Free SOA Exam PA (Predictive Analytics) Formula Sheet (2026)

Every Exam PA formula you need on the test, grouped by topic and rendered with full math notation. 53 formulas across 5 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

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All Exam PA Formulas

Predictive Analytics Problem Definition 7 items
Profit per policy
Profit=P−L−E\text{Profit} = P - L - E, P = premium (price) per policy, L = expected loss (claim severity times frequency), E = expense or acquisition cost per policy
Value as price minus cost
V=P−CV = P - C, V = value of the item, P = price (e.g. sale price per building), C = cost (e.g. construction cost per building)
Base rate of a binary target
p=neventNp = \dfrac{n_{event}}{N}, p = base rate (positive-class proportion), neventn_{event} = number of positive-class records, N = total records
Majority-class (no-information) accuracy
accmaj=1−pacc_{maj} = 1 - p, accmajacc_{maj} = accuracy of always predicting the majority class, p = base rate of the rare (positive) class
Business impact of a predictive model
Impact=V×f\text{Impact} = V \times f, V = dollar value of the decision the prediction improves, f = how often that decision is made
Additive error data-generating model
y=f(x)+εy = f(x) + \varepsilon, y = observed response, f(x) = true signal function, ε\varepsilon = random noise with variance Var⁡(ε)\operatorname{Var}(\varepsilon)
Bias-variance decomposition of expected test MSE
E[(y0−f^(x0))2]=Var⁡(f^(x0))+[Bias⁡(f^(x0))]2+Var⁡(ε)E\left[(y_0 - \hat f(x_0))^2\right] = \operatorname{Var}(\hat f(x_0)) + \left[\operatorname{Bias}(\hat f(x_0))\right]^2 + \operatorname{Var}(\varepsilon), f^\hat f = fitted model, x0x_0 = test point, y0y_0 = true response, Var⁡(ε)\operatorname{Var}(\varepsilon) = irreducible error
Data Exploration and Visualization 8 items
Variance of a factor-level estimate
Var⁡(μ^level)∝1nlevel\operatorname{Var}(\hat\mu_{\text{level}}) \propto \frac{1}{n_{\text{level}}}, μ^level\hat\mu_{\text{level}} = estimated mean for that level, nleveln_{\text{level}} = number of rows in that level
Minority rows needed to oversample to a target share
f=pN1−pf = \frac{pN}{1-p} from p=fN+fp = \frac{f}{N+f}, p = target minority proportion, N = majority-class row count, f = minority rows after oversampling
Boxplot outlier fences
lower=Q1−1.5×IQR, upper=Q3+1.5×IQR\text{lower} = Q1 - 1.5\times\text{IQR},\ \text{upper} = Q3 + 1.5\times\text{IQR}, Q1 = first quartile, Q3 = third quartile, IQR = interquartile range
Interquartile range
IQR=Q3−Q1\text{IQR} = Q3 - Q1, Q1 = 25th percentile (first quartile), Q3 = 75th percentile (third quartile)
Shared variance between two predictors
r2=rxy2r^2 = r_{xy}^2, rxyr_{xy} = Pearson correlation between the two variables; r2r^2 = proportion of one variable's variation explained by the other
Sample variance
s2=1n−1∑i=1n(xi−xˉ)2s^{2} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}, xix_i = observation, xˉ\bar{x} = sample mean, n = sample size
Pearson correlation coefficient
rxy=∑(xi−xˉ)(yi−yˉ)∑(xi−xˉ)2∑(yi−yˉ)2r_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2}\sqrt{\sum (y_i - \bar{y})^2}}, x,y = paired values, xˉ,yˉ\bar{x},\bar{y} = means, i = observation index
Sample standard deviation
s=s2=1n−1∑i=1n(xi−xˉ)2s = \sqrt{s^{2}} = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^{2}}, s2s^2 = sample variance, xix_i = observation, xˉ\bar{x} = mean, n = sample size
Data Transformations and Unsupervised Learning Techniques 12 items
Standardized feature (z-score for clustering)
z=x−μσz = \dfrac{x - \mu}{\sigma}, x = raw feature value, μ = feature mean, σ = feature standard deviation
Complete linkage between-cluster distance
