Equation sheet

Statistical analysis and probability

$\bar{x}$ = sample mean · $n$ = sample size · $s$ = sample standard deviation (the sample-based estimate of the standard deviation of the population) · $o$ = observed results · $e$ = expected results · $\Sigma$ = sum of all · $\text{degrees of freedom}$ = the number of distinct possible outcomes minus one

• $\displaystyle \bar{x} = \frac{1}{n}\sum_{i=1}^{n}x_i$
  Mean7.2 Natural selection acts on phenotypes
• $\displaystyle s = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$
  Standard deviationnot shown yet
• $\displaystyle SE_{\bar{x}} = \frac{s}{\sqrt{n}}$
  Standard error of the meannot shown yet
• $\displaystyle \chi^2 = \sum\frac{(o-e)^2}{e}$
  Chi-square5.3 Testing a ratio: chi-square
• $P(A \text{ or } B) = P(A) + P(B)$
  Probability, mutually exclusive events5.3 Two genes, and the rules of probability
• $P(A \text{ and } B) = P(A) \times P(B)$
  Probability, independent events5.3 Two genes, and the rules of probability5.2 Independent assortment and random fertilization
• $\text{range} = \text{maximum} - \text{minimum}$
  Range (mode: the most frequent value; median: the middle value, with as many above as below)not shown yet

Hardy–Weinberg equilibrium

$p$ = frequency of allele 1 in a population · $q$ = frequency of allele 2 in a population

• $p^2 + 2pq + q^2 = 1 \qquad p + q = 1$
  Hardy–Weinberg equations7.5 Hardy–Weinberg equilibrium7.4 Population genetics

Rate and growth

$dY$ = amount of change · $dt$ = change in time · $B$ = birth rate · $D$ = death rate · $N$ = population size · $K$ = carrying capacity · $r_{\max}$ = maximum per capita growth rate of the population · $n$ = total number of organisms of a particular species · $N$ = total number of organisms of all species

• $\displaystyle \text{rate} = \frac{dY}{dt}$
  Rate3.2 Concentration and inhibitors8.3 Population growth
• $\displaystyle \frac{dN}{dt} = B - D$
  Population growth8.3 Population growth
• $\displaystyle \frac{dN}{dt} = r_{\max}N$
  Exponential growth8.3 Population growth
• $\displaystyle \frac{dN}{dt} = r_{\max}N\Big(\frac{K-N}{K}\Big)$
  Logistic growth8.4 Density limits growth
• $\displaystyle \text{diversity index} = 1 - \sum\Big(\frac{n}{N}\Big)^2$
  Simpson’s diversity index8.6 Biodiversity

Water potential

$\Psi$ = water potential · $\Psi_P$ = pressure potential · $\Psi_S$ = solute potential · $i$ = ionization constant (1.0 for sucrose, which does not ionize in water) · $C$ = molar concentration · $R$ = pressure constant · $T$ = temperature in kelvin

• $\Psi = \Psi_P + \Psi_S$
  Water potential (in an open container Ψ_P = 0, so Ψ = Ψ_S)2.7 Tonicity and water potential
• $\Psi_S = -iCRT$
  Solute potential of a solution2.7 Tonicity and water potential

Surface area and volume

$r$ = radius · $\ell$ = length · $w$ = width · $h$ = height · $s$ = length of one side of a cube · $SA$ = surface area · $V$ = volume

• $V = \tfrac43\pi r^3 \qquad SA = 4\pi r^2$
  Sphere2.2 Why cells are small
• $V = s^3 \qquad SA = 6s^2$
  Cube2.2 Why cells are small
• $V = \ell wh \qquad SA = 2\ell h + 2\ell w + 2wh$
  Rectangular solid2.2 Why cells are small
• $V = \pi r^2h \qquad SA = 2\pi rh + 2\pi r^2$
  Cylinder2.2 Why cells are small