This documentation is for astroML version 0.2

This page


astroML Mailing List

GitHub Issue Tracker


Scipy 2012 (15 minute talk)

Scipy 2013 (20 minute talk)


If you use the software, please consider citing astroML.

Example of central limit theoremΒΆ

Figure 3.20.

An illustration of the central limit theorem. The histogram in each panel shows the distribution of the mean value of N random variables drawn from the (0, 1) range (a uniform distribution with \mu = 0.5 and W = 1; see eq. 3.39). The distribution for N = 2 has a triangular shape and as N increases it becomes increasingly similar to a Gaussian, in agreement with the central limit theorem. The predicted normal distribution with \mu = 0.5 and \sigma = 1/ \sqrt{12 N} is shown by the line. Already for N = 10, the “observed” distribution is essentially the same as the predicted distribution.

# Author: Jake VanderPlas
# License: BSD
#   The figure produced by this code is published in the textbook
#   "Statistics, Data Mining, and Machine Learning in Astronomy" (2013)
#   For more information, see
#   To report a bug or issue, use the following forum:
import numpy as np
from matplotlib import pyplot as plt
from scipy.stats import norm

# This function adjusts matplotlib settings for a uniform feel in the textbook.
# Note that with usetex=True, fonts are rendered with LaTeX.  This may
# result in an error if LaTeX is not installed on your system.  In that case,
# you can set usetex to False.
from astroML.plotting import setup_text_plots
setup_text_plots(fontsize=8, usetex=True)

# Generate the uniform samples
N = [2, 3, 10]

x = np.random.random((max(N), 1E6))

# Plot the results
fig = plt.figure(figsize=(5, 5))

for i in range(len(N)):
    ax = fig.add_subplot(3, 1, i + 1)

    # take the mean of the first N[i] samples
    x_i = x[:N[i], :].mean(0)

    # histogram the data
    ax.hist(x_i, bins=np.linspace(0, 1, 101),
            histtype='stepfilled', alpha=0.5, normed=True)

    # plot the expected gaussian pdf
    mu = 0.5
    sigma = 1. / np.sqrt(12 * N[i])
    dist = norm(mu, sigma)
    x_pdf = np.linspace(-0.5, 1.5, 1000)
    ax.plot(x_pdf, dist.pdf(x_pdf), '-k')

    ax.set_xlim(0.0, 1.0)
    ax.set_ylim(0.001, None)


    ax.text(0.99, 0.95, r"$N = %i$" % N[i],
            ha='right', va='top', transform=ax.transAxes)

    if i == len(N) - 1: