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# Introduction

The process of selecting a sample or a portion of elements from a population or a process using a specific method is called sampling. For example,

1. A political analyst selects specific or random set of people for interviews to estimate the proportion of the votes that each candidate may get from the population of voters;
2. An auditor selects a sample of vouchers and calculates the sample mean for estimating population average amount; and
3. A doctor examines a few drops of blood to draw conclusions about the nature of disease or blood constitution of the whole body.

Samples drawn from population are analysed and sample results are called sample statistic (such as sample mean, sample proportion, sample standard deviation, etc.).

Statistical inferences are drawn about the population characteristics. These sample statistics are treated as estimator of population parameters, such as ยต, ฯ, p, etc.

# Comparison between Sampling Survey and Complete Enumeration

Complete enumeration means to collect complete information for all the units belonging to a population.

Let us compare between the two methods of survey which are as follows:

# Types of Sampling Errors

The deviation between the value of population parameter as obtained from a sample and its observed value is known as Error.

There are two types of errors:

1. Sampling error

This type of error occurs due to the error in sampling technique. Since, only a part of the population is investigated in a sampling design, every sampling design is subjected to this type of error.

The factors contributing to sampling error are listed below:
• Error arising due to defective sampling design
• Error arising due to substitution
• Error owing to wrong choice of statistic
• Variability in the population
• Error owing to faulty demarcation of units
1. Non-sampling error

This type of error happens both in sampling and complete enumeration.

The factors behind non-sampling errors are given below:
• Due to negligence and carelessness on the part of investigator
• Due to faulty planning of sampling
• Due to incomplete investigation and sample survey
• Due to faulty selection of sample unit
• Due to framing of wrong questionnaire

# Some Important Terms Associated with Sampling

1. Population: The total number of units lying under the area of study is called population. If the number of observations in a population is countable, then it is called finite population and if the observations are uncountable, then it is called infinite population.
2. Sample: A group of units of a population selected with a view of representing the population in all its characteristics is called a sample.
3. Population size and sample size: Number of units in a population is called population size (N) and the number of units in a sample is called sample size (n).
4. Parameter: It may be defined as a characteristic of a population based on all units of the population. Statistical inferences are drawn about population parameters based on the sample observations drawn from the population.
5. Statistic: Any statistical measure computed from a sample, is known as statistic. It may be defined as a statistical measure of sample observation or as a function of sample observations.
6. Sampling fluctuations: The value of a statistic varies from sample to sample. For example, if we compute the value of a statistic, say standard deviation, it is natural that its value varies from sample to sample as the sampling units are different for different samples. This variation in the value of the statistics is termed as sampling fluctuations.
7. Sampling distribution: We can obtain the values of statistic for all possible samples of a given size with their corresponding probabilities. The value of the statistic can be arranged in the form of a probability distribution. This probability distribution is known as sampling distribution.

The calculation of sampling distribution statistic is based on the following properties:
• The mean  of sampling distribution of mean values is equal to the population mean, regardless of the form of population distribution.
• The sampling distribution has a standard deviation which is also called as standard error equal to the population standard deviation divided by the square root of the sample size, i.e.,
• It may be noted that, a standard deviation is the spread of the values around the average in a single sample, whereas, the standard error is the spread of the averages around the average of averages in a sampling distribution.
• The sampling distribution of sample mean values from normally distributed populations is the normal distribution for samples.
• The mean of the statistic obtained from the sampling distribution is known as expectation.
• The standard deviation of the statistic is known as standard error.
• For sampling with replacement, the standard error is calculated using the formula,
• For sampling without replacement, the standard error is calculated using the formula:

where,

ฯ is the standard deviation of the statistic

n is sample size

N is the population size

Note: Values of a statistic are different for different samples, but the parameter always remains unknown constant.

