normal distribution
Enable an agent to identify a normal probability law, interpret its parameters, assess whether its use is justified, and perform probability calculations with explicit assumptions.
Research draft, second pass
A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.
recalled by Codex without web access - no source was read
Researched by: Codex
Purpose and description
Enable an agent to identify a normal probability law, interpret its parameters, assess whether its use is justified, and perform probability calculations with explicit assumptions.
The normal distribution is a continuous probability distribution on the real line with density f(x) = exp(-(x-μ)²/(2σ²))/(σ√(2π)), where μ is its mean and σ² > 0 is its variance.
It can be Instantiate a normal law from valid parameters and an explicit parameter convention.; Standardise values and convert standard scores back into the original units.; Calculate interval probabilities, tail probabilities, and quantiles.; Propagate a normal law through affine transformations and justified combinations of normal variables.; Assess a normal assumption or approximation against support constraints, diagnostics, and the intended calculation.; Generate simulated draws while recording the parameters and reproducibility settings..
Distinguishing features
A nondegenerate member has density f(x) = exp(-(x-mu)^2/(2*sigma^2))/(sigma*sqrt(2*pi)) for every real x, with sigma > 0.
Its log-density is exactly quadratic in x; symmetry and a bell-shaped appearance alone do not establish normality.
The mean, median, and unique mode coincide at mu, and variance is sigma^2; these properties are necessary but not sufficient for normality.
Standardising X by Z = (X-mu)/sigma yields the standard normal law, independent of the original location and scale.
It assigns positive probability to every nonempty open interval on the real line, distinguishing it from bounded, truncated, discrete, and strictly positive distributions.
Scope
+ The univariate Gaussian family, its density, support, and parameter restrictions
+ Location, scale, standardisation, and parameter conventions
+ Probabilities, quantiles, moments, and transformations within the normal family
+ Exact normality assumptions, empirical adequacy, and normal approximations
+ Conditions governing normal sums and conditional interpretation of calculated results
- Multivariate Gaussian distributions and Gaussian processes
- Probability theory and continuous distributions in general
- Complete models of datasets, measurement systems, or data collection
- Complete statistical procedures such as regression, hypothesis testing, and experimental design
- Lognormal, truncated normal, folded normal, and Gaussian mixture distributions as separate families
Characteristics
- Location parameter
- mu: any finite real value, in the units of the represented variable Fixes the centre, expectation, median, and mode of the distribution.
- Standard deviation
- sigma > 0, in the units of the represented variable Controls dispersion and converts deviations into dimensionless standard scores.
- Variance
- sigma^2 > 0, in squared units of the represented variable Enters the density and variance propagation formulas; confusing it with standard deviation changes every calculation.
- Parameter notation
- Explicit declaration of whether the second parameter denotes variance, standard deviation, or precision Prevents incompatible interpretations of normal-distribution notation and software arguments.
- Represented quantity
- Random variable, measurement error, residual, estimator, or other explicitly identified quantity Normality of one quantity does not establish normality of related observations or derived quantities.
- Conditioning context
- Unconditional law or law conditional on specified variables, groups, and conditions Conditional normal distributions can combine into a marginal distribution that is not normal.
- Modelling status
- Exact by mathematical construction, assumed, empirically assessed, approximate, or rejected for a stated use Separates properties of the mathematical law from evidence supporting an application.
- Parameter knowledge
- Specified, estimated with documented method and uncertainty, or unknown Plug-in probability calculations can conceal uncertainty arising from estimated parameters.
- Use-specific adequacy
- Unassessed, adequate within stated tolerances, or inadequate for a specified probability region and task A model that describes the centre acceptably may still misrepresent decision-relevant tails.
Also called
Where this came from
wikidata · CC0 1.0
Drafted structure
Bundle to layer to finding to question, as the second pass will find it: 6 bundles · 11 layers · 17 findings · 29 questions.
Normal family identity Establishes which mathematical law is represented and distinguishes it from visually similar distributions.
An agent must recognise the family through its defining structure rather than through an informal bell-curve description.
Defining law
Records the density, support, and admissible parameters of a univariate normal law.
