

.. _sphx_glr_gallery_statistics_histogram_cumulative.py:


==================================================
Using histograms to plot a cumulative distribution
==================================================

This shows how to plot a cumulative, normalized histogram as a
step function in order to visualize the empirical cumulative
distribution function (CDF) of a sample. We also use the ``mlab``
module to show the theoretical CDF.

A couple of other options to the ``hist`` function are demonstrated.
Namely, we use the ``normed`` parameter to normalize the histogram and
a couple of different options to the ``cumulative`` parameter.
The ``normed`` parameter takes a boolean value. When ``True``, the bin
heights are scaled such that the total area of the histogram is 1. The
``cumulative`` kwarg is a little more nuanced. Like ``normed``, you
can pass it True or False, but you can also pass it -1 to reverse the
distribution.

Since we're showing a normalized and cumulative histogram, these curves
are effectively the cumulative distribution functions (CDFs) of the
samples. In engineering, empirical CDFs are sometimes called
"non-exceedance" curves. In other words, you can look at the
y-value for a given-x-value to get the probability of and observation
from the sample not exceeding that x-value. For example, the value of
225 on the x-axis corresponds to about 0.85 on the y-axis, so there's an
85% chance that an observation in the sample does not exceed 225.
Conversely, setting, ``cumulative`` to -1 as is done in the
last series for this example, creates a "exceedance" curve.

Selecting different bin counts and sizes can significantly affect the
shape of a histogram. The Astropy docs have a great section on how to
select these parameters:
http://docs.astropy.org/en/stable/visualization/histogram.html





.. image:: /gallery/statistics/images/sphx_glr_histogram_cumulative_001.png
    :align: center





.. code-block:: python


    import numpy as np
    import matplotlib.pyplot as plt
    from matplotlib import mlab

    np.random.seed(19680801)

    mu = 200
    sigma = 25
    n_bins = 50
    x = np.random.normal(mu, sigma, size=100)

    fig, ax = plt.subplots(figsize=(8, 4))

    # plot the cumulative histogram
    n, bins, patches = ax.hist(x, n_bins, normed=1, histtype='step',
                               cumulative=True, label='Empirical')

    # Add a line showing the expected distribution.
    y = mlab.normpdf(bins, mu, sigma).cumsum()
    y /= y[-1]

    ax.plot(bins, y, 'k--', linewidth=1.5, label='Theoretical')

    # Overlay a reversed cumulative histogram.
    ax.hist(x, bins=bins, normed=1, histtype='step', cumulative=-1,
            label='Reversed emp.')

    # tidy up the figure
    ax.grid(True)
    ax.legend(loc='right')
    ax.set_title('Cumulative step histograms')
    ax.set_xlabel('Annual rainfall (mm)')
    ax.set_ylabel('Likelihood of occurrence')

    plt.show()

**Total running time of the script:** ( 0 minutes  0.028 seconds)



.. container:: sphx-glr-footer


  .. container:: sphx-glr-download

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



  .. container:: sphx-glr-download

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

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

    `Generated by Sphinx-Gallery <https://sphinx-gallery.readthedocs.io>`_
