D00 / Foundation layer — under all 49 territories
Financial Mathematics, Statistics, and Data Foundations
The quantitative language, statistical judgment, and implementation discipline every later domain assumes.
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Every card below is included in the current governed release with its article, sources, Python, TypeScript, and available labs.
Search the released libraryMathematical Language and Quantitative Reasoning
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Variables, Constants, Expressions, and Equations
Learn to read a financial formula as a small sentence about money.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Functions, Domains, Ranges, and Graphs
See a function as a rule that turns an allowed input into one output.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Ratios, Proportions, Rates, and Percentages
Compare amounts fairly by making the denominator and the unit visible.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Percentage Change, Percentage Points, and Basis Points
Use three different languages for change without mixing them up.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Exponents, Roots, and Logarithms
Understand growth, undoing a power, and asking “which power?” in plain language.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Summation, Products, and Index Notation
Read a compact list instruction as “add these” or “multiply these.”Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Linear Equations and Systems
Find unknown amounts when several simple balances must be true at once.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Inequalities, Bounds, and Constraints
Describe what is allowed when the answer is a safe range, not one exact number.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Units, Dimensions, Scale, and Normalization
Keep the meaning attached to every number, even when the scale changes.Financial Mathematics, Statistics, and Data Foundations / Mathematical Language and Quantitative Reasoning
Rounding, Precision, Tolerance, and Significant Digits
Show only as many digits as the information can honestly support.Financial Arithmetic, Time Value, and Returns
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Simple Interest
See how a fixed rate adds interest to the original amount without adding yesterday's interest to today's base.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Compound Interest
See how each period can earn interest on the amount already built up, not only on the starting amount.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Present Value and Future Value
Translate money at one date into an equivalent amount at another date using a stated growth rate.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Discount Factors and Discount Rates
Use one small factor to show how much of a future amount is worth at an earlier date.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Cash-Flow Timelines and Net Present Value
Place each cash flow on a timeline, translate each one to the same date, and add the signed values.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Simple Return
Measure the change from a starting investment to an ending value, including cash received along the way.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Log Return
Use the natural logarithm of the ending value divided by the starting value to describe multiplicative change on an additive scale.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Holding-Period and Cumulative Return
Combine several period returns by multiplying their growth factors, then compare the result with the start of the whole path.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Arithmetic versus Geometric Average Return
Use a simple average to describe a typical one-period rate, and a geometric average to summarize a compounded path.Financial Mathematics, Statistics, and Data Foundations / Financial Arithmetic, Time Value, and Returns
Return Annualization and Deannualization
Translate a return over one length of time into an equivalent yearly rate, or reverse that translation, using compounding.Data, Variables, Samples, and Measurement
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Observations, Entities, Variables, and Datasets
An entity is the thing we study, an observation is one recorded case, a variable is a named property, and a dataset is an organized collection of observations.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Numeric, Categorical, Ordinal, and Binary Variables
Numeric values support arithmetic, categorical values name groups, ordinal values have an order but not a trustworthy distance, and binary values represent two states.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Population, Sample, Census, and Sampling Frame
The population is the full group of interest, a sample is the observed subset, a census attempts to measure every population member, and a sampling frame is the operational list from which selection occurs.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Cross-Sectional, Time-Series, Panel, and Event Data
Cross-sectional data varies mainly across entities, time-series data follows one unit across time, panel data follows multiple entities across time, and event data records occurrences at event times.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Identifiers, Keys, Joins, and Data Grain
An identifier names an entity, a key uniquely identifies a record at the declared grain, and a join combines records according to matching keys.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Timestamps, Time Zones, Calendars, and Observation Time
A timestamp needs an instant or timezone context, while a calendar defines valid business dates; observation time should remain distinct from ingestion and publication time.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Missing, Nonfinite, Censored, and Truncated Values
Missing means no usable value is present, nonfinite includes infinities and NaN-like numerical states, censoring preserves a bound without the exact value, and truncation removes cases outside a selection boundary.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Measurement Error, Resolution, Accuracy, and Precision
