:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.statistics``:
descriptive statistics, hypothesis tests, confidence intervals,
ordinary least-squares regression, correlation, maximum likelihood,
nonparametric tests, bootstrap and jackknife resampling, shrinkage
estimation, and multiple-testing corrections.

See also the narrative tutorial:

- :doc:`/tutorials/randomness_three_ways`

Each script in this gallery is self-contained and can be run directly with
``python examples/statistics/<section>/<script>.py``.

Sections
--------

- **descriptive** -- the five-number summary, skewness, and kurtosis,
  and Gauss's normal law of measurement errors.
- **hypothesis_tests** -- z-tests, t-tests, chi-square tests, ANOVA, and
  Fisher's exact test.
- **confidence_intervals** -- intervals for means, proportions, and
  variances.
- **regression** -- ordinary least squares with residual diagnostics.
- **correlation** -- Pearson's and Spearman's correlation coefficients.
- **likelihood** -- maximum-likelihood fitting and Wilks's
  likelihood-ratio test.
- **nonparametric** -- the Kolmogorov-Smirnov, Wilcoxon signed-rank, and
  Mann-Whitney U tests.
- **bootstrap** -- bootstrap confidence intervals via
  ``scipy.stats.bootstrap``, and the jackknife.
- **shrinkage** -- the James-Stein estimator.
- **multiple_testing** -- Bonferroni and Benjamini-Hochberg corrections.


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Bootstrap resampling
----------------------

Bootstrap confidence intervals via ``scipy.stats.bootstrap``, and the
leave-one-out jackknife.


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares a bootstrap confidence interval for the median (which has no simple closed form) against the parametric t-based interval for the mean, on the same skewed dataset.">

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  .. image:: /api/gallery/statistics/bootstrap/images/thumb/sphx_glr_plot_01_bootstrap_vs_parametric_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/bootstrap/plot_01_bootstrap_vs_parametric`

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      <div class="sphx-glr-thumbnail-title">Bootstrap vs. parametric confidence intervals</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Leaves out one observation at a time and recomputes the statistic. For the plug-in variance, the jackknife&#x27;s bias correction recovers the unbiased n - 1 variance exactly, and for the mean its standard error is exactly the classical s / sqrt(n). For a non-smooth statistic such as the midrange, the jackknife standard error is unreliable and disagrees with the bootstrap&#x27;s -- one of the shortcomings Efron&#x27;s bootstrap was designed to overcome.">

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  .. image:: /api/gallery/statistics/bootstrap/images/thumb/sphx_glr_plot_02_jackknife_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/bootstrap/plot_02_jackknife`

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      <div class="sphx-glr-thumbnail-title">The jackknife</div>
    </div>


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Confidence intervals
-----------------------

Intervals for means, proportions, and variances.


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    <div class="sphx-glr-thumbcontainer" tooltip="Builds many 95% confidence intervals from repeated samples of a known population, and checks that close to 95% of them actually contain the true parameter -- the frequentist interpretation of &quot;95% confidence.&quot;">

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  .. image:: /api/gallery/statistics/confidence_intervals/images/thumb/sphx_glr_plot_01_coverage_simulation_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/confidence_intervals/plot_01_coverage_simulation`

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      <div class="sphx-glr-thumbnail-title">Confirming a 95% confidence interval's coverage by simulation</div>
    </div>


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Correlation
-----------

Pearson's product-moment and Spearman's rank correlation coefficients.


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    <div class="sphx-glr-thumbcontainer" tooltip="Simulates parent and child heights in the spirit of Francis Galton&#x27;s 1880s data: the two are positively correlated, but tall parents have children who are on average less extreme -- &quot;regression toward the mean&quot;. Pearson&#x27;s r measures the strength of that linear relation.">

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  .. image:: /api/gallery/statistics/correlation/images/thumb/sphx_glr_plot_01_galton_pearson_correlation_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/correlation/plot_01_galton_pearson_correlation`

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      <div class="sphx-glr-thumbnail-title">Galton's heights and Pearson's correlation</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Spearman&#x27;s rho is Pearson&#x27;s r applied to ranks. It equals 1 for any increasing relationship, however curved, and a single wild outlier barely moves it, while Pearson&#x27;s r can swing a long way.">

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  .. image:: /api/gallery/statistics/correlation/images/thumb/sphx_glr_plot_02_spearman_rank_correlation_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/correlation/plot_02_spearman_rank_correlation`

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      <div class="sphx-glr-thumbnail-title">Spearman's rank correlation</div>
    </div>


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Descriptive statistics
------------------------

The five-number summary, skewness, and kurtosis, and Gauss's normal law
of measurement errors.


