Sampling and Sampling distributions

Sampling is a technique to draw samples from a population, that is, Sampling is a process. The sampling distribution is a probability distribution of a statistic (such as mean, variance, proportion, etc.). This category contains posts about sampling and sampling distribution.

Sampling Frame and Sampling Unit: A Quick Reference

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The post is about the concept of Sampling Frame and Sampling Unit.

Sampling Unit

The population divided into a finite number of distinct and identifiable units is called sampling units. OR

The individuals whose characteristics are to be measured in the analysis are called elementary or sampling units. OR

Before selecting the sample, the population must be divided into parts called sampling units or simply sample units.

Sampling Frame

The list of all the sampling units with a proper identification (which represents the population to be covered is called the sampling-frame). The frame may consist of either a list of units or a map of the area (in case a sample of the area is being taken), such that every element in the population belongs to one and only one unit.

The frame should be accurate, free from omission and duplication (overlapping), adequate, and up-to-date units must cover the whole of the population and should be well identified.

In improving the sampling design, supplementary information for the field covered by the sampling frame may also be valuable.

Sampling Frame and Sampling Unit

Sampling Frame and Sampling Unit: Examples

  1. List of households (and persons) enumerated in the population census.
  2. A map of areas of a country showing the boundaries of area units.
  3. In sampling an agricultural crop, the unit might be a field, a farm, or an area of land whose shape and dimensions are at our disposal.

An ideal sampling frame will have the following qualities/characteristics:

  • all sampling units have a logical and numerical identifier
  • all sampling units can be found i.e. contact information, map location, or other relevant information about sampling units is present
  • the frame is organized in a logical and systematic manner
  • the sampling frame has some additional information about the units that allow the use of more advanced sampling frames
  • every element of the population of interest is present in the frame
  • every element of the population is present only once in the frame
  • no elements from outside the population of interest are present in the frame
  • the data is up-to-date

Classification of Sampling Frame

A sampling frame can be classified as subject to several types of defects as follows:

A frame may be inaccurate: where some of the sampling units of the population are listed inaccurately or some units that do not exist are included in the list.

A frame may be inadequate: when it does not include all classes of the population that are to be taken in the survey.

A frame may be incomplete: when some of the sampling units of the population are either completely omitted or include more than once.

A frame may be out of date: when it has not been updated according to the demand of the occasion, although it was accurate, complete, and adequate at the time of construction.

Imagine you are interested in studying the eating habits of people in your city. The entire population of the city would be too big to survey, so you decide to take a sample. The sampling-frame would be like a phone book of everyone in the city. The sampling unit would be each person listed in the phone book.

Summary

Remember that the quality of the sampling-frame directly affects the representativeness of the sample. If the frame does not accurately reflect the population, the results may be biased.

In short, the quality of the sampling-frame directly affects the validity of the study. Ideally, the frame should be complete (including everyone in the target population) and accurate (with no duplicates or errors). In reality, perfect frames can be difficult to achieve, but researchers strive to get as close as possible.

FAQs about Samling Frames and Sampling Units

  1. Define Sampling frame.
  2. Define Sampling unit.
  3. How a sampling frame should be?
  4. What is the classification of the sampling frame?
  5. Give some examples of sampling frames and sampling units.

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What is Standard Error of Sampling? (2012)

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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.

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.

  1. 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.
  2. 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 $\sigma_{\bar{x}}=\frac{S}{\sqrt{n}}$

Standard Error of sampling

(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 $n_2$.

FAQs about Standard Error

  1. Define the Standard Error of Mean.
  2. Standard Error is affected by which two values?
  3. Write the formula of the standard error of mean, proportion, and difference between means.
  4. What is the application of standard error of mean in Sampling?
  5. Discuss the importance of standard error?
https://itfeature.com Standard Error

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Sampling Error Definition, Example, Formula

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In Statistics, sampling error also called estimation error is the amount of inaccuracy in estimating some value that is caused by only a portion of a population (i.e. sample) rather than the whole population. It is the difference between the statistic (value of the sample, such as sample mean) and the corresponding parameter (value of population, such as population mean) is called the sampling error. If $\bar{x}$ is the sample statistic and $\mu$ is the corresponding population parameter then it is defined as \[\bar{x} – \mu\].

Exact calculation/ measurements of sampling error are not feasible generally as the true value of the population is unknown usually, however, it can often be estimated by probabilistic modeling of the sample.

Sampling Error

Causes of Sampling-Error

  • The cause of the Error discussed may be due to the biased sampling procedure. Every research should select sample(s) that is free from any bias and the sample(s) are representative of the entire population of interest.
  • Another cause of this Error is chance. The process of randomization and probability sampling is done to minimize the sampling process error but it is still possible that all the randomized subjects/ objects are not representative of the population.

Eliminate/ Reduce the Sampling Error

The elimination/ Reduction of sampling-error can be done when a proper and unbiased probability sampling technique is used by the researcher and the sample size is large enough.

  • Increasing the sample size
    The sampling-error can be reduced by increasing the sample size. If the sample size $n$ is equal to the population size $N$, then the sampling error will be zero.
  • Improving the sample design i.e. By using the stratification
    The population is divided into different groups containing similar units.

The potential Sources of Errors are:

Potential Sources of Sampling and Non-Sampling

Also Read: Sampling and Non-Sampling Errors

Read more about Sampling Error on Wikipedia

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Sampling and Non Sampling Errors

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Before Differentiating the Sampling and Non Sampling Errors, let us define the Error term first.

The difference between an estimated value and the population’s true value is called an error. Since a sample estimate is used to describe a characteristic of a population. A sample being only a part of the population cannot provide a perfect representation of the population, no matter how carefully the sample is selected. Generally, it is seen that an estimate is rarely equal to the true value and we may think about how close will the sample estimate be to the population’s true value.

Two Kinds of Errors: Sampling and Non Sampling Errors

There are two kinds of errors, namely (I) Sampling Errors and (II) Non Sampling Errors

Sampling and Non Sampling Errors
  1. Sampling Errors (random error)
  2. Non-Sampling Errors (non-random errors)

  1. Sampling Errors
    A Sampling Error     is the difference between the value of a statistic obtained from an observed random sample and the value of the corresponding population parameter being estimated. Let $T$ be the sample statistic used to estimate the population parameter, the sampling error denoted by $E$  is  $E = T −\theta$. The value of Sampling Errors reveals the precision of the estimate. The smaller the sampling error, the greater will be the precision of the estimate. The sampling error can be reduced:

    i)   By increasing the sample size
    ii)  By improving the sampling design
    iii) By using the supplementary information

  2. Non Sampling Error
    The errors that are caused by sampling the wrong population of interest and by response bias, as well as those made by an investigator in collecting analysis and reporting the data, are all classified as non-sampling errors or non-random errors. These errors are present in a complete census as well as in the sampling survey.

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