Density and full real support
The ordinary nondegenerate normal family has the stated Gaussian density on the entire real line, with finite mu and strictly positive sigma.
- Does the proposed law have the Gaussian density for every real x, with sigma strictly positive? definition
- Do bounds or discreteness of the represented quantity restrict normality to an approximation? boundary
Nearby family boundaries
Separates a normal law from related distributions and exceptional conventions.
Normal versus derived laws
Truncation, taking an absolute value, exponentiation, and mixing can produce laws outside the univariate normal family; a zero-variance point mass requires an explicit convention.
- Has truncation, censoring, folding, exponentiation, or mixing changed the law or the observation process? boundary
- Does the source use the ordinary sigma > 0 definition or an extended convention admitting a degenerate Gaussian point mass? provenance
Parameters and standardisation Makes location, dispersion, units, and parameter knowledge unambiguous.
Normal calculations depend on correctly interpreting two parameters and distinguishing their specified values from estimates.
Location and scale conventions
Connects normal parameters to their mathematical meanings and physical units.
Parameter meaning
Mu is the expectation and centre; sigma is standard deviation; sigma squared is variance; precision, when used, is reciprocal variance.
- Does each parameter field or notation denote location, standard deviation, variance, or precision? definition
- Are location and scale expressed in compatible units, with variance expressed in squared units? measurement
Standard scores and estimates
Records standardisation and the evidential status of the values used.
Standardisation with parameter status
For X distributed as N(mu, sigma squared), (X-mu)/sigma is standard normal. Substituting estimated parameters does not automatically preserve that exact sampling law.
- Are mu and sigma specified population parameters or estimates, and where did their values come from? provenance
- Does the intended use require accounting for parameter estimation when interpreting a standardised value? action
Probabilities and numerical use Connects the normal law to events, quantiles, and reliable numerical evaluation.
An agent needs to translate a practical question into the correct probability operation and retain accuracy in extreme tails.
Events and quantiles
Defines the event or probability threshold being evaluated.
Probability versus density
Interval probabilities come from cumulative probabilities or integration, not density heights; every individual point has probability zero.
- Is the request for a density, interval probability, one-sided tail, two-sided event, or quantile? definition
- What thresholds and units define the event, or what cumulative probability defines the requested quantile? measurement
Tail computation
Captures numerical precision requirements where direct probability subtraction may fail.
Stable extreme-tail evaluation
Extreme upper-tail probabilities can lose numerical accuracy when computed as one minus a rounded cumulative probability; survival and log-probability functions support more stable evaluation.
- How small a probability or how extreme a quantile must the calculation resolve? measurement
- Should the implementation use a survival function, log-probability, or an inverse-tail routine to meet that precision requirement? action
Transformations and combinations States when operations preserve normality and how parameters propagate.
Normal inputs alone do not justify treating every transformed or aggregated output as normal.
Affine transformations
Handles changes of origin, scale, and units.
Affine closure
For nonzero a, if X is normal then aX+b is normal with mean a*mu+b and variance a squared times sigma squared; a=0 produces a point mass.
- Is the proposed transformation affine, and what are its coefficients? definition
- What transformed mean and variance follow, and does the result remain nondegenerate? measurement
Sums and dependence
Records the assumptions required to combine normal variables.
Justified normal combinations
Independent normal variables have normal sums. Joint normality also supports normal linear combinations, but marginal normality alone is insufficient; covariance terms affect the resulting variance.
- What establishes independence or joint normality of the variables being combined? provenance
- Which weights and covariance terms determine the combination's mean and variance? measurement
Normality and approximation warrant Assesses why a normal law applies and where the justification stops.
The mathematical family's exact properties must be separated from the adequacy of using it for observations, errors, or sampling distributions.
Empirical normality
Relates observed evidence to a specific normality claim and intended use.
Target-specific diagnostics
Quantile plots, distributional tests, and substantive constraints can assess departures from a normal model. Finite data cannot establish exact normality, and failure to reject a test is not proof.
