WORKED EXAMPLE
A potato chip has a mass of 4.20 g. After an hour in distilled water its mass is 5.10 g. Calculate the percentage change in mass.
Step 1: there is a clear starting value, so use percentage change(5.10 − 4.20) = 0.90 g gainedStep 2: divide by the initial mass and multiply by 100(0.90 ÷ 4.20) × 100 = 21.4285…+21.4% (3 s.f.)Percentage change is used here rather than raw mass change because chips do not all start the same size.
WORKED EXAMPLE
Site A has 38 species and site B has 22. Calculate the percentage difference in species number.
Step 1: neither site is a baseline, so use percentage difference38 − 22 = 16Step 2: find the mean of the two values(38 + 22) ÷ 2 = 30Step 3: divide and multiply by 100(16 ÷ 30) × 100 = 53.333…53.3% (3 s.f.)Read the question carefully: if it names one value as the starting point, switch to percentage change instead.
Discrete and continuous data
Both are quantitative — both are numbers — but they behave differently, and the difference decides which graph you draw.
Discrete data is counted: separate whole values with nothing in between. The number of woodlice in a quadrat, the number of offspring with brown eyes. You cannot have 3.5 woodlice.
Continuous data is measured: any value within a range, including decimals. Temperature during an enzyme reaction, the volume of oxygen given off, leaf length.
The quick test: ask whether a value halfway between two of your readings makes sense. Half a degree, yes — continuous. Half a beetle, no — discrete.
Reciprocals: turning time into rate
Plenty of experiments measure how long something took — how long until the cross disappears, how long until the colour changes. Time is the wrong thing to plot, because a longer time means a slower reaction, so the graph comes out backwards.
Rate from a time
rate = 1 ÷ time
WORKED EXAMPLE
A starch and iodine mixture loses its colour after 25 seconds. Express this as a rate.
Take the reciprocal of the timerate = 1 ÷ 25 = 0.040.04 s−1The units are “per second” because you divided 1 by a time. A faster reaction now gives a bigger number, which is what you want on a graph.
Statistical tests you should recognise
You will be given the formulae in the exam. What you need is to know which test does what, and to be able to apply it. Which one fits depends on the size of the sample, whether the data is discrete or continuous, and what the question is actually asking.
Test
What it does
Typical question
Simpson’s reciprocal index
Measures biodiversity, taking in both the number of species and how evenly individuals are spread between them
Which of two habitats is more biodiverse?
Lincoln index
Estimates the size of an animal population from a capture, mark and recapture study
How many beetles live in this field?
Chi-squared test
Compares observed results with the results you expected
Do these offspring ratios fit the predicted genetic cross?
t-test
Compares the means of two sets of data to see if they differ significantly
Are leaves in the shade genuinely wider than leaves in the sun?
Conditions for the t-test: the two data sets should be roughly normally distributed, continuous, and have similar standard deviations. If your data is counts of categories rather than measurements, you want chi-squared instead.
A higher Simpson’s reciprocal index means more biodiversity. That is one of those facts that is quick to learn and regularly worth a mark.
💡 Exam tip
Simplify ratios so the second number is 1. “3 : 1” beats “24 : 8”.
Show your working. Method marks survive an arithmetic slip; a bare wrong answer does not.
Decide between percentage change and percentage difference by asking whether one value is a starting point.
Keep the minus sign on a decrease. It is part of the answer.
Quote a mean with its standard deviation whenever you have repeats.
If one reading sits far from the others, mention the median — it shows you noticed.
When a question gives you a time, check whether it wants a rate. That usually means 1 ÷ time.
⚠ Common mix-up
Forgetting the six faces when calculating the surface area of a cube.
Saying a big organism has a big SA : V ratio. It has a big surface area but an even bigger volume, so the ratio is small.
Percentage change dividing by the final value. It is always the initial value on the bottom.
Quoting a mean for lopsided data without mentioning that one extreme reading has pulled it.
Confusing SD and SE. SD describes the spread of the data; SE describes how reliable the mean is.
Calling counted data continuous. Numbers of organisms are discrete, however large the number.
Plotting time when the question wants rate. The graph will slope the wrong way.
Up next: Units, Symbols & Values — SI units, prefixes, standard form and significant figures, and how to convert between them without losing a factor of a thousand.
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