AP Biology formula sheet: Every equation with a worked example

The AP (Advanced Placement) Biology formula sheet is a two-page reference you get on both sections of the exam. It has the statistical formulas, population growth equations, water potential, Simpson's Diversity Index and a few geometry basics that show up in FRQs (free-response questions). The sheet doesn't do the thinking for you, though. If you don't know when to reach for chi-square versus Hardy-Weinberg, having them printed in front of you doesn't help.

This guide walks through every equation on the current sheet with a worked example, so you know both what the symbols mean and when the exam is actually asking for that formula. If you want structured practice on the topics behind these equations, AP Biology on Cognito covers the underlying content in short video lessons and quiz questions.

The current College Board formula sheet is split into blocks covering statistical analysis and probability, Simpson's Diversity Index, rate and growth, water potential, pH, surface area and volume, and metric prefixes. There's also a chi-square critical values table and a short reminder of the laws of probability. Nothing else. If a question needs an equation that isn't on this list, either it's given in the stem or you're expected to reason without one.

A quick note: The sheet was revised in 2019, and older prep materials sometimes list dilution (C₁V₁ = C₂V₂), Q10, or primary productivity as being on it. None of those are on the current sheet. If a question needs one, the formula will be given in the stem.

The mean is the average value across your data set: Sum all values, divide by the number of values (n). Standard deviation (s) tells you how spread out the data is around that mean. Standard error of the mean (SEM) tells you how confident you can be in the mean itself, and it shrinks as your sample size grows.

The formulas on the sheet are:

Mean: x̄ = Σxi / n

Standard deviation: s = √[ Σ(xi − x̄)² / (n − 1) ]

Standard error of the mean: SEM = s / √n

Worked example: You measure the height of 5 bean seedlings and get 8, 9, 10, 11, 12 cm. The mean is 50 / 5 = 10 cm. The squared deviations are 4, 1, 0, 1, 4, summing to 10. Standard deviation = √(10 / 4) = √2.5 ≈ 1.58 cm. Standard error = 1.58 / √5 ≈ 0.71 cm. So you'd report a mean height of 10 cm ± 0.71 (SEM).

Chi-square (χ²) tests whether your observed data fit a predicted (expected) pattern. It shows up in genetics problems (does this cross fit a 9:3:3:1 ratio?) and in ecology (are these organisms distributed randomly?).

The formula is:

χ² = Σ [ (observed − expected)² / expected ]

Degrees of freedom (df) = number of categories − 1.

Worked example: You cross two heterozygous pea plants and expect a 3:1 ratio of tall to short. Out of 80 offspring you observe 55 tall, 25 short. Expected values are 60 tall, 20 short. So χ² = (55−60)²/60 + (25−20)²/20 = 25/60 + 25/20 = 0.417 + 1.25 = 1.67. With df = 1, the critical value at p = 0.05 is 3.84. Because 1.67 < 3.84, you fail to reject the null hypothesis. In plain English: The data fit the expected 3:1 ratio.

Hardy-Weinberg predicts allele and genotype frequencies in a population that isn't evolving. The two equations are:

p + q = 1 (allele frequencies)

p² + 2pq + q² = 1 (genotype frequencies, where p² = homozygous dominant, 2pq = heterozygous, q² = homozygous recessive)

Because you can usually see the recessive phenotype directly, most problems start there.

Worked example: In a population of 400 mice, 64 have white fur (recessive homozygous, aa). So q² = 64/400 = 0.16, which gives q = 0.4. Then p = 1 − 0.4 = 0.6. Frequency of heterozygotes (Aa) = 2pq = 2(0.6)(0.4) = 0.48, so 48% of the population, or 192 mice, are carriers.

Hardy-Weinberg is the most tested equation on the sheet in Unit 7 FRQs. It's worth knowing the five assumptions cold: No mutation, no migration, no natural selection, random mating, large population.

The generic rate equation is dY/dt = change in Y per unit time. That's it. In practice, the growth equations underneath are what you'll use.

Exponential growth (unlimited resources): dN/dt = rmax × N

Logistic growth (limited by carrying capacity K): dN/dt = rmax × N × (K − N) / K

N is population size, rmax is the maximum per-capita growth rate, K is carrying capacity.

