5.3Theme A — Unity and diversity

Tool 3: Mathematics — IB Biology SL Revision Notes & Topical Questions

10 min read 11 parts Lesson 4 of 11 Unit quiz

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Lesson summary

5.3 Tool 3: Mathematics — IB BIOLOGY SL notes: Apply general mathematics, units, uncertainty, statistics and graphing skills to biological data.Calculations in biology use decimals, fractions, percentages, ratios, proportions, frequencies, densities, approximations and reciprocal…

Scientific notation
Significant figures
Estimation
Percentage change
Percentage difference relative to A
Symmetrical percentage difference

Part 1 of 9Numbers, notation and significant figures

Calculations in biology use decimals, fractions, percentages, ratios, proportions, frequencies, densities, approximations and reciprocals

Ratios include the surface area to volume ratio, which for a cube of side L is 6L² / L³ = 6 / L.
Frequencies are used for allele frequency, and densities for population density or stomatal density.
Reciprocals such as 1/T are used to turn time taken into a rate.

Scientific notation

Scientific notation a × 10ⁿ, where 1 ≤ a < 10 and n is an integer

n is positive for large numbers and negative for small numbers.

Significant figures

Significant figures the digits that are reliable and necessary to show the quantity
All non-zero digits and zeros between them are significant, so 4107 has 4 s.f.
Leading zeros are not significant, so 0.00079 has 2 s.f.
Trailing zeros in a whole number without a decimal point are not significant, so 640 has 2 s.f.
Trailing zeros after the decimal point are significant.

Rounding

Rounding if the next digit is 5 or more, round the previous digit up
Final answers should match the least precise data, usually 2-3 s.f., and intermediate values should not be rounded.

Approximation

Approximation a value close to the real value when an exact calculation is unnecessary or difficult

Estimation

Estimation a reasoned judgement of a value that cannot be measured directly, such as using a molecular clock to date divergence

Part 2 of 9SI units and prefixes

Quantity	SI base unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Temperature	kelvin	K
Amount of substance	mole	mol
Current	ampere	A
Luminous intensity	candela	cd

Prefixes

Prefixes kilo (10³), centi (10^-2), milli (10^-3), micro µ (10^-6) and nano (10^-9)
1 mm = 1000 µm.
1 cm³ = 1 ml and 1 dm³ = 1 litre.

Wording

Wording amount in science means moles, so refer to mass, volume or concentration instead

Symbols

Symbols use the correct symbol with each unit, such as m³ for cubic metres

Part 3 of 9Percentages, ratios and rates

Percentages, ratios and rates

Percentage change

(final value - initial value) / initial value × 100

It allows comparison when starting values differ, such as potato cylinders of different mass in osmosis.
A positive value is an increase and a negative value is a decrease.

Percentage difference relative to A

(A - B) / A × 100, used when one value is the reference

Symmetrical percentage difference

(A - B) / [(A + B) / 2] × 100, used when there is no baseline

Ratio

Ratio expressed in simplest form, such as a 3:1 phenotype ratio

Rate of change

Rate of change change in the dependent variable divided by change in the independent variable

Units are quantity per unit time, for example cm³ min^-1.

Proportionality

Proportionality directly proportional means that doubling x doubles y
Inversely proportional means that doubling x halves y.

Data types

Data types discrete data are countable separate values and continuous data are measurements that can take any value in a range
The data type affects the choice of graph and statistical test.

Part 4 of 9Averages and spread

Averages and spread

Mean

Mean the sum of values divided by the number of values, x̄ = Σx / n

It is affected by outliers.

Median

Median the middle value when data are in order, and it is not affected by outliers

Mode

Mode the most frequent value, useful for categorical data

Range

Range the largest value minus the smallest value

Standard deviation (SD)

Standard deviation (SD) a measure of spread about the mean, with the mean calculated first

A small SD means high precision.
About 68% of normally distributed values lie within ±1 SD and 95% within ±2 SD.

Standard error of the mean

Standard error of the mean an estimate of how far the sample mean is from the true mean, and it is always smaller than the SD

Interquartile range

Interquartile range the 75th percentile minus the 25th percentile

It describes the middle 50% of the data and is shown by the box of a box-and-whisker plot.

Part 5 of 9Uncertainty, error and error bars

Uncertainty

Uncertainty the ± range around a measurement within which the true value is expected to lie

It is normally half the smallest scale division, so a scale in 0.1 cm³ gives ±0.05 cm³.

Error

Error the difference between a measured value and the true value, caused by equipment or technique

Level of precision

Level of precision processed values should not be more precise than the raw data
All raw values should use the same number of decimal places.

