# Standard Error of Estimate

Standard error (SE) is a statistical term used to measure the accuracy within a sample taken from population of interest. The standard error of the mean measures the variation in the sampling distribution of the sample mean, usually denoted by $\sigma_\overline{x}$ is calculated as

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

Drawing (obtaining) different samples from the same population of interest usually results in different values of sample means, indicating that there is a distribution of sampled means having its own mean (average values) and variance. The standard error of the mean is considered as the standard deviation of all those possible sample drawn from the same population.

The size of the standard error is affected by standard deviation of the population and number of observations in a sample called the sample size. The larger the standard deviation of the population ($\sigma$), the larger the standard error will be, indicating that there is more variability in the sample means. However larger the number of observations in a sample smaller will be the standard error of estimate, indicating that there is less variability in the sample means, where by less variability we means that the sample is more representative of the population of interest.

If the sampled population is not very larger, we need to make some adjustment in computing the SE of the sample means. For a finite population, in which total number of objects (observations) is $N$ and the number of objects (observations) in a sample is $n$, then the adjustment will be $\sqrt{\frac{N-n}{N-1}}$. This adjustment is called the finite population correction factor. Then the adjusted standard error will be

$\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}$

The SE is used to:

1. measure the spread of values of statistic about the expected value of that statistic
2. construct confidence intervals
3. test the null hypothesis about population parameter(s)

The standard error is computed from sample statistics. To compute SE for simple random samples, assuming that the size of population ($N$) is at least 20 times larger than that of the sample size ($n$).
\begin{align*}
Sample\, mean, \overline{x} & \Rightarrow SE_{\overline{x}} = \frac{n}{\sqrt{n}}\\
Sample\, proportion, p &\Rightarrow SE_{p} \sqrt{\frac{p(1-p)}{n}}\\
Difference\, b/w \, means, \overline{x}_1 – \overline{x}_2 &\Rightarrow SE_{\overline{x}_1-\overline{x}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\\
Difference\, b/w\, proportions, \overline{p}_1-\overline{p}_2 &\Rightarrow SE_{p_1-p_2}=\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}
\end{align*}

The standard error is identical to the standard deviation, except that it uses statistics whereas the standard deviation uses the parameter.