d(A,B)=max⁡a∈A, b∈B∥a−b∥d(A,B) = \max_{a \in A,\, b \in B} \lVert a - b \rVert, A, B = clusters, a, b = member points, distance = farthest pair
Proportion of variance explained by a principal component
PVEm=Var(zm)∑jVar(zj)\text{PVE}_m = \frac{\text{Var}(z_m)}{\sum_j \text{Var}(z_j)}, PVEm = proportion for PC m, Var(zm) = variance of PC m, denominator = total variance across all PCs
Variance of a principal component from its standard deviation
Var(zm)=sm2\text{Var}(z_m) = s_m^2, Var(zm) = variance captured by PC m, sm = reported standard deviation of PC m; total variance = number of standardized variables
Within-cluster sum of squares (K-means objective)
WCSS=∑k=1K∑x∈Ck∥x−μk∥2\text{WCSS} = \sum_{k=1}^{K} \sum_{x \in C_k} \lVert x - \mu_k \rVert^{2}, K = number of clusters, CkC_k = cluster k, x = observation, μkμ_k = mean vector of cluster k
Single linkage between-cluster distance
d(A,B)=min⁡a∈A, b∈B∥a−b∥d(A,B) = \min_{a \in A,\, b \in B} \lVert a - b \rVert, A, B = clusters, a, b = member points, distance = closest pair
First principal component score
z1=ϕ11x1+ϕ21x2+ϕ31x3z_1 = \phi_{11}x_1 + \phi_{21}x_2 + \phi_{31}x_3, z1 = PC1 score, φj1 = loading of variable j on PC1, xj = standardized feature j
Loading vector normalization constraint
∑jϕjm2=1\sum_{j} \phi_{jm}^2 = 1, φjm = loading of variable j on PC m; squared loadings of a component sum to one (unit-length loading vector)
Number of dummy columns for a categorical variable
d=k−1d = k - 1, k = number of category levels, d = number of dummy (indicator) columns needed to avoid redundancy in a GLM
Log transform for a skewed non-negative variable
x′=ln⁡(x+1)x' = \ln(x + 1), x = raw strictly non-negative value, x' = transformed value; the +1 lets x = 0 be handled
Min-max scaling
xmm=x−xmin⁡xmax⁡−xmin⁡x_{\text{mm}} = \dfrac{x - x_{\min}}{x_{\max} - x_{\min}}, x = raw value, xmin⁡x_{\min} = minimum, xmax⁡x_{\max} = maximum, xmmx_{\text{mm}} = scaled value in [0,1]
Z-score standardization
z=x−xˉsz = \dfrac{x - \bar{x}}{s}, x = raw value, xˉ\bar{x} = sample mean, s = sample standard deviation, z = standardized value
Generalized Linear Models 15 items
Poisson log-link model with a log-exposure offset
log⁡(E[count])=log⁡(exposure)+β0+β1x1+⋯\log(E[\text{count}]) = \log(\text{exposure}) + \beta_0 + \beta_1 x_1 + \cdots, exposure = units of exposure (offset, coef fixed at 1), β0\beta_0 = intercept, βj\beta_j = slope, xjx_j = predictor
Weighted least squares objective
min⁡β∑iwi(yi−y^i)2\min_{\beta} \sum_{i} w_i (y_i - \hat{y}_i)^2, wiw_i = weight of row i, yiy_i = observed response, y^i\hat{y}_i = fitted value, β\beta = coefficients
Group-size weighted mean of grouped rates
yˉw=∑iniyi∑ini\bar{y}_w = \dfrac{\sum_i n_i y_i}{\sum_i n_i}, nin_i = group size (weight) of row i, yiy_i = group-average rate for row i
Expected count as exposure times a fitted rate
E[count]=exposure×eβ0+β1x1E[\text{count}] = \text{exposure} \times e^{\beta_0 + \beta_1 x_1}, exposure = units of exposure, eβ0+β1x1e^{\beta_0 + \beta_1 x_1} = rate per unit exposure, β0\beta_0 = intercept, β1\beta_1 = slope, x1x_1 = predictor
GLM log link function
log⁡(μ)=β0+β1x1+…\log(\mu) = \beta_0 + \beta_1 x_1 + \dots, so eβ1e^{\beta_1} is the factor by which a one-unit rise in x1x_1 scales μ\mu; μ\mu = expected response, β\beta = coefficients
Pearson residual for a GLM
ri=yi−μ^iV(μ^i)r_i = \frac{y_i - \hat{\mu}_i}{\sqrt{V(\hat{\mu}_i)}}, yiy_i = observed, μ^i\hat{\mu}_i = fitted mean, V(μ^i)V(\hat{\mu}_i) = variance function at the fitted mean
Lognormal mean back-transformation (smearing correction)
E^[Y]=eμ^log⁡ eσ2/2\hat{E}[Y] = e^{\hat{\mu}_{\log}}\, e^{\sigma^2/2}, μ^log⁡\hat{\mu}_{\log} = fitted value on log scale, σ2\sigma^2 = residual variance on log scale
Estimated dispersion parameter