- Does the normality claim concern raw observations, conditional errors, residuals, or a sampling distribution? boundary
- What diagnostics and support constraints bear on the claim, especially in the probability region used for decisions? measurement
- Which observed departures would make the normal model inadequate for the intended calculation? action
Limit-based approximations
Makes central-limit and related approximation claims conditional on their actual assumptions.
Central-limit conditions and error
Central limit theorems can justify normal limits for appropriately standardised sums under specified conditions; they neither make the original observations normal nor guarantee adequate finite-sample tail accuracy.
- Which limit theorem supports this approximation, and are its moment and dependence assumptions satisfied? provenance
- What evidence or error bound supports accuracy at the actual sample size and requested probability region? measurement
- For a discrete target, is a continuity correction or direct calculation needed to meet the required accuracy? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- The sense covered is the probability distribution in mathematics and statistics.
- The definition uses the nondegenerate univariate convention; some treatments also admit zero-variance Gaussian distributions as point masses.
- These are recalled mathematical facts, not findings from sources consulted for this response; suitability for any particular dataset requires checking.
- Which of these check these first hold for the sense of normal distribution this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Standard normal distribution, with μ = 0 and σ² = 1
- General univariate normal distribution
- Multivariate normal distribution, extending the concept to jointly Gaussian random vectors
- Which of these kinds and varieties hold for the sense of normal distribution this model covers, and on what evidence? provenance
Identifiers and schemes
Recalled without web access and unsourced; every item is a lead to verify.
- Conventional mathematical notation - N(μ, σ²) - Commonly specifies mean and variance; some sources instead use the standard deviation as the second parameter, so conventions must be checked.
- Which of these identifiers and schemes hold for the sense of normal distribution this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Modelling approximately symmetric measurement errors and variation around a mean
- Approximating sampling distributions through central limit theorems when their conditions hold
- Constructing statistical confidence intervals and hypothesis tests
- Modelling conditional errors in Gaussian linear regression
- Calculating probabilities and control limits in statistical process control when a normal model is appropriate
- Which of these real-world use hold for the sense of normal distribution this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Mean μ - Any real number - Same unit as the random variable
- Standard deviation σ - Any positive real number - Same unit as the random variable
- Variance σ² - Any positive real number - Square of the random variable's unit
- Skewness - Exactly 0 - Dimensionless
- Excess kurtosis - Exactly 0 - Dimensionless
- Probability within one, two, and three standard deviations of the mean - Approximately 68.27%, 95.45%, and 99.73%, respectively - Percent
- Which of these typical measurements hold for the sense of normal distribution this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Applying a normal model to strongly skewed, multimodal, or heavy-tailed data can distort probabilities and uncertainty estimates.
- Normal tail estimates can substantially understate extreme-event probabilities when the actual distribution has heavier tails.
- Its unbounded support assigns probability to impossible values when used for strictly positive or bounded quantities.
- A central limit theorem concerns suitably normalised sums or averages under stated conditions; it does not establish that individual observations are normal.
- Marginal normality does not establish joint multivariate normality or independence.
- Which of these failure modes and hazards hold for the sense of normal distribution this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Gaussian distribution - A synonym for normal distribution, rather than a separate probability family.
- Student's t-distribution - Has heavier tails for finite degrees of freedom and approaches the standard normal distribution as the degrees of freedom increase.
- Lognormal distribution - A positive random variable is lognormal when its logarithm is normal; the variable itself is generally asymmetric.
- Truncated normal distribution - Conditions a normal variable to lie within a specified interval and renormalises its density.
- Central limit theorem - A class of convergence results explaining when normal limits arise, rather than a probability distribution.
- Bell-shaped distribution - A descriptive shape shared by several distributions; a normal distribution must satisfy the specific Gaussian density.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of normal distribution this model covers, and on what evidence? provenance
What the second pass must settle
- Should this registry entry explicitly admit zero-variance Gaussian point masses, or reserve normal distribution for strictly positive variance?
- Which authoritative references and software interfaces should anchor the catalogue's parameter notation and precision conventions?
- What evidence and tolerances should an agent require before accepting a normal approximation for central probabilities versus extreme-tail decisions?
- Where should catalogue ownership fall between this family model and neighbouring models of normality diagnostics, parameter estimation, and sampling distributions?