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    <div class="sphx-glr-thumbcontainer" tooltip="Summarizes a skewed dataset (exponential-ish waiting times) and visualizes its five-number summary as a boxplot.">

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  .. image:: /api/gallery/statistics/descriptive/images/thumb/sphx_glr_plot_01_five_number_summary_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/descriptive/plot_01_five_number_summary`

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      <div class="sphx-glr-thumbnail-title">Descriptive statistics and the five-number summary</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Repeated measurements of one quantity scatter around its true value. Gauss (1809) asked which error law makes the arithmetic mean of the measurements the most probable value of the quantity, and found the normal law. This example simulates noisy measurements, compares their errors with the normal curve, and checks that the mean both maximizes the normal error probability and minimizes the sum of squared errors.">

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  .. image:: /api/gallery/statistics/descriptive/images/thumb/sphx_glr_plot_02_gauss_error_law_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/descriptive/plot_02_gauss_error_law`

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      <div class="sphx-glr-thumbnail-title">Gauss's normal law of errors</div>
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Hypothesis tests
------------------

z-tests, Student's t-tests, chi-square tests (goodness-of-fit and
independence), Fisher's exact test, and one-way ANOVA.


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    <div class="sphx-glr-thumbcontainer" tooltip="With only a handful of measurements, the standardized mean t = (\bar x - \mu_0)/(s/\sqrt n) is not normal: estimating the spread from the same small sample gives it heavier tails. Gosset (&quot;Student&quot;, 1908) derived its exact distribution. This example shows the heavy tails, how a normal cutoff misleads for small samples, and the one- and two-sample t-tests built on Student&#x27;s distribution.">

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  .. image:: /api/gallery/statistics/hypothesis_tests/images/thumb/sphx_glr_plot_01_student_t_test_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/hypothesis_tests/plot_01_student_t_test`

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      <div class="sphx-glr-thumbnail-title">Student's t-distribution and the small-sample t-test</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Pearson&#x27;s chi-square statistic applied two ways: testing whether a die&#x27;s observed roll counts match a fair-die model (goodness-of-fit), and testing whether two categorical variables in a contingency table are independent.">

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  .. image:: /api/gallery/statistics/hypothesis_tests/images/thumb/sphx_glr_plot_02_chi_square_tests_thumb.png
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  :doc:`/api/gallery/statistics/hypothesis_tests/plot_02_chi_square_tests`

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      <div class="sphx-glr-thumbnail-title">Chi-square goodness-of-fit and independence tests</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Fisher&#x27;s thought experiment: a lady claims she can tell whether milk or tea was poured first. She is given 8 cups, 4 of each, and told so. If she is only guessing, the number of milk-first cups she identifies correctly follows a hypergeometric distribution, and getting all 4 right happens with probability exactly 1/70.">

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  .. image:: /api/gallery/statistics/hypothesis_tests/images/thumb/sphx_glr_plot_03_fisher_exact_lady_tea_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/hypothesis_tests/plot_03_fisher_exact_lady_tea`

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      <div class="sphx-glr-thumbnail-title">Fisher's exact test and the lady tasting tea</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Compares three fertilizer treatments&#x27; effect on plant growth, as in Fisher&#x27;s Rothamsted field trials. ANOVA splits the total variability into a between-groups and a within-groups sum of squares; their ratio, the F-statistic, tests all group means for equality in one test.">

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  .. image:: /api/gallery/statistics/hypothesis_tests/images/thumb/sphx_glr_plot_04_one_way_anova_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/hypothesis_tests/plot_04_one_way_anova`

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      <div class="sphx-glr-thumbnail-title">Fisher's one-way analysis of variance</div>
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Likelihood
----------

Maximum-likelihood fitting and the likelihood-ratio test.


