AP® Biology Equations and Formulas - Statistical Analysis and Probability - Chi-Square

Sep 29, 2026

AP® Biology Equations and Formulas compiles key quantitative relationships used in biology coursework and exam preparation. It summarizes statistical measures such as sample mean, sample standard deviation, standard error, degrees of freedom, and chi-square with p-values. The sheet also includes probability laws for mutually exclusive and independent events, Hardy-Weinberg population genetics equations, and core data statistics (mode, median, mean, range). Additional formulas cover metabolic and environmental calculations including growth rates, water potential, solute potential, temperature effects (Q10), Gibbs free energy, pH, and dilution.

LanguageEnglish

CategoryExam

FormatPDF

Pages92

Reading Time3 h 52 min

AccessFree to download

Statistical Analysis and Probability

Measures of Central Tendency and Dispersion

Degrees of Freedom and Chi-Square

Probability Laws

Population Genetics

Hardy-Weinberg Equations

Rate, Growth, and Energy

Rate and Population Growth Models

Temperature Coefficient Q10

Gibbs Free Energy

Water, Solute, and pH Calculations

Water Potential and Solute Potential

Geometry and Solution Calculations

Surface Area and Volume

Dilution

AP® BIOLOGY EQUATIONS AND FORMULAS | Mean
~~x~~ = 1~~n ~~ Âi 1 xi | | Statistical Analysis and Probability
Standard Deviation
s = x~~n 1 ~~ | | | | | | | | ~~x~~ = sample mean n = size of the sample
s = sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population)
o = observed results e = expected results
Degrees of freedom are equal to the number of distinct possible outcomes minus one. | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | | Standard Error of the Mean
SE~~x~~ = ~~ ~~sn | | | | | Chi-Square
c 2 = Â ~~ ~~(o~~ ~~~~ ~~e)2~~ ~~ | | | | | | | Chi-Square Table | | | | | | | | | | | | p
value | Degrees of Freedom | | | | | | | | | | | | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | | | | 0.05 | 3.84 | 5.99 | 7.82 | 9.49 | 11.07 | 12.59 | 14.07 | 15.51 | | | | 0.01 | 6.64 | 9.21 | 11.34 | 13.28 | 15.09 | 16.81 | 18.48 | 20.09 | | | | 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)
Hardy-Weinberg Equations | | | | | | | | | | | | | | | | | | | | | | Metric Prefixes
Factor Prefix Symbol
109 giga G
106 mega M
103 kilo k 10–2 centi c 10–3 milli m 10–6 micro μ 10–9 nano n 10–12 pico p | | p2 + 2pq + q2 = 1 p + q = 1 | | | p = frequency of the dominant allele in a population
q = frequency of the recessive allele in a population | | | | | | | | | 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 = r N
dt max
Logistic Growth
dNdt = rmaxN (~~ ~~KK-N)
Temperature Coefficient Q10
10
􀂧 k2 􀂷 T2 􀀐T1
Q10 􀀠 􀂨 􀂸
􀂩 k1 􀂹
Primary Productivity Calculation
mg O2 × 0.698 mL = mL O2
L mg L
mLLO~~2 ~~ × ~~ ~~0.536~~ ~~mg~~ ~~C~~m~~L~~ ~~O~~ ~~2~~ ~~fixed = mg~~ ~~CLfixed (at standard temperature and pressure) | dY = amount of changedt = change in time B = birth rate
D = death rate
N = population size K = carrying capacity
rmax = maximum per capita growth rate of population | | Water Potential ( Y ) Y = YP + YS
YP = pressure potential YS = solute potential
The water potential will be equal to the solute potential of a solution in an open container because the pressure potential of the solution in an open container is zero.
The Solute Potential of a Solution
YS = -iCRT
i = ionization constant (this is 1.0 for sucrose because sucrose does not ionize in water)
C = molar concentration
R = pressure constant
(R = 0.0831 liter bars/mole K) T = temperature in Kelvin (ºC + 273) | | --- | --- | --- | --- | | | T2 = higher temperature T1 = lower temperature k2 = reaction rate at T2
k1 = reaction rate at T1
Q 10 = the factor by which thereaction rate increases when the temperature is raised by ten degrees | | | | Surface Area and Volume Volume of a Sphere
V = 4~~3 ~~ pr3
Volume of a Rectangular Solid
V = 􀁁wh
Volume of a Right Cylinder
V = pr2h
Surface Area of a Sphere
A = 4pr2
Surface Area of a Cube
A = 6s2
Surface Area of a Rectangular Solid
A 􀀠 S surface area of each side | r = radius
􀁁 = length
h = height
w = width
s = length of one side of a cube
A = surface area
V = volume
􀀶 = sum of all | Dilution (used to create a dilute solution from a concentrated stock solution)
CiVi = CfVf
i = initial (starting) C = concentration of solute
f = final (desired) V = volume of solution | | | | | Gibbs Free Energy
ΔG = ΔH – TΔS
ΔG = ch

FAQs

What are the sample mean and sample standard deviation used for?

The sheet defines x̄ as the sample mean and s as the sample standard deviation, described as the sample-based estimate of the population standard deviation.

How do mutually exclusive and independent events relate to probability?

For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B).

What do water potential and solute potential represent?

Water potential is expressed as Y = YP + YS, where YP is pressure potential and YS is solute potential. The solute potential is given by YS = -iCRT.

Sep 29, 2026

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