AP Biology Formula Sheet

Sample mean

Add up every measurement and divide by the number of measurements $n$. The average value that describes the center of a data set.

Sample standard deviation

A measure of how spread out the data are around the mean. Larger $s$ means more variability. Note the denominator is $n-1$, not $n$, because it is a sample.

Standard error of the mean

How much the sample mean is expected to vary from the true population mean. Used to draw error bars: means whose error bars do not overlap are likely significantly different.

Chi-square statistic

Sum, over every category, of observed minus expected squared, divided by expected ($o$ = observed count, $e$ = expected count). A bigger $\chi^2$ means observations stray further from the prediction. Find degrees of freedom as $df = (\text{categories} - 1)$, then compare $\chi^2$ to the critical value at $p = 0.05$: if $\chi^2$ is greater than or equal to the critical value, reject the null hypothesis (the deviation is statistically significant). Otherwise, the data are consistent with the prediction.

Allele frequencies

For a gene with two alleles, the frequency of the dominant allele $p$ plus the frequency of the recessive allele $q$ accounts for every allele in the pool, so they sum to 1.

Genotype frequencies

Distributes the population into genotypes: $p^2$ is homozygous dominant, $2pq$ is heterozygous, and $q^2$ is homozygous recessive. The recessive phenotype frequency equals $q^2$, which is usually the easiest starting point. Equilibrium requires five conditions: no mutation, no gene flow (migration), no natural selection, random mating, and a very large population. A measured deviation from these predicted frequencies is evidence the population is evolving.

Water potential

Total water potential $\Psi$ is the pressure potential $\Psi_p$ plus the solute potential $\Psi_s$. Water moves from a region of higher (less negative) water potential to a region of lower (more negative) water potential.

Solute potential

Solute lowers water potential, so $\Psi_s$ is negative. Here $i$ is the ionization constant (1 for sucrose, 2 for NaCl), $C$ is molar concentration, $R$ is the pressure constant $0.0831\ \text{L·bar/(mol·K)}$, and $T$ is temperature in kelvin (K = °C + 273). More solute makes $\Psi_s$ more negative, pulling water in.

Gibbs free energy

Combines enthalpy change $\Delta H$, temperature $T$ (in kelvin), and entropy change $\Delta S$ to predict spontaneity. A negative $\Delta G$ is spontaneous and exergonic (releases energy); a positive $\Delta G$ is endergonic and needs an energy input, such as ATP coupling.

pH

The negative base-10 log of the hydrogen ion concentration. Each whole pH unit is a tenfold change in $[\text{H}^+]$: lower pH is more acidic. Central to enzyme activity and homeostasis questions.

Rate

The change in a measured quantity $\Delta y$ divided by the change in time $\Delta t$. The general "how fast" formula for enzyme assays, diffusion, primary productivity, and any slope-of-a-graph question.

Exponential growth

Population growth rate when resources are unlimited, where $N$ is population size and $r_{max}$ is the maximum per-capita growth rate. Produces a J-shaped curve that keeps steepening.

Logistic growth

Adds a carrying capacity $K$, the maximum population the environment can sustain. As $N$ approaches $K$ the $\frac{(K-N)}{K}$ term shrinks toward zero and growth levels off, giving the S-shaped curve.

Sphere

Volume and surface area of a sphere of radius $r$. The go-to shape for modeling a cell.

Cube

Volume and surface area of a cube of side length $s$. Common in the classic agar-block diffusion lab.

Cylinder

Volume and surface area of a cylinder of radius $r$ and height $h$. The surface area includes the curved wall plus the two circular ends.

Rectangular solid

Volume of a rectangular solid with length $l$, width $w$, and height $h$. Use these shapes to compute a surface-area-to-volume ratio: as a cell grows, volume rises faster than surface area, so a large cell cannot exchange materials fast enough across its membrane. That constraint is why cells stay small, divide, or evolve folds and villi.