.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/analytical/statistics/plot_01_student_t_confidence_interval.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_analytical_statistics_plot_01_student_t_confidence_interval.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_analytical_statistics_plot_01_student_t_confidence_interval.py:


Student's t: confidence intervals from a few replicates
==========================================================

With only `n` replicates the standard deviation `s` is itself uncertain,
so the interval :math:`\bar x\pm t\,s/\sqrt n` from
:func:`~chemistrykit.analytical.t_confidence_interval` uses Student's
`t` instead of the normal 1.96. We simulate many triplicate analyses and
show that the `t` interval covers the true value 95% of the time, while
the naive :math:`\pm1.96\,s/\sqrt n` interval falls well short.

.. GENERATED FROM PYTHON SOURCE LINES 14-26

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from chemistrykit.analytical import t_confidence_interval

    rng = np.random.default_rng(1908)
    true_value, sigma, n = 50.00, 0.40, 3  # e.g. % analyte, triplicate analyses

    ci = t_confidence_interval([50.21, 49.83, 50.36])
    print(f"Triplicate: mean = {ci.mean:.3f}, s = {ci.std:.3f}, t(2 dof, 95%) = {ci.t_critical:.3f}")
    print(f"95% interval: {ci.lower:.3f} to {ci.upper:.3f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Triplicate: mean = 50.133, s = 0.273, t(2 dof, 95%) = 4.303
    95% interval: 49.455 to 50.812




.. GENERATED FROM PYTHON SOURCE LINES 27-40

.. code-block:: Python

    trials = 5000
    t_hits = z_hits = 0
    intervals = []
    for i in range(trials):
        data = rng.normal(true_value, sigma, size=n)
        ci = t_confidence_interval(data, confidence=0.95)
        t_hits += ci.lower <= true_value <= ci.upper
        z_half = 1.96 * ci.std / np.sqrt(n)
        z_hits += abs(ci.mean - true_value) <= z_half
        if i < 40:
            intervals.append(ci)
    print(f"\nCoverage over {trials} triplicates: t interval {t_hits / trials:.3f}, naive 1.96 interval {z_hits / trials:.3f}")





.. rst-class:: sphx-glr-script-out

 .. code-block:: none


    Coverage over 5000 triplicates: t interval 0.945, naive 1.96 interval 0.814




.. GENERATED FROM PYTHON SOURCE LINES 41-62

.. code-block:: Python

    fig, axes = plt.subplots(1, 2, figsize=(12, 4.5))
    for i, ci in enumerate(intervals):
        color = "C0" if ci.lower <= true_value <= ci.upper else "crimson"
        axes[0].plot([ci.lower, ci.upper], [i, i], color=color)
        axes[0].plot([ci.mean], [i], "o", color=color, markersize=3)
    axes[0].axvline(true_value, color="black", linewidth=0.8)
    axes[0].set_xlabel("analyte (%)")
    axes[0].set_ylabel("triplicate analysis")
    axes[0].set_title("95% t intervals (red: miss the true value)")

    dof = np.arange(1, 31)
    t_vals = [t_confidence_interval(np.arange(d + 1, dtype=float)).t_critical for d in dof]
    axes[1].plot(dof, t_vals, "o-", label="Student's t (95%)")
    axes[1].axhline(1.96, color="gray", linestyle="--", label="normal z = 1.96")
    axes[1].set_yscale("log")
    axes[1].set_xlabel("degrees of freedom n - 1")
    axes[1].set_ylabel("critical value")
    axes[1].set_title("t approaches 1.96 only for large n")
    axes[1].legend()
    plt.tight_layout()
    plt.show()



.. image-sg:: /api/gallery/analytical/statistics/images/sphx_glr_plot_01_student_t_confidence_interval_001.png
   :alt: 95% t intervals (red: miss the true value), t approaches 1.96 only for large n
   :srcset: /api/gallery/analytical/statistics/images/sphx_glr_plot_01_student_t_confidence_interval_001.png
   :class: sphx-glr-single-img






.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.215 seconds)


.. _sphx_glr_download_api_gallery_analytical_statistics_plot_01_student_t_confidence_interval.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_student_t_confidence_interval.ipynb <plot_01_student_t_confidence_interval.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_student_t_confidence_interval.py <plot_01_student_t_confidence_interval.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_student_t_confidence_interval.zip <plot_01_student_t_confidence_interval.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_