Worked example: A yeast culture has N = 500 cells, rmax = 0.5 per hour, and K = 2,000 cells. Exponential growth would predict dN/dt = 0.5 × 500 = 250 cells per hour. Logistic growth accounts for crowding: dN/dt = 0.5 × 500 × (2000 − 500) / 2000 = 0.5 × 500 × 0.75 = 187.5 cells per hour. As N approaches K, that logistic rate drops toward zero, which is the whole point of a carrying capacity.

Water potential (Ψ, psi) tells you which way water will move by osmosis. Water always moves from higher to lower water potential. Pure water at atmospheric pressure has Ψ = 0.

Total water potential: Ψ = Ψp + Ψs

Solute potential: Ψs = −iCRT

Where i = ionization constant (1 for sucrose, 2 for NaCl, etc.), C = molar concentration, R = pressure constant (0.0831 liter·bar / mole·K), T = temperature in Kelvin (add 273 to Celsius).

Worked example: A 0.5 M sucrose solution at 27 °C in an open beaker has Ψp = 0 (open to air). Solute potential Ψs = −(1)(0.5)(0.0831)(300) = −12.47 bars. So total Ψ = 0 + (−12.47) = −12.47 bars. If you drop a potato core with Ψ = −8 bars into this solution, water flows from the potato (higher Ψ) into the solution (lower Ψ), and the potato loses mass.

Simpson's Diversity Index measures the diversity of a community, factoring in both the number of species and how evenly individuals are spread across them. The formula on the sheet is:

Diversity Index = 1 − Σ (n / N)²

Where n = number of individuals of one species and N = total number of individuals of all species. Values run from 0 (no diversity) to just under 1 (very high diversity).

Worked example: A meadow has 3 species of wildflower with counts of 40, 30 and 30, for N = 100. Σ(n/N)² = (0.4)² + (0.3)² + (0.3)² = 0.16 + 0.09 + 0.09 = 0.34. Diversity Index = 1 − 0.34 = 0.66. Compare that to a monoculture of 100 individuals from one species: (1.0)² = 1, so index = 0. The higher the value, the more diverse the community.

The surface area to volume ratio (SA:V) drives cell size, diffusion rate and heat exchange. Small cells have high SA:V, which is why they exchange materials efficiently. As cells grow, volume increases faster than surface area, and diffusion becomes the limiting factor.

The formulas on the sheet:

Cube: SA = 6s², V = s³

Rectangular solid: SA = 2(lw + lh + wh), V = lwh

Cylinder: SA = 2πr² + 2πrh, V = πr²h

Sphere: SA = 4πr², V = (4/3)πr³

Worked example: Compare a cube of side 2 cm to a cube of side 4 cm. Small cube: SA = 24, V = 8, ratio = 3:1. Large cube: SA = 96, V = 64, ratio = 1.5:1. The bigger cube has half the SA:V, so materials diffuse in and out half as efficiently per unit volume. That's why real cells stay small (or fold their membranes, or split).

The pH equation on the sheet is:

pH = −log[H+], and [H+] = 10^(−pH)

A one-unit change in pH is a tenfold change in hydrogen ion concentration. Worked example: A solution of pH 5 has [H+] = 10^(−5) = 1 × 10⁻⁵ M. A solution of pH 3 has [H+] = 10^(−3) = 1 × 10⁻³ M, which is 100 times more acidic than pH 5. This shows up on enzyme optimum and buffer questions.

The best practice for the formula sheet is timed FRQ work with the sheet open on the desk, exactly how you'll have it in the exam. Pull released FRQs from AP Central going back five years and sort them by which equation they use. You'll find that Hardy-Weinberg and chi-square between them account for most of the quantitative marks, with water potential appearing in maybe one paper in three.

If you want the underlying content behind these equations laid out cleanly, enzyme kinetics for rate, cell membranes for water potential, natural selection for Hardy-Weinberg, Cognito's AP Biology course has short lessons for each. And if you're taking multiple AP sciences, the full AP course catalogue covers Chemistry, Environmental Science and Psychology too.