ϕ^=1n−p∑i=1n(yi−μ^i)2V(μ^i)\hat{\phi} = \frac{1}{n-p}\sum_{i=1}^{n}\frac{(y_i - \hat{\mu}_i)^2}{V(\hat{\mu}_i)}, n = observations, p = parameters, yiy_i = observed, μ^i\hat{\mu}_i = fitted mean, VV = variance function
Elastic net penalized regression objective
min⁡β12n∑i(yi−β0−∑jβjxij)2+λ[1−α2∑jβj2+α∑j∣βj∣]\min_{\beta} \frac{1}{2n}\sum_i (y_i - \beta_0 - \sum_j \beta_j x_{ij})^2 + \lambda[\frac{1-\alpha}{2}\sum_j \beta_j^2 + \alpha\sum_j |\beta_j|], λ = penalty strength, α = L1/L2 mix, β = coefficients
Variance-inflation factor for a predictor
VIFj=11−Rj2\text{VIF}_j = \frac{1}{1 - R_j^2}, Rj2R_j^2 = R-squared from regressing predictor j on all other predictors
One-standard-error threshold for lambda.1se selection
threshold=MSEmin⁡+SEmin⁡\text{threshold} = \text{MSE}_{\min} + \text{SE}_{\min}, MSEminMSE_{min} = minimum CV mean squared error, SEminSE_{min} = its standard error; lambda.1se = largest λ with CV MSE ≤ threshold
Log-link multiplicative effect of a coefficient
μnew=μ×eβ\mu_{new} = \mu \times e^{\beta} per unit, μ = expected response, β = coefficient on the link scale; eβ−1e^{\beta}-1 is the percent change
Logit-link odds ratio from a coefficient
OR=eβOR = e^{\beta}, OR = multiplicative change in the odds of the event per unit increase, β = logit-link coefficient
Interaction-adjusted slope of a predictor at a level
slope of x1 at level ℓ=β1+βint,ℓ\text{slope of } x_1 \text{ at level } \ell = \beta_1 + \beta_{\text{int},\ell}, β1 = main coefficient, βintβ_{int},ℓ = interaction coefficient (zero for baseline level)
GLM linear predictor with link function
g(μ)=η=β0+β1x1+…g(\mu) = \eta = \beta_0 + \beta_1 x_1 + \dots, g = link function, μ = expected response, η = linear predictor, β = coefficients, x = predictors
Tree-Based Models 11 items
Default mtry for random forest classification
mtry=pm_{try} = \sqrt{p}, p = total number of predictors, mtrym_{try} = predictors sampled at each split
Out-of-bag probability a row is left out of a bootstrap sample
P=(1−1/n)n→e−1≈0.368P = (1-1/n)^{n} \to e^{-1} \approx 0.368, n = number of training rows, P = chance a given row is omitted from one bootstrap (~37%)
Default mtry for random forest regression
mtry=p/3m_{try} = p/3, p = total number of predictors, mtrym_{try} = predictors sampled at each split
Gradient boosting additive update
Fm(x)=Fm−1(x)+λ hm(x)F_m(x) = F_{m-1}(x) + \lambda\, h_m(x), F = ensemble prediction, m = boosting round, λ = learning rate (shrinkage), hmh_m = tree fit to current pseudo-residuals
k-fold cross-validation error
CV error=1k∑i=1kError(foldi)\text{CV error} = \frac{1}{k}\sum_{i=1}^{k} \text{Error}(\text{fold}_i), k = number of folds, foldifold_i = i-th held-out validation fold
Variance of an averaged tree ensemble
Var⁡=ρσ2+1−ρBσ2\operatorname{Var} = \rho\sigma^{2} + \frac{1-\rho}{B}\sigma^{2}, ρ = pairwise tree correlation, σ² = single-tree prediction variance, B = number of trees
Learning-rate to tree-count inverse tradeoff
nnew≈nold×λoldλnewn_{new} \approx n_{old} \times \frac{\lambda_{old}}{\lambda_{new}}, n = number of trees, λ = learning rate; halving λ roughly doubles the trees needed
Cost-complexity pruning criterion
Rcp(T)=R(T)+cp⋅∣T∣R_{cp}(T) = R(T) + cp \cdot |T|, R(T) = tree total error, cp = complexity parameter, |T| = number of terminal nodes (leaves)
Gini impurity of a node
G=1−∑kpk2G = 1 - \sum_{k} p_k^{2}, G = Gini impurity, pkp_k = proportion of class k in the node; G = 0 at a pure node (rpart default)
Entropy of a node
H=−∑kpklog⁡2pkH = -\sum_{k} p_k \log_2 p_k, H = entropy in bits, pkp_k = proportion of class k in the node; H = 0 at a pure node
Root mean square error
RMSE=1n∑i(yi−y^i)2\text{RMSE} = \sqrt{\frac{1}{n}\sum_i (y_i - \hat{y}_i)^2}, yiy_i = actual value, y^i\hat{y}_i = predicted value, n = number of observations
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