Statistics - Standard Error ( SE )

The standard deviation of a sampling distribution is called as standard error. In sampling, the three most important characteristics are: accuracy, bias and precision. It can be said that:

The estimate derived from any one sample is accurate to the extent that it differs from the population parameter. Since the population parameters can only be determined by a sample survey, hence they are generally unknown and the actual difference between the sample estimate and population parameter cannot be measured.
The estimator is unbiased if the mean of the estimates derived from all the possible samples equals the population parameter.
Even if the estimator is unbiased an individual sample is most likely going to yield inaccurate estimate and as stated earlier, inaccuracy cannot be measured. However it is possible to measure the precision i.e. the range between which the true value of the population parameter is expected to lie, using the concept of standard error.

Formula

$SE_\bar{x} = \frac{s}{\sqrt{n}}$

Where −

${s}$ = Standard Deviation
and ${n}$ = No.of observations

Example

Problem Statement:

Calculate Standard Error for the following individual data:

Items	14	36	45	70	105

Solution:

Let's first compute the Arithmetic Mean $\bar{x}$

$\bar{x} = \frac{14 + 36 + 45 + 70 + 105}{5} \\[7pt] \, = \frac{270}{5} \\[7pt] \, = {54}$

Let's now compute the Standard Deviation ${s}$

$s = \sqrt{\frac{1}{n-1}((x_{1}-\bar{x})^{2}+(x_{2}-\bar{x})^{2}+...+(x_{n}-\bar{x})^{2})} \\[7pt] \, = \sqrt{\frac{1}{5-1}((14-54)^{2}+(36-54)^{2}+(45-54)^{2}+(70-54)^{2}+(105-54)^{2})} \\[7pt] \, = \sqrt{\frac{1}{4}(1600+324+81+256+2601)} \\[7pt] \, = {34.86}$

Thus the Standard Error $SE_\bar{x}$

$SE_\bar{x} = \frac{s}{\sqrt{n}} \\[7pt] \, = \frac{34.86}{\sqrt{5}} \\[7pt] \, = \frac{34.86}{2.23} \\[7pt] \, = {15.63}$

The Standard Error of the given numbers is 15.63.

The smaller the proportion of the population that is sampled the less is the effect of this multiplier because then the finite multiplier will be close to one and will affect the standard error negligibly. Hence if the sample size is less than 5% of population, the finite multiplier is ignored.

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