Error is measured minus reference, accuracy concerns closeness to an accepted reference, precision concerns repeatability, and resolution is the smallest display or recording increment.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Revisions, Vintages, and Point-in-Time Availability
A revision changes a previously reported value, a vintage is the dataset as known at a particular time, and point-in-time retrieval selects only records available by the as-of cutoff.Financial Mathematics, Statistics, and Data Foundations / Data, Variables, Samples, and Measurement
Data Provenance, Lineage, Ownership, and Licensing
Provenance records origin, lineage records transformations, ownership assigns stewardship, and licensing states permitted use.Location, Ranking, and Exploratory Summaries
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Count, Sum, Minimum, Maximum, and Range
Reduces one numeric series to the five headline aggregates: how many observations there are, their total, the smallest and largest value, and the distance between those extremes.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Arithmetic Mean
Divides the sum of a numeric series by the number of observations and reports both the mean and the count it was divided by.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Weighted Mean
Computes a weighted average of values using the parallel weights series, dividing the sum of value-times-weight products by the total weight.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Median and Mode
Reports the middle of a numeric series and its most frequent values, tallying exact equality so a tie produces several modes rather than one arbitrary winner.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Trimmed and Winsorized Means
Cuts the same number of observations off each tail of the sorted series and averages what is left, then averages again with those tails replaced by the surviving boundary values instead of discarded.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Percentiles, Quantiles, and Quartiles
Returns the three quartiles of a numeric series and the interquartile range between the first and third, using linear interpolation between neighbouring order statistics.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Ranks, Ties, and Percentile Rank
Assigns each observation its position in the sorted series, giving tied observations the average of the positions they span, and rescales those ranks onto a 0-to-100 percentile scale.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Frequency Tables and Relative Frequency
Counts how often each distinct value occurs in a numeric series and expresses those counts as shares of the total.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Histograms and Empirical Distribution Functions
Splits the observed span into equal-width bins and counts the observations in each, alongside the empirical distribution function evaluated at every sorted observation.Financial Mathematics, Statistics, and Data Foundations / Location, Ranking, and Exploratory Summaries
Five-Number Summary and Box Plot
Produces the minimum, three quartiles, and maximum of a numeric series, the Tukey 1.5-IQR fences drawn from them, and the observations lying outside those fences.Dispersion, Shape, and Robust Statistics
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Deviation, Absolute Deviation, and Squared Deviation
Centres a numeric series on its arithmetic mean and returns the three per-observation distance forms built from that centring: signed, absolute, and squared.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Range and Interquartile Range
Measures spread two ways on the same series: the full distance from smallest to largest, and the distance covering the middle half between the first and third quartiles.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Mean Absolute Deviation
Averages the absolute distances of the observations from their arithmetic mean, giving a spread figure in the same units as the data and without squaring.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Median Absolute Deviation
Takes the median of the absolute distances from the series median, and rescales it by 1.4826 so it lines up with a standard deviation under normality.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Population and Sample Variance
Averages the squared deviations from the mean twice over the same series: once dividing by the observation count, once by that count less one.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Population and Sample Standard Deviation
Takes the square root of both the population and the sample variance of one series, returning spread in the same units as the observations.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Coefficient of Variation and Scale Comparability
Divides the sample standard deviation by the absolute value of the mean, turning spread into a unit-free figure that can be compared across series measured on different scales.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Z-Score, Robust Z-Score, and Standardization
Standardises every observation twice: once against the mean and sample standard deviation, and once against the median and median absolute deviation scaled by 0.67448975.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Skewness and Tail Asymmetry
Divides the third central moment by the second raised to the power 1.5, giving the moment coefficient of skewness, and labels which tail is the longer one.Financial Mathematics, Statistics, and Data Foundations / Dispersion, Shape, and Robust Statistics
Kurtosis, Excess Kurtosis, and Tail Weight
Divides the fourth central moment by the square of the second, subtracts 3 to give excess kurtosis, and says whether the tails are heavier or lighter than a normal distribution's.Probability and Random Variables
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Experiments, Outcomes, Sample Spaces, and Events
