Formula Sheet for Algebra

Formula Sheet for Algebra Statistical Analysis and Probability Standard Error 𝑆𝐸𝑥̅ = 𝑆 √𝑛 Mean 𝑛 1 𝑥̅ = ∑ 𝑥𝑖 𝑛 𝑖=1 Standard Deviation 𝑆=√ ∑(𝑥𝑖 − 𝑥̅ )2 𝑛−1 Chi-Square 𝜒2 = ∑ (𝑜 − 𝑒)2 𝑒 s = sample standard deviation (i.e., the sample based estimate of the standard deviation of the population) 𝑥̅ = mean N = size of the sample O = observed individuals with observed genotype e = expected individuals with observed genotype Degrees of freedom equals the number of distinct possible outcomes minus one. Chi-Square Table p 0.05 0.01 1 3.84 6.64 2 5.99 9.32 3 7.82 11.34 Degrees of Freedom 4 5 9.49 11.07 13.28 15.09 6 12.59 16.81 Laws of Probability If A and B are mutually exclusive, then P(A or B) = 𝑃(𝐴) + 𝑃(𝐵) If A and B are independent, then P(A and B) = 𝑃(𝐴) × 𝑃(𝐵) 7 14.07 18.48 8 15.51 20.09 Hardy-Weinberg Equations 𝑝2 + 2𝑝𝑞 + 𝑞 2 = 1 𝑝=𝑞=1 p = frequency of the dominant allele in a population q = frequency of the recessive allele in a population Metric Prefixes Factor 109 106 103 10−2 10−3 10−6 10−9 10−12 Prefix giga mega kilo centi milli micro nano pico Symbol G M k c m 𝜇 n p Mode = value that occurs most frequently in a data set Median = middle value that separates the greater and lesser halves of a data set Mean = sum of all data points divided by number of data points Range = value obtained by subtracting the smallest observation (sample minimum) from the greatest (sample maximum) Rate and Growth Rate 𝑑𝑌/𝑑𝑡 Population Growth 𝑑𝑁/𝑑𝑡 = 𝐵 – 𝐷 Exponential Growth 𝑑𝑁 = 𝑟𝑚𝑎𝑥 𝑁 𝑑𝑡 Logistic Growth 𝑑𝑁 𝐾−𝑁 = 𝑟𝑚𝑎𝑥 𝑁 ( ) 𝑑𝑡 𝐾 dY = amount of change t = time B = birth rate D = death rate N = population size K = carrying capacity 𝑟𝑚𝑎𝑥 aximum per capita growth rate of population Water Potential (Ψ) Ψ = Ψ𝑝 + Ψ𝑠 Ψ𝑝 = pressure potential Ψ𝑠 = solute potential The water potential will be equal to the solute potential of a solution in an open container, since the pressure potential of the solution in an open container is zero. The Solution Potential of the Solution Ψ𝑠 = −𝑖𝐶𝑅𝑇 I = ionization constant (For sucrose this is 1.0 because sucrose does not ionize in water) C = molar concentration R = pressure constant (R= 0.0831 liter bars/mole K) T= temperature in Kelvin (273 = ∘C) Surface Area and Volume Volume of Sphere 𝑉 = 4/3𝜋𝑟 3 Volume of a cube (or square column) 𝑉=𝑙𝑤ℎ Volume of a column 𝑉 = 𝜋𝑟 2 ℎ Surface area of a sphere 𝐴 = 4𝜋𝑟 2 Surface area of a cube 𝐴 = 6𝑎 Surface area of a rectangular solid 𝐴 = ∑ (surface area of each side) r = radius l = length h = height w = width A = surface area V = volume ∑ = Sum of all A = surface area of one side of the cube Dilution – used to create a dilute solution from a concentrated stock solution 𝐶𝑖 𝑉𝑖 = 𝐶𝑓 𝑉𝑓 i = initial (starting) f = final (desired) C = concentration of solute V = volume of solution Gibbs Free Energy Δ𝐺 = Δ𝐻 = 𝑇Δ𝑆 Δ𝐺 = change in Gibbs free energy Δ𝑆 = change in entropy Δ𝐻 = change in enthalpy T = absolute temperature (in Kelvin) 𝒑𝑯 = − log[𝐻`]