Standard Error (statistics)

Standard errors provide simple measures of uncertainty in a value and are often used because:

If the standard error of several individual quantities is known then the standard error of some Function of the quantities can be easily calculated in many cases;

Where the Probability Distribution of the value is known, they can be used to calculate an exact Confidence Interval ; and

Where the probability distribution is unknown, relationships like Chebyshev 's or the Vysochanskiï-Petunin Inequality can be used to calculate a conservative confidence interval

As the sample size tends to infinity the Central Limit Theorem guarantees that the sampling distribution of the mean is asymptotically Normal .

The standard error of a sample from a Population is
the Standard Deviation of the Sampling Distribution and may be estimated by
the formula:

:

rac{\sigma}{\sqrt{n}}

where

\sigma

is the standard deviation of the population distribution
and

n

is the size (number of items) in the sample.

STANDARD ERRORS

Single sample

$\sigma_\overline{x} = rac{\sigma}{\sqrt{n}}$

$\sigma_\widehat p= \sqrt{rac{p(1-p)}{n}}$

Two samples

$\sigma_{\overline{x}_1-\overline{x}_2}=\sqrt$

$\sigma_{\widehat p_1-\widehat p_2}=\sqrt{rac{\widehat p_1(1-\widehat p_1)}{n_1}+rac{\widehat p_2(1-\widehat p_2)}{n_2}}$

A very important implication of this formula is that it is possible to halve the measurement error by quadrupling the sample size. When designing statistical studies where cost is a factor, this may have a factor in understanding cost-benefit tradeoffs.

SEE ALSO

Sampling Distribution

Standard Deviation

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Standard Error (statistics)