Sizes a sample space and a candidate event, and checks that every outcome named in the event actually appears in the sample space.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Probability Rules, Complements, Unions, and Intersections
Applies the complement and addition rules to a pair of events, deriving the probability that A does not happen and the probability that at least one of A or B happens.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Conditional Probability
Rescales the joint probability of two events by each marginal in turn, giving the probability of A once B is known and the probability of B once A is known.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Independence and Dependence
Compares the observed joint probability of two events against the product of their marginals, and reports both the verdict and the size of the gap.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Bayes' Theorem and Base Rates
Turns a prior, a true-positive rate, and a false-positive rate into the posterior probability of the condition given a positive result, along with the unconditional probability of that result.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Discrete and Continuous Random Variables
Validates a discrete random variable given as a support and a matching probability vector, echoing back the declared kind, the support, and the total mass it carries.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Expected Value
Weights each value of a discrete random variable by its probability and sums the result, keeping the individual products so the total can be read term by term.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Variance, Moments, and Moment-Generating Intuition
Computes the first two raw moments of a discrete random variable and derives its variance as the second raw moment less the square of the first.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Joint, Marginal, and Conditional Distributions
Collapses a joint probability table over each axis in turn to give the two marginal distributions, then renormalises one slice of it to give the conditional distribution of X at a chosen value of Y.Financial Mathematics, Statistics, and Data Foundations / Probability and Random Variables
Covariance and Correlation of Random Variables
Computes the probability-weighted covariance of the two variables in a joint distribution table and divides it by the product of their standard deviations to give the correlation.Probability Distributions and Simulation Basics
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
PMF, PDF, CDF, Survival, and Quantile Functions
Reads an empirical sample from input.values and reports where the threshold input.x sits inside it, plus the sample quantile at probability input.p.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Bernoulli and Binomial Distributions
Computes the Bernoulli mean and the binomial probability mass and cumulative probability for k successes in n independent trials with success probability p.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Poisson Distribution and Event Counts
Evaluates the Poisson probability of exactly k events, the cumulative probability of at most k, and the distribution's mean and variance, for a rate of lambda events per interval.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Uniform Distribution and Random Sampling
Draws sampleCount values on the unit interval from a seeded linear congruential generator and reports the draws with their support and realised mean.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Normal Distribution and Standard Normal
Evaluates the normal density and cumulative probability at input.x for a distribution with mean mu and standard deviation sigma, and returns the standardised score that maps x onto the standard normal.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Lognormal Distribution and Positive Quantities
Evaluates the lognormal density and cumulative probability at input.x for a variable whose logarithm is normal with mean mu and standard deviation sigma, and returns the distribution's median.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Student-t Distribution and Heavy Tails
Evaluates the Student t density at input.x for df degrees of freedom and reports the distribution's variance, which only exists once df passes two.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Exponential, Gamma, and Weibull Waiting-Time Models
Evaluates three waiting-time models at the same point input.x: the exponential survival function, the gamma density, and the Weibull survival function, all sharing shape and scale.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Mixture Distributions, Multimodality, and Fat Tails
Evaluates a weighted mixture of normal components at input.x, which is how a single series can show more than one mode and fatter tails than any of its parts.Financial Mathematics, Statistics, and Data Foundations / Probability Distributions and Simulation Basics
Random Sampling and Monte Carlo Intuition
Estimates the probability p by counting how many of sampleCount seeded uniform draws land at or below it, and reports how far that Monte Carlo estimate lands from the known answer.Sampling, Estimation, and Statistical Inference
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Parameters, Statistics, Estimands, and Estimators
Places the known population parameter from input.populationMean alongside the statistic computed from input.sample, and names which side is the estimand and which is the estimator.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Sampling Distributions
Summarises the replicated estimates in input.estimates as a sampling distribution: where it centres, how wide it is, and how many replications produced it.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Estimator Bias, Consistency, Efficiency, and Robustness
Scores an estimator against a known truth: the average error of the replicated estimates, their mean squared error, a robust median estimate, and whether the mean squared error clears mseBenchmark.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Law of Large Numbers
