.. _interpreting-results:

Interpreting Results
====================

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

.. contents:: On this page
   :local:
   :depth: 2

----

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

**Standard deviation** :math:`\sigma`
  The square root of variance, in the same units as X.

**Skewness**
  The normalised third central moment:

  .. math::

     \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:

  .. math::

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

  A Normal distribution has kurtosis = 3.

**Excess kurtosis**
  Kurtosis minus 3:

  .. math::

     \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
:doc:`/troubleshooting/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
--------

* :doc:`/theory/statistical-moments` — mathematical definitions of each measure
* :doc:`/theory/raw-vs-central` — when to use each moment reference
* :doc:`/theory/skewness-kurtosis` — detailed discussion of shape measures