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    <div class="sphx-glr-thumbcontainer" tooltip="Fits an exponential and a gamma distribution to the same skewed data by maximum likelihood, and plots the exponential model&#x27;s log-likelihood as a function of its scale: the curve peaks exactly at the sample mean, the closed-form MLE.">

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  .. image:: /api/gallery/statistics/likelihood/images/thumb/sphx_glr_plot_01_maximum_likelihood_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/likelihood/plot_01_maximum_likelihood`

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      <div class="sphx-glr-thumbnail-title">Fisher's maximum likelihood</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Repeatedly tests the true null hypothesis &quot;the normal mean is 0&quot; with the likelihood-ratio statistic 2(l1 - l0). Wilks&#x27;s theorem says the statistic&#x27;s null distribution approaches chi-square with one degree of freedom, the number of parameters the null hypothesis fixes.">

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  .. image:: /api/gallery/statistics/likelihood/images/thumb/sphx_glr_plot_02_wilks_likelihood_ratio_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/likelihood/plot_02_wilks_likelihood_ratio`

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      <div class="sphx-glr-thumbnail-title">Wilks's theorem and the likelihood-ratio test</div>
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Multiple testing
----------------

Bonferroni and Benjamini-Hochberg corrections for many simultaneous tests.


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    <div class="sphx-glr-thumbcontainer" tooltip="Runs 1000 one-sample t-tests, 100 of which have a real effect. With no correction, dozens of the &quot;discoveries&quot; are false; Bonferroni avoids false discoveries but misses most real effects; Benjamini-Hochberg keeps the fraction of false discoveries near the chosen 5% while finding far more of the real ones.">

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  .. image:: /api/gallery/statistics/multiple_testing/images/thumb/sphx_glr_plot_01_false_discovery_rate_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/multiple_testing/plot_01_false_discovery_rate`

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      <div class="sphx-glr-thumbnail-title">Benjamini-Hochberg and the false discovery rate</div>
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Nonparametric tests
-------------------

The Kolmogorov-Smirnov test and the Wilcoxon and Mann-Whitney rank tests.


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    <div class="sphx-glr-thumbcontainer" tooltip="Draws a sample from a Student t distribution with 3 degrees of freedom and compares its empirical CDF with the standard normal CDF. The KS statistic is the largest vertical gap between the two curves.">

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  .. image:: /api/gallery/statistics/nonparametric/images/thumb/sphx_glr_plot_01_kolmogorov_smirnov_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/nonparametric/plot_01_kolmogorov_smirnov`

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      <div class="sphx-glr-thumbnail-title">The Kolmogorov-Smirnov test</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Rank tests throw away the raw values and keep only their order, so a few extreme observations cannot dominate. With heavy-tailed data they detect a shift that the t-test misses.">

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  .. image:: /api/gallery/statistics/nonparametric/images/thumb/sphx_glr_plot_02_wilcoxon_mann_whitney_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/nonparametric/plot_02_wilcoxon_mann_whitney`

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      <div class="sphx-glr-thumbnail-title">Wilcoxon and Mann-Whitney rank tests</div>
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Linear regression
--------------------

Ordinary least squares with residual diagnostics.


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    <div class="sphx-glr-thumbcontainer" tooltip="Fits a noisy linear relationship and inspects the fit: coefficient standard errors/p-values, R^2, and a residuals-vs-fitted-values plot.">

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  .. image:: /api/gallery/statistics/regression/images/thumb/sphx_glr_plot_01_ols_diagnostics_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/regression/plot_01_ols_diagnostics`

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      <div class="sphx-glr-thumbnail-title">OLS regression with residual diagnostics</div>
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Shrinkage estimation
--------------------

The James-Stein estimator and Stein's paradox.


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    <div class="sphx-glr-thumbcontainer" tooltip="Estimates p unrelated normal means from one observation each. Shrinking all the observations toward zero by the James-Stein factor lowers the total squared error for every p &gt;= 3, even though no single observation carries information about the others.">

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  .. image:: /api/gallery/statistics/shrinkage/images/thumb/sphx_glr_plot_01_james_stein_thumb.png
    :alt:

  :doc:`/api/gallery/statistics/shrinkage/plot_01_james_stein`

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      <div class="sphx-glr-thumbnail-title">Stein's paradox</div>
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.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/statistics/bootstrap/index.rst
   /api/gallery/statistics/confidence_intervals/index.rst
   /api/gallery/statistics/correlation/index.rst
   /api/gallery/statistics/descriptive/index.rst
   /api/gallery/statistics/hypothesis_tests/index.rst
   /api/gallery/statistics/likelihood/index.rst
   /api/gallery/statistics/multiple_testing/index.rst
   /api/gallery/statistics/nonparametric/index.rst
   /api/gallery/statistics/regression/index.rst
   /api/gallery/statistics/shrinkage/index.rst



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