Interpreting Results

After clicking Compute Moments, ToFUL displays results across five tabs. This page explains each tab and how to read the outputs.

Moments Tab

The primary results are shown as metric cards, one per moment order, arranged in rows of up to five.

Each card displays:

  • μᵣ — the moment symbol with Unicode subscript (μ₁, μ₂, …)

  • Value — the computed moment to your chosen display precision

  • Method — the numerical algorithm that produced the result

Method labels:

Method label

Meaning

sympy-exact

Closed-form symbolic result via SymPy. No numerical error.

term-magnitude

Series converged: last 10 terms all below tolerance.

wynn-epsilon

Wynn ε-algorithm extracted the limit from the partial-sum sequence.

aitken-delta2

Aitken Δ² three-point extrapolation.

cohen-villegas-zagier

Optimal alternating-series accelerator.

ratio-bound

Geometric tail correction added to partial sum.

gauss-laguerre+quad

Gauss-Laguerre cross-checked with SciPy adaptive quad.

gauss-hermite+quad

Gauss-Hermite cross-checked with SciPy adaptive quad.

quad-adaptive

SciPy adaptive Gauss-Kronrod integration.

mpmath-tanh-sinh

mpmath double-exponential quadrature (high precision mode).

partial-sum

Convergence uncertain; raw partial sum shown. Increase max terms or precision.

Statistics Tab

This tab is active only when the moment reference is set to Mean (a = μ), because derived statistics require central moments.

Variance \(\sigma^2\)

The second central moment \(E[(X-\mu)^2]\). Measures the average squared deviation from the mean. Always non-negative.

Standard deviation \(\sigma\)

The square root of variance, in the same units as X.

Skewness

The normalised third central moment:

\[\text{Skewness} = \frac{\mu_3}{\sigma^3}\]

  • Negative — the distribution has a longer left tail (left-skewed).

  • Zero — the distribution is symmetric.

  • Positive — the distribution has a longer right tail (right-skewed).

The threshold used in the interpretation badge is abs(skewness) < 0.5 for “symmetric”. This is a display heuristic, not a statistical test.

Kurtosis

The normalised fourth central moment:

\[\text{Kurtosis} = \frac{\mu_4}{\sigma^4}\]

A Normal distribution has kurtosis = 3.

Excess kurtosis

Kurtosis minus 3:

\[\text{Excess kurtosis} = \frac{\mu_4}{\sigma^4} - 3\]

  • Negative — platykurtic: lighter tails than Normal (e.g. Uniform).

  • Zero — mesokurtic: tails like Normal.

  • Positive — leptokurtic: heavier tails than Normal (e.g. Cauchy, Student-t).

Distribution Tab

Two side-by-side Plotly charts:

Left — Distribution shape

For CRVs: a filled area plot of the PDF with the mean marked as a vertical dashed line. If a custom reference point a was set, it is also shown.

For DRVs: a stem plot (vertical lines with dots) showing P(X = x) for the first 80 support values. The mean is marked as above.

Hovering over the plot shows the exact x and f(x) or P(X=x) values.

Right — Moment magnitudes

A bar chart comparing the magnitude of each moment. The bars are colour-coded by order and labelled with the numeric value. This is useful for seeing how quickly moments grow with order.

Table Tab

A structured table with one row per moment order, containing:

  • Order (r) — integer moment order

  • μ(subscript) — Unicode symbol

  • Value — computed value at display precision

  • Method — algorithm used

  • Converged — Yes/No convergence status

  • Nodes/Terms — number of quadrature nodes (CRV) or series terms (DRV)

A Download CSV button exports the full table.

Convergence Tab

Diagnostic information for each moment:

  • A pie chart showing how many moments used each method (appears when more than one method was used).

  • One row per moment with the convergence status pill (green = converged, red = uncertain), the method name, and the full info string from the convergence engine.

If any moment shows partial-sum as its method, see Convergence Issues.

Auto-Correction Notice

If the parser modified your input (e.g. converted ^ to ** or |x| to abs(x)), a collapsible notice appears above the results listing each change. The interpreted expression is shown in the display form (with Unicode superscripts and subscripts) for verification.

See also