.. _book_fig_chapter3_fig_uniform_mean:


Convergence of mean for uniformly distributed values
----------------------------------------------------
Figure 3.21.

A comparison of the sample-size dependence of two estimators for the location
parameter of a uniform distribution, with the sample size ranging from
N = 100 to N =10,000. The estimator in the top panel is the sample mean,
and the estimator in the bottom panel is the mean value of two extreme values.
The theoretical 1-, 2-, and 3-sigma contours are shown for comparison. When
using the sample mean to estimate the location parameter, the uncertainty
decreases proportionally to 1/ N, and when using the mean of two extreme
values as 1/N. Note different vertical scales for the two panels.

The two methods of estimating the mean :math:`\mu` are:

- :math:`\bar\mu = \mathrm{mean}(x)`, with an error that scales as
  :math:`1/\sqrt{N}`.
- :math:`\bar\mu = \frac{1}{2}[\mathrm{max}(x) + \mathrm{min}(x)]`, with an
  error that scales as :math:`1/N`.

The shaded regions on the plot show the expected 1, 2, and 3-:math:`\sigma`
error.  Notice the difference in scale between the y-axes of the two plots.



.. image:: ../images/chapter3/fig_uniform_mean_1.png
    :scale: 100
    :align: center


.. raw:: html
    
    <div class="toggle_trigger"><a href="#">

**Code output:**

.. raw:: html

    </a></div>
    <div class="toggle_container">

.. literalinclude:: fig_uniform_mean.txt

.. raw:: html

    </div>
    <div class="toggle_trigger" id="start_open"><a href="#">

**Python source code:**

.. raw:: html

    </a></div>
    <div class="toggle_container">

.. literalinclude:: fig_uniform_mean.py
    :lines: 25-

.. raw:: html

    </div>
    <div align="right">

:download:`[download source: fig_uniform_mean.py] <fig_uniform_mean.py>`

.. raw:: html
    
    </div>