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Loic Guegan 2023-10-18 22:09:21 +02:00
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7
source/statistics/bessel_correction.rst
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@ -8,7 +8,8 @@ Bessel's Correction
Bessel's correction is the use of :math:`n-1` instead of :math:`n` in the formulas for sample
variance and sample standard deviation.
In fact, using :math:`n` as a denominator leads to a biased estimator.
This variance estimator is noted :math:`s^2_n`.
The biased estimator for the sample standard deviation is noted :math:`s_n`.
The biased estimator for the sample variance is noted :math:`s^2_n`.
Lets compute the discrepency between population variance and the biased sample variance:
.. math::
@ -33,9 +34,9 @@ Lets compute the discrepency between population variance and the biased sample v
&= \frac{\sigma^2}{n}
This result shows us that the discrepency between the population and sample variance is :math:`\frac{\sigma^2}{n}`.
This result shows that the discrepency between the population and sample variance is :math:`\frac{\sigma^2}{n}`.
It is simply, the :ref:`Standard Error of the Mean <SEM>`.
From this result we can deduce how :math:`S_n^2` must be adjusted:
From this result we can deduce how :math:`s_n^2` must be adjusted to be unbiased:
.. math::
\mathbb{E} \left[ s^2_n \right] = \sigma^2 - \frac{\sigma^2}{n} = \frac{n-1}{n} \sigma^2