Standard Error 2

Standard error (SE) is a statistical term used to measure the accuracy within a sample taken from a 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 mean (average values) and variance. The standard error of the mean is considered the standard deviation of all those possible samples drawn from the same population.

The size of the standard error is affected by the standard deviation of the population and the 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 the standard error of estimate, indicating that there is less variability in the sample means, whereas by less variability we mean that the sample is more representative of the population of interest.

If the sampled population is not very large, we need to make some adjustments in computing the SE of the sample means. For a finite population, in which the 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

\[\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N-n}{N-1}}\]

Standard Error is used to

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

The SE is computed from sample statistic. To compute SE for simple random samples, assuming that the size of the 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 SE is identical to the standard deviation, except that it uses statistics whereas the standard deviation uses the parameter.

For more about SE follow the link Standard Error of Estimate

R for Data Analysis

MCQs Mathematics Intermediate