Sampling and Sampling distributions

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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Introduction to Sampling Theory

Often we are interested in drawing some valid conclusions (inferences) about a large group of individuals or objects (called population in statistics). Instead of examining (studying) the entire group (population, which may be difficult or even impossible to examine), we may examine (study) only a small part (portion) of the population (an entire group of objects or people). Our objective is to draw valid inferences about certain facts about the population from results found in the sample; a process known as statistical inferences. The process of obtaining samples is called sampling and the theory concerning the sampling is called sampling theory.

Example

Example: We may wish to conclude the percentage of defective bolts produced in a factory during a given 6-day week by examining 20 bolts each day produced at various times during the day. Note that all bolts produced in this case during the week comprise the population, while the 120 selected bolts during 6 days constitute a sample.

In business, medical, social, and psychological sciences, etc., research, sampling theory is widely used for gathering information about a population. The sampling process comprises several stages:

• Defining the population of concern
• Specifying the sampling frame (set of items or events possible to measure)
• Specifying a sampling method for selecting the items or events from the sampling frame
• Determining the appropriate sample size
• Implementing the sampling plan
• Sampling and data collecting
• Data that can be selected

Reasons to Study a Sample

When studying the characteristics of a population, there are many reasons to study a sample (drawn from the population under study) instead of the entire population such as:

1. Time: it is difficult to contact every individual in the whole population
2. Cost: The cost or expenses of studying all the items (objects or individuals) in a population may be prohibitive
3. Physically Impossible: Some populations are infinite, so it will be physically impossible to check all items in the population, such as populations of fish, birds, snakes, and mosquitoes. Similarly, it is difficult to study the populations that are constantly moving, being born, or dying.
4. Destructive Nature of items: Some items, objects, etc. are difficult to study as during testing (or checking) they are destroyed, for example, a steel wire is stretched until it breaks and the breaking point is recorded to have a minimum tensile strength. Similarly different electric and electronic components are checked and they are destroyed during testing, making it impossible to study the entire population as time, cost and destructive nature of different items prohibit to study of the entire population.
5. Qualified and expert staff: For enumeration purposes, highly qualified and expert staff is required which is sometimes impossible. National and International research organizations, agencies, and staff are hired for enumeration purposive which is sometimes costly, needs more time (as a rehearsal of activity is required), and sometimes it is not easy to recruit or hire highly qualified staff.
6. Reliability: Using a scientific sampling technique the sampling error can be minimized and the non-sampling error committed in the case of a sample survey is also minimal because qualified investigators are included.

Sampling Methods

The sampling methods can be classified into probability sampling and non-probabilit sampling methods.

• Probability Sampling: Every member has a known chance of being selected (e.g., random, stratified, systematic, cluster sampling).
• Non-Probability Sampling: Selection is not random (e.g., convenience, purposive, snowball sampling).

Summary

Every sampling system is used to obtain some estimates having certain properties of the population under study. The sampling system should be judged by how good the estimates obtained are. Individual estimates, by chance, may be very close or may differ greatly from the true value (population parameter) and may give a poor measure of the merits of the system.

A sampling system is better judged by the frequency distribution of many estimates obtained by repeated sampling, giving a frequency distribution having a small variance and a mean estimate equal to the true value.

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