Note

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# Using histograms to plot a cumulative distributionΒΆ

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 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

```
import numpy as np
import matplotlib.pyplot as plt
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, density=True, histtype='step',
cumulative=True, label='Empirical')
# Add a line showing the expected distribution.
y = ((1 / (np.sqrt(2 * np.pi) * sigma)) *
np.exp(-0.5 * (1 / sigma * (bins - mu))**2))
y = y.cumsum()
y /= y[-1]
ax.plot(bins, y, 'k--', linewidth=1.5, label='Theoretical')
# Overlay a reversed cumulative histogram.
ax.hist(x, bins=bins, density=True, 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()
```

References

The use of the following functions, methods, classes and modules is shown in this example:

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