Standard Error 2: A Quick Guide

Introduction to Standard Errors (SE)

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.

Size of the Standard Error

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 population’s standard deviation ($\sigma$), the larger the standard error will be, indicating more variability in the sample means. However, the larger the number of observations in a sample, the smaller the estimate’s SE, indicating less variability in the sample means. In contrast, by less variability, we mean that the sample is more representative of the population of interest.

Adjustments in Computing SE of Sample Means

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

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

Uses of Standard Error

1. It measures the spread of values of statistics about the expected value of that statistic. It helps us understand how well a sample represents the entire population.
2. It is used to construct confidence intervals, which provide a range of values likely to contain the true population parameter.
3. It helps to test the null hypothesis about population parameter(s), such as t-tests and z-tests. It helps determine the significance of differences between sample means or between a sample mean and a population mean.
4. It helps in determining the required sample size for a study to achieve the desired level of precision.
5. By comparing standard errors of different samples or estimates, one can assess the relative variability and reliability of those estimates.

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*}

Summary

The SE provides valuable insights about the reliability and precision of sample-based estimates. By understanding SE, a researcher can make more informed decisions and draw more accurate conclusions from the data under study. The SE is identical to the standard deviation, except that it uses statistics whereas the standard deviation uses the parameter.

FAQS about SE

1. What is the SE, and how it is computed?
2. What are the uses of SE?
3. From which is the size of the SE affected?
4. When will the SE be large?
5. When will the SE be small?
6. What will be the standard error for proportion?

For more about SE follow the link Standard Error of Estimate

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MCQs Mathematics Intermediate Second Year