Walks the running mean of input.sample one observation at a time and reports how far it has drifted from input.populationMean by the end.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Central Limit Theorem
Standardises the replicated estimates in input.estimates by subtracting populationMean and dividing by their spread, then checks how close the standardised values sit to zero.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Standard Error
Reports the standard error of the mean for input.sample — its sample standard deviation divided by the square root of the observation count.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Confidence Intervals and Coverage
Builds a two-sided interval around the mean of input.sample and reports whether populationMean falls inside it.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Null and Alternative Hypotheses
States a two-sided test of the mean of input.sample against input.nullMean, and returns the z statistic with the reject-or-not decision taken at the fixed critical value 1.959963984540054.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
P-Values, Significance, Type I/II Errors, and Power
Turns the two-sided z statistic for input.sample against input.nullMean into a p-value, compares it with alpha, and approximates the test's power at input.alternativeMean.Financial Mathematics, Statistics, and Data Foundations / Sampling, Estimation, and Statistical Inference
Effect Size, Practical Significance, and Multiple Comparisons
Reports a standardised effect size for the gap between the mean of input.sample and input.nullMean, tests that gap against a practical threshold, and splits alpha across input.comparisons tests Bonferroni-style.Dependence, Regression, and Model Foundations
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Scatter Plots, Association, and Nonlinear Patterns
Pairs input.x and input.y into plottable points and reports the Pearson correlation over them along with its sign.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Sample Covariance Calculation and Interpretation
Computes the sample covariance between input.x and input.y using the n-1 denominator, and reports which way the two series move together.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Pearson Correlation Calculation and Interpretation
Computes the Pearson correlation between input.x and input.y and its absolute value as a sign-free strength score.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Spearman Rank Correlation and Kendall Tau
Replaces input.x and input.y with their ranks, averaging ranks across ties, then reports the Spearman correlation of those ranks and Kendall's tau-a from the concordant and discordant pair counts.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Correlation, Causation, Confounding, and Spurious Relationships
Contrasts the pooled correlation of input.x and input.y with the correlation recomputed inside each label in input.groups, which is where a confounded or reversed relationship shows itself.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Simple Ordinary Least Squares Regression
Fits a single-predictor ordinary least squares line of input.y on input.x and returns the fitted coefficients with the in-sample fitted values.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Intercepts, Slopes, Coefficients, and Predictions
Fits the least squares line of input.y on input.x and evaluates it at one new predictor value, input.predictX.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Residuals, MAE, MSE, and RMSE
Returns the least squares residuals of input.y on input.x together with the three usual summaries of their size.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
R-Squared and Adjusted R-Squared
Reports the share of the variation in input.y that the least squares fit on input.x accounts for, and the same figure adjusted for the one estimated slope.Financial Mathematics, Statistics, and Data Foundations / Dependence, Regression, and Model Foundations
Regression Assumptions, Heteroskedasticity, and Multicollinearity
Runs three diagnostics on the least squares fit of y on x: the mean residual, a stand-in for non-constant error variance, and a stand-in for collinearity with a second predictor.Financial Time-Series Foundations
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Time Order, Frequency, Regularity, and Financial Calendars
Checks whether a timestamped series runs in chronological order and whether its observations are evenly spaced, reporting the common spacing in seconds when there is one.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Levels, Changes, Differences, and Returns
Converts a level series into its period-over-period differences and its simple returns, keeping the original levels alongside them.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Lags, Leads, and Temporal Dependence
Pairs every observation with the value lag positions behind it, and exposes the same series shifted forward as the lead.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Rolling and Expanding Windows
Computes a fixed-length rolling mean of the series next to its expanding mean, so a window that forgets can be compared with one that never does.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Resampling, Aggregation, and Time Alignment
Cuts the series into consecutive fixed-size buckets and averages each one, coarsening the sampling frequency.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Trend, Seasonality, Cycles, and Remainder
Separates a series into a rolling-mean trend, a seasonal difference taken period steps back, and the remainder left once the trend is removed.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Autocovariance and Autocorrelation
Measures how far a series moves with its own past at a given lag, using the population convention that divides by the full observation count.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Stationarity and Differencing Intuition
Compares the average level of the second half of a series with the first half, and reports the first differences, as a rough read on whether the level drifts.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Smoothing, Baselines, and Naive Forecasts
