Pearson Correlation
x_i and y_i are paired values, σ_x and σ_y are standard deviations, Cov is covariance.
r = Cov(x, y) / (σ_x σ_y) IB Diploma Programme 2026
📊 IBDP Maths AI Formula Sheet
Data, finance, and modelling equations for SL students with HL analytics and calculus add-ons clearly marked.
Statistics Financial Math Modelling
Command Key Modelling Relationships
AI focuses on technology-backed modelling. This sheet emphasises interpretation statements so you can justify calculator output and score reasoning marks.
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Finance & annuity reminders
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Regression parameter meaning
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Probability distributions
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HL calculus for modeling rates
Data Analysis & Regression (SL Core)
Regression parameters, correlation interpretation, and spread measures that underpin calculator-based IA style tasks.
Least Squares Line
Line of best fit ŷ = a + bx, where b is slope and a intercept.
Slope
b = r (σ_y / σ_x) Intercept
a = ȳ − b x̄ Coefficient of Determination
Explains percentage of variation in y explained by the model.
R² = r² Standard Deviation (sample)
x_i data points, x̄ sample mean, n sample size.
s = √[ Σ (x_i − x̄)² / (n − 1) ] Topic Focus
Correlation Storytelling
- After calculating r, comment on strength, direction, and context (e.g., moderate positive link).
- Flag potential outliers and mention they may weaken correlation reliability.
Least Squares Line
- Write ŷ = a + bx with parameter values before substituting test x-values.
- Clarify interpolation vs. extrapolation when predicting.
Spread & z-values
- Quote whether you are using population or sample standard deviation (n vs. n−1).
- State what a positive vs. negative z-score means for the context variable.
Financial Mathematics (SL Core)
Compound growth, annuities, loans, and depreciation models frequently tested on Paper 2 calculator sections.
Compound Interest
P principal, r periodic rate, n number of compounding periods.
A = P (1 + r)^n Annuity Future Value
PMT regular payment, i interest per period, n number of payments.
FV = PMT · [ (1 + i)^n − 1 ] / i Loan Amortisation Payment
PV loan principal, i rate per period, n total payments.
PMT = PV · [ i (1 + i)^n ] / [ (1 + i)^n − 1 ] Depreciation (reducing balance)
V₀ initial value, k depreciation rate per year, t years elapsed.
V_t = V₀ (1 − k)^t Topic Focus
Interest Rate Conventions
- Convert annual nominal rates to per-period rates before inserting into formulas.
- State compounding frequency when explaining why growth differs from simple interest.
Annuities vs. Loans
- Clarify whether you are solving for payment, number of periods, or present value.
- Include a sentence about cash flow direction (savings vs. repayments) to gain reasoning marks.
Depreciation Models
- Mention that reducing-balance depreciation assumes constant percentage loss each period.
- Compare linear vs. exponential depreciation when discussing suitability.
Probability & Distributions (SL Core)
Discrete models, conditional probability, and normal distribution density function used across Paper 2 modelling prompts.
Poisson Model
λ is average rate, r is number of events in interval.
P(X = r) = (λ^r e^{−λ}) / r! Normal Distribution
X ~ N(μ, σ²) is symmetric; use z = (x − μ)/σ to standardise.
f(x) = (1 / (σ √(2π))) e^{−(x − μ)² / (2σ²)} Conditional Probability
Event A given B.
P(A|B) = P(A ∩ B) / P(B) Expected Value (discrete)
x_i outcomes, p_i probabilities.
E(X) = Σ x_i p_i Topic Focus
Model Selection
- State why Poisson (events in interval) or binomial (fixed trials) applies before substituting.
- Use cumulative probabilities whenever the question asks for 'at most' or 'at least' scenarios.
Normal Distribution
- Sketch the curve and mark μ ± σ to contextualize probability statements.
- Convert raw values to z before using calculator, and cite symmetry when appropriate.
Conditional Probability
- Write the definition P(A|B) = P(A ∩ B)/P(B) explicitly before substituting values.
- Mention independence test P(A ∩ B) = P(A)P(B) when examining events.
Higher Level Modelling Toolkit
HL-only relationships for optimisation and advanced statistics.
Logistic growth, inference tests, numerical methods, and extended correlation tools used in HL Paper 3.
Logistic Growth Model
L carrying capacity, k growth rate, A constant from initial condition.
y = L / [1 + A e^{−kt}] Chi-squared Goodness of Fit
O_i observed frequency, E_i expected frequency.
χ² = Σ (O_i − E_i)² / E_i Numerical Derivative (Euler step)
For dy/dx = f(x, y), step size h, point (x_n, y_n).
y_{n+1} = y_n + h · f(x_n, y_n) Spearman Rank Correlation
d_i rank differences, n number of pairs.
r_s = 1 − [6 Σ d_i²] / [n(n² − 1)] Topic Focus
Logistic Growth
- State carrying capacity L and initial condition to solve for constant A before modelling.
- Compare early exponential behaviour vs. later saturation when interpreting graphs.
Chi-squared & Tests
- List degrees of freedom when quoting critical values from tables or calculator.
- Explain whether results support or reject the stated hypothesis.
Numerical Methods
- Mention step size h and show at least two iterations when using Euler's method.
- Highlight error accumulation and why smaller h improves accuracy.
Rank Correlation
- Show ranking tables before computing d_i differences to avoid transcription mistakes.
- Interpret r_s qualitatively (strong/weak) rather than just providing a number.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
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Contextualise r Values
After computing correlation, add a one-sentence interpretation about strength and direction to secure communication marks.
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Name Models in IA
State whether you used exponential, logistic, or piecewise models so moderators see precise mathematical reasoning.
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Pre-set Calculator Templates
Save annuity, amortisation, and poisson calculations as stored functions to reduce keying errors in Paper 2.
Need Guided Practice for Maths AI?
We pair technology fluency with written justifications so you can narrate what your GDC outputs mean, even on HL papers.
Formulas map to the 2026 Applications & Interpretation guide; HL-only tools are grouped separately for clarity.
Always include variable definitions when interpreting regression or distribution results to secure communication marks.