Normal Distribution Calculator mechanism

Model scope

The z-score and ordinary normal-CDF concepts are standard, but the instrument claims inverse-CDF, confidence intervals, and a bell-curve plot without supplying the corresponding formulas, required inputs, or validity constraints. The specification therefore cannot be accepted as complete or fully verifiable.

• The z-score formula z=(x-μ)/σ is correct only when σ>0 and x, μ, and σ use compatible units.
• The normal CDF formula is underspecified: “rational approximation of erf” does not identify the approximation, its domain handling, or its numerical error tolerance.
• The claimed inverse-CDF output requires an explicit inverse operation x=μ+σΦ⁻¹(p), which is not provided.
• The claimed confidence-interval bounds require additional inputs and a stated method. For a known population standard deviation, the formulas are μ±z_(1−α/2)σ/√n; for an estimated standard deviation, a t critical value and sample standard deviation are required.
• The bell-curve plot requires an explicit density formula f(x)=exp(−(x−μ)^2/(2σ^2))/(σ√(2π)); no plotting formula or curve-feature definitions are provided.

Units and conventions

• σ must be strictly positive; σ=0 is invalid because the z-score divides by zero.
• x and μ must have the same units, and σ must use the corresponding units; for example, mixing Celsius with Fahrenheit without conversion is invalid.
• Probability and percentile inputs are dimensionless and must satisfy 0<p<1 for a finite inverse CDF.
• Confidence-interval inputs must specify n, confidence level or α, and whether σ is known or estimated.
• The CDF is continuous, so P(X<x)=P(X≤x) for a normal distribution.
• The test plot values below assume the plotted quantity is the probability density, not the CDF or shaded probability.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "mu": 100,
    "sigma": 15,
    "x": 130
  },
  "expected_outputs": {
    "z_score": 2,
    "cdf_P_X_less_than_x": 0.9772498681,
    "percentile_percent": 97.72498681,
    "plot_sample": {
      "x": 130,
      "density": 0.0035993978
    },
    "curve_features": {
      "mode_x": 100,
      "maximum_density": 0.026596152
    }
  },
  "working": "1. z=(x−μ)/σ=(130−100)/15=30/15=2. 2. P(X<130)=Φ(2)=0.9772498681, so the percentile is 0.9772498681×100=97.72498681%. 3. Density at x=130: f(130)=exp(−2^2/2)/(15√(2π))=0.1353352832/37.59942412=0.0035993978. 4. The curve maximum occurs at x=μ=100: f(100)=1/(15√(2π))=1/37.59942412=0.026596152."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification exposes two hard failure inputs: sigma=0 causes undefined division, and inverse-CDF probability exactly 1 has no finite result. A robust implementation must explicitly validate sigma>0 and handle endpoint probabilities as limits or rejected inputs. The sign-convention probe can also produce a materially wrong tail probability if the deviation is accidentally absolutized.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json