Runs simple exponential smoothing over the series and reports both the naive last-value forecast and the smoothed one.Financial Mathematics, Statistics, and Data Foundations / Financial Time-Series Foundations
Look-Ahead Leakage and Time-Aware Data Splits
Splits a timestamped series at a given position and reports whether the training block reaches into the testing block's time range.Financial Risk and Performance Statistics
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Volatility and Annualized Volatility
Takes the sample standard deviation of a return series and rescales it to an annual horizon by the square root of the observations per year.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Downside Deviation and Target Shortfall
Measures dispersion of returns below a target only, so upside swings do not count as risk.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Drawdown and Maximum Drawdown
Compounds a return series into a wealth path and tracks how far that path sits below its own running peak.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Loss Distributions and Loss Quantiles
Flips a return series into losses, sorts them, and reads off the quantile at the requested confidence by linear interpolation between neighbours.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Value at Risk Intuition
Reads historical value at risk straight off the sorted loss series and counts how many observed losses ran past it.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Expected Shortfall Intuition
Averages the losses that sit at or beyond the value-at-risk threshold, describing the tail rather than only its edge.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Beta and Market-Relative Risk
Divides the sample covariance between a return series and its benchmark by the benchmark's sample variance.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Sharpe, Sortino, and Information Ratio Intuition
Computes three reward-per-unit-of-risk ratios from the same series, each dividing an average by a different notion of risk.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Active Return and Tracking Error
Subtracts the benchmark from the portfolio period by period, then reports the average and the annualized volatility of that difference.Financial Mathematics, Statistics, and Data Foundations / Financial Risk and Performance Statistics
Covariance Matrices, Portfolio Variance, and Diversification
Combines portfolio weights with a covariance matrix into portfolio variance, and isolates the share of it that comes from the off-diagonal terms.Statistical Computing and Reproducibility
10 buildsFinancial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Floating-Point Representation, Overflow, and Underflow
Inspects one double-precision number: whether it is finite, how large one representable step at its magnitude is, and whether squaring it would stay inside a stated magnitude ceiling.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Stable Summation and Mean Calculation
Adds a list twice, once straight left to right and once carrying a running compensation term, so the two totals can be put side by side.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Stable Online Variance and Welford's Algorithm
Runs Welford's single-pass update to produce the mean and sample variance without ever accumulating a sum of raw squares.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Batch, Rolling, and Streaming Statistic Equivalence
Computes the same fixed-window mean two ways, by re-averaging each window slice and by a running add-and-drop sum, then checks that they agree.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Missing, Infinite, Invalid, and Unsupported-State Policies
Screens a mixed raw list down to the entries that are genuinely finite numbers and reports how many were turned away.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Pseudorandom Numbers, Seeds, and Reproducibility
Draws a run of unit-interval numbers from a seeded linear congruential generator, so the same seed always replays the same stream.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Vectorization, Index Alignment, and Shape Safety
Inner-joins two labelled vectors on their index field and reports what each side lost in the process.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Leakage-Free Fitting, Scaling, and Preprocessing
Fits a standardizing centre and scale on the training split alone, then applies that one transform to both splits.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Fixtures, Numerical Tolerances, and Property Tests
Compares an actual vector against an expected one under an absolute tolerance, the way a fixture check does.Financial Mathematics, Statistics, and Data Foundations / Statistical Computing and Reproducibility
Reproducible Analysis, Metadata, and Audit Trails
Serializes a run's inputs, method, version and seed into a deterministic JSON string and fingerprints it with a 32-bit FNV-1a hash.The foundation ladder
12 modules,
in dependency order.
This domain is a course, not a catalog: 120 topics that every later territory quietly assumes you already have. Read it in order and the rest of the atlas stops requiring leaps of faith.
- D00-F01Mathematical Language and Quantitative Reasoning10 topics
- D00-F02Financial Arithmetic, Time Value, and Returns10 topics
- D00-F03Data, Variables, Samples, and Measurement10 topics
- D00-F04Location, Ranking, and Exploratory Summaries10 topics
- D00-F05Dispersion, Shape, and Robust Statistics10 topics
- D00-F06Probability and Random Variables10 topics
- D00-F07Probability Distributions and Simulation Basics10 topics
- D00-F08Sampling, Estimation, and Statistical Inference10 topics
- D00-F09Dependence, Regression, and Model Foundations10 topics
- D00-F10Financial Time-Series Foundations10 topics
- D00-F11Financial Risk and Performance Statistics10 topics
- D00-F12Statistical Computing and Reproducibility10 topics
Territory coverage
What this domain
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The published builds above are the released slice of this map. Every subject below belongs to this territory's charter.