Error bars

Error bars show uncertainty, range, SD or standard error on a graph
Know which one is plotted.
If SD error bars of two means overlap substantially, the difference is probably not significant.

Part 6 of 9Correlation and hypothesis testing

Correlation

Correlation an association between two variables, positive or negative

Correlation does not prove causation, as a third variable may be responsible.

Correlation coefficient r

Correlation coefficient r ranges from -1 to +1
Values near +1 or -1 show a strong correlation.
A value of 0 shows no correlation.

Coefficient of determination R2

Coefficient of determination R2 a measure of how well a line or curve fits, from 0 to 1
R² near 1 means the trend line is a good fit.

Null hypothesis

Null hypothesis there is no significant difference or association, so any difference is due to chance

Alternative hypothesis

Alternative hypothesis there is a significant difference or association

Significance level

Significance level if p < 0.05 there is less than a 5% probability that the difference is due to chance, so reject the null hypothesis

Part 7 of 9Statistical tests and indices

1t-test

t-test compares the means of two sets of data
Data must be roughly normally distributed and continuous with similar standard deviations.

2Chi-squared test

χ² = Σ[(O - E)² / E], where O is observed and E is expected frequency
It tests species association from quadrat data and genetic ratios.
It needs raw frequencies, not percentages, with expected values of at least 5.
Expected value in a contingency table = (row total × column total) / grand total.

3Simpson's reciprocal index

D = N(N - 1) / Σn(n - 1)
What the symbols mean: D = Simpson’s index of diversity · n = number of individuals of one species · N = total number of individuals of all species
It includes species richness and evenness, and a higher D means higher biodiversity.
Its minimum value is 1 and it has no units.

4Lincoln index

N = (M × C) / R, with population density equal to number divided by area

5Magnification

Magnification image size divided by actual size

6Logarithms

Logarithms bacterial growth is exponential, so log₁₀ of numbers against time gives a straight line in the log phase
pH is also a logarithmic scale.

Part 8 of 9Drawing graphs

Drawing graphs

Graph types

Graph types bar charts, histograms, scatter graphs, line or curve graphs, logarithmic graphs, pie charts and box-and-whisker plots

Sketch graph

Sketch graph shows a qualitative trend, often proportional or inversely proportional

Raw data tables

Raw data tables independent variable in the first column
Headings give the quantity, unit and uncertainty.
Units are not repeated in the body of the table.

Plotting rules

Independent variable on the x-axis and dependent variable on the y-axis.
Label both axes with units.
Choose a linear scale that uses more than half of the grid.
Plot points accurately in pencil.

Line of best fit

Line of best fit a straight line or smooth curve showing the trend, with points evenly spread on each side
It does not have to pass through every point and is not drawn dot-to-dot unless intermediate values cannot be assumed.
It goes through the origin only if the data allow it.

Line graphs

Line graphs for continuous data

Bar charts

Bar charts for grouped or categorical data

Part 9 of 9Reading graphs

1Gradient of a straight line

Δy / Δx

2Tangent

Tangent a straight line that just touches a curve at one point, so its gradient is the rate at that point
The initial rate of reaction is the gradient of the tangent at t = 0.

3Changing gradient

Changing gradient a curve has many gradients, showing how the rate changes

4Intercepts

Intercepts the points where a line or curve crosses an axis

5Maxima and minima

Maxima and minima peaks where the gradient goes from positive through 0 to negative, and troughs where it goes from negative through 0 to positive

6Interpolation

Interpolation reading values between plotted points from the line of best fit

7Extrapolation

Extrapolation extending the line beyond the data, which is less reliable because the trend may not continue

8Biology-specific graphics

Biology-specific graphics dichotomous keys, food chains and webs, pyramids of energy and pedigree charts

Key terms

Term	Meaning
Mean	The sum of all values divided by the number of values.
Standard deviation	A measure of the spread of data about the mean.
Null hypothesis	The statement that there is no significant difference or association, so differences are due to chance.
Percentage change	(final value - initial value) / initial value × 100.
Scientific notation	A number written as a × 10ⁿ, with 1 ≤ a < 10.
Correlation	An association between two variables that does not prove cause.
Simpson's reciprocal index	A diversity index, D = N(N - 1) / Σn(n - 1), that rises with richness and evenness.
Tangent	A straight line touching a curve at one point whose gradient gives the rate there.

Avoid these mistakes

Rounding intermediate values — round only the final answer

Using percentages in a chi-squared test — use raw frequencies

Saying correlation proves cause — a third variable may explain the link

Confusing uncertainty with error — uncertainty is the ± range and error is the difference from the true value

Last updated: 18 October 2026