The standard error (SE) of a statistic is the standard deviation of the sampling distribution of that statistic. The standard error of sampling reflects how much sampling fluctuation a statistic will show. The inferential (deductive) statistics involved in constructing confidence intervals and significance testing are based on standard errors. Increasing the sample size decreases the standard error.

## Table of Contents

In practical applications, the true value of the standard deviation of the error is unknown. As a result, the term standard error is often used to refer to an estimate of this unknown quantity.

The size of the SE is affected by two values.

- The Standard Deviation of the population affects the standard errors. The larger the population’s standard deviation ($\sigma$), the larger is SE i.e. $\frac {\sigma}{\sqrt{n}}$. If the population is homogeneous (which results in a small population standard deviation), the SE will also be small.
- The standard errors are affected by the number of observations in a sample. A large sample will result in a small SE of estimate (indicates less variability in the sample means)

**Application of Standard Error** **of Sampling**

The SEs are used in different statistical tests such as

- to measure the distribution of the sample means
- to build confidence intervals for means, proportions, differences between means, etc., for cases when population standard deviation is known or unknown.
- to determine the sample size
- in control charts for control limits for means
- in comparison tests such as
*z*-test,*t*-test, Analysis of Variance, - in relationship tests such as Correlation and Regression Analysis (standard error of regression), etc.

**(1) Standard Error Formula Means**

The SE for the mean or standard deviation of the sampling distribution of the mean measures the deviation/ variation in the sampling distribution of the sample mean, denoted by $\sigma_{\bar{x}}$ and calculated as the function of the standard deviation of the population and respective size of the sample i.e

$\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$ (used when population is finite)

If the population size is infinite then ${\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}} \times \sqrt{\frac{N-n}{N}}}$ because $\sqrt{\frac{N-n}{N}}$ tends towards 1 as *N* tends to infinity.

When the population’s standard deviation ($\sigma$) is unknown, we estimate it from the sample standard deviation. In this case SE formula is $\sigma_{\bar{x}}=\frac{S}{\sqrt{n}}$

**(2) Standard Error Formula for Proportion**

The SE for a proportion can also be calculated in the same manner as we calculated the standard error of the mean, denoted by $\sigma_p$ and calculated as $\sigma_p=\frac{\sigma}{\sqrt{n}}\sqrt{\frac{N-n}{N}}$.

In case of finite population $\sigma_p=\frac{\sigma}{\sqrt{n}}$

in case of infinite population $\sigma=\sqrt{p(1-p)}=\sqrt{pq}$, where $p$ is the probability that an element possesses the studied trait and $q=1-p$ is the probability that it does not.

**(3) Standard Error Formula for Difference Between Means**

The SE for the difference between two independent quantities is the square root of the sum of the squared standard errors of both quantities i.e $\sigma_{\bar{x}_1+\bar{x}_2}=\sqrt{\frac{\sigma_1^2}{n_1}+\frac{\sigma_2^2}{n_2}}$, where $\sigma_1^2$ and $\sigma_2^2$ are the respective variances of the two independent population to be compared and $n_1+n_2$ are the respective sizes of the two samples drawn from their respective populations.

**Unknown Population Variances**

Suppose the variances of the two populations are unknown. In that case, we estimate them from the two samples i.e. $\sigma_{\bar{x}_1+\bar{x}_2}=\sqrt{\frac{S_1^2}{n_1}+\frac{S_2^2}{n_2}}$, where $S_1^2$ and $S_2^2$ are the respective variances of the two samples drawn from their respective population.

** Equal Variances are assumed**

In case when it is assumed that the variance of the two populations are equal, we can estimate the value of these variances with a pooled variance $S_p^2$ calculated as a function of $S_1^2$ and $S_2^2$ i.e

\[S_p^2=\frac{(n_1-1)S_1^2+(n_2-1)S_2^2}{n_1+n_2-2}\]

\[\sigma_{\bar{x}_1}+{\bar{x}_2}=S_p \sqrt{\frac{1}{n_1}+\frac{1}{n_2}}\]

**(4) Standard Error for Difference between Proportions**

The SE of the difference between two proportions is calculated in the same way as the SE of the difference between means is calculated i.e.

\begin{eqnarray*}

\sigma_{p_1-p_2}&=&\sqrt{\sigma_{p_1}^2+\sigma_{p_2}^2}\\

&=& \sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}

\end{eqnarray*}

where $p_1$ and $p_2$ are the proportion for infinite population calculated for the two samples of sizes $n_1$ and $n_2$.