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AP Biology — Equations & Formulas

8 sections · 34 entries · print it and keep it beside your practice sets

AP Biology hands you a short formula sheet, and roughly a quarter of the exam is quantitative. The sheet gives you the equation but never tells you which variable is which — so the graded skill is reading a scenario and mapping it onto one of these lines, with units.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

Statistical analysis & probability

Quantity / relationshipWhat it means & how to use it

x̄

x̄ = (Σ xᵢ) / n

Sample mean of n measurements

s = √( Σ(xᵢ − x̄)² / (n − 1) )

Sample standard deviation

Divide by n − 1, not n. Measures spread of the individuals.

SE_x̄

SE_x̄ = s / √n

Standard error of the mean

Measures uncertainty in the mean. Error bars of ±2 SE approximate a 95% confidence interval: if two sets of bars do not overlap, the means differ significantly.

χ²

χ² = Σ ( (o − e)² / e )

Chi-square goodness of fit; o = observed, e = expected

df = (number of categories) − 1

Degrees of freedom for a chi-square test

Compare χ² to the critical value at p = 0.05. Larger than critical ⇒ reject the null hypothesis.

Multiplication rule

P(A and B) = P(A) × P(B)

Probability of two independent events both happening

Addition rule

P(A or B) = P(A) + P(B)

Probability of either of two mutually exclusive events

Mode / median / range

mode = most frequent value; median = middle value; range = max − min

Other descriptive measures

Genetics & evolution

Quantity / relationshipWhat it means & how to use it

Hardy–Weinberg

p² + 2pq + q² = 1

Genotype frequencies for a two-allele locus

Allele frequencies

p + q = 1

p = frequency of the dominant allele, q = recessive

Reading it

p² is the homozygous dominant frequency, 2pq the heterozygous frequency, q² the homozygous recessive frequency

Start from q² — the recessive phenotype is the only genotype you can count directly. Then q = √(q²).

Conditions

Equilibrium requires a large population, no mutation, no migration, no selection, and random mating

If observed genotype frequencies differ from the predicted ones, one of these five assumptions is being violated.

Rates & population growth

Quantity / relationshipWhat it means & how to use it

Rate

rate = ΔY / Δt

Any rate is a change per unit time

dN/dt

dN/dt = B − D

Population growth from births B and deaths D

Exponential growth

dN/dt = r_max N

r_max is the maximum per-capita growth rate

Logistic growth

dN/dt = r_max N ( (K − N) / K )

K is carrying capacity

Growth RATE is fastest at N = K/2, and approaches zero as N approaches K.

Q₁₀

Q₁₀ = (k₂ / k₁)^( 10 / (t₂ − t₁) )

Temperature coefficient: how much a rate changes per 10 °C

k₂ is the rate at the higher temperature t₂. Q₁₀ near 2–3 is typical for enzyme-catalysed reactions.

Primary productivity (dissolved oxygen method)

Quantity / relationshipWhat it means & how to use it

mL O₂/L

mg O₂/L × 0.698 = mL O₂/L

Convert oxygen mass to volume

mg C fixed/L

mL O₂/L × 0.536 = mg carbon fixed/L

Convert oxygen volume to carbon fixed

GPP, NPP

NPP = light bottle − initial; respiration = initial − dark bottle; GPP = light bottle − dark bottle

Gross and net primary productivity from light/dark bottles

Surface area & volume

Quantity / relationshipWhat it means & how to use it

S/V

S/V = surface area / volume

Surface-area-to-volume ratio

As a cell grows, volume rises with the cube of length but surface area only with the square — so S/V falls and diffusion can no longer supply the interior. This is the answer to “why are cells small?”.

Cube

V = s³, A = 6 s²

Side length s

Rectangular solid

V = ℓ w h, A = 2(ℓh + ℓw + wh)

Length ℓ, width w, height h

Cylinder

V = π r² h, A = 2π r h + 2π r²

Radius r, height h (closed both ends)

Sphere

V = (4/3) π r³, A = 4π r²

Radius r

Water potential

Quantity / relationshipWhat it means & how to use it

Ψ = Ψ_p + Ψ_s

Water potential is pressure potential plus solute potential

Water always moves from higher (less negative) Ψ to lower (more negative) Ψ.

Ψ_s

Ψ_s = − i C R T

Solute potential; i = ionization constant, C = molarity, R = 0.0831 L·bar/(mol·K), T in kelvin

Ψ_pure

Ψ of pure water in an open container = 0 bars

Reference point

Adding solute makes Ψ_s negative; adding pressure makes Ψ_p positive.

i = 1 for sucrose; i = 2 for NaCl; i = 3 for CaCl₂

Ionization constant

Energy, pH & dilution

Quantity / relationshipWhat it means & how to use it

ΔG

ΔG = ΔH − T ΔS

Gibbs free energy change; T in kelvin

ΔG < 0 is exergonic and spontaneous; ΔG > 0 is endergonic and must be coupled to an exergonic reaction such as ATP hydrolysis.

pH = − log₁₀ [H⁺]

Acidity

Dilution

C_i V_i = C_f V_f

Making a working solution from a stock; C = concentration, V = volume

Magnification

total magnification = objective power × ocular power

Microscopy

Metric prefixes (also supplied)

Quantity / relationshipWhat it means & how to use it

Prefixes

giga G 10⁹ · mega M 10⁶ · kilo k 10³ · centi c 10⁻² · milli m 10⁻³ · micro μ 10⁻⁶ · nano n 10⁻⁹ · pico p 10⁻¹²

Factor by which the base unit is multiplied

Practise with the sheet open Exam-skill drills Cheatsheet