The number of samples that can be drawn in case of sampling with replacement is Nn

The number of samples that can be drawn in case of sampling without replacement is NCn

# Basic Principles of Sample Survey

1. Law of statistical regularity: According to this law, if we draw a sample of fairly large size from the population under discussion at random, then on an average the sample would possess the characteristics of that population. The larger the sample size, better is the idea of the population. It is not always possible to increase the sample size, as it would put an extra burden on the available resources.
2. Principle of inertia: According to this principle, if all factors other than the sample size are kept constant, then the results derived from a sample are likely to be more reliable, accurate and precise as the sample size increases.
3. Principle of optimization: The principle of optimization ensures that if we have an appropriate sampling design, then an optimum level of efficiency at a minimum cost or the maximum efficiency at a given level of cost can be achieved.
4. Principle of validity: The principle of validity states that a sampling design is valid, only if it is possible to obtain valid estimates and valid tests about population parameters. Only a probability sampling ensures this validity.

Example
Consider a population comprising of the following units: 1, 3, 5, 7. Draw all possible samples of size 2,
(a) with replacement
(b) without replacement
Also, show the sampling distribution and calculate the standard error in both the cases.
Solution
Here, N = 4 and n = 2.
The total number of possible samples that can be drawn with replacement is given by
N n = 42 = 16
To show the sampling distribution and to calculate the standard error, we will need the sample mean.

So let us calculate the sample mean of every sample.

 Serial No. Sample Sample Mean (x) 1 1, 1 1 2 1, 3 2 3 1, 5 3 4 1, 7 4 5 3, 1 2 6 3, 3 3 7 3, 5 4 8 3, 7 5 9 5, 1 3 10 5, 3 4 11 5, 5 5 12 5, 7 6 13 7, 1 4 14 7, 3 5 15 7, 5 6 16 7, 7 7

Sampling distribution of sample mean is as shown below:

The total number of possible samples that can be drawn without replacement is given as follows:

To show the sampling distribution and to calculate the standard error, we will need the sample mean.

So, let us also calculate the sample mean of every sample.

 Serial No. Sample Sample Mean 1 1, 3 2 2 1, 5 3 3 1, 7 4 4 3, 5 4 5 3, 7 5 6 5, 7 6

Sampling distribution of sample mean is as shown below:

 Sample Mean 2 3 4 5 6 Total Probability (P) 1/6 1/6 2/6 1/6 1/6 1

Since we are not replacing the units during sampling, we cannot consider the entire population while calculating the variance. Hence, we cannot calculate the variance by the method we used previously.

# Different Methods of Sampling

There are three different methods of sampling. They are:

1. Probability sampling methods

In this type of sampling, there is always a fixed, pre-assigned probability for each member of the population to be a part of the sample taken from that population.

Types of Probablity Sampling Methods

• Simple random sampling: Under this method, each item of the population has an equal probability of being selected to be a part of the sample.

If the units are drawn one by one and returned to the population before the next unit is drawn, so that the composition of the original population remains unchanged at any stage of the sampling, then the sampling procedure is known as simple random sampling with replacement.

If the units that are drawn one by one are not returned to the population before drawing the next unit, it is called as sampling without replacement.

Advantages

• Each item in the population has an equal chance of being selected
• The data obtained is more reliable

Disadvantages

This method is not suitable if the population size is very small.

• Stratified sampling: This method is very suitable for a heterogeneous population. Under this method, firstly the population is sub-divided into several groups called strata on the basis of certain pre-determined criteria fixed by the sampler. Then samples of desired size are selected from each of these groups by the method of random sampling.

Advantages

• Estimates for different strata can be prepared independently
• It eliminates the difference between strata and thereby reduces the sampling error

Disadvantages

If the basis of stratification is not properly decided, the results can be misleading.

• Multi-stage sampling: Under this method, sampling is done in several stages. The population is supposed to be made up of very large units that are the first stage sampling units which in turn are supposed to be made of smaller units. These are the second stage sampling units and is made up of even smaller units which make the third stage of sampling and so on till we reach the ultimate sampling units.

The coverage in case of multi-stage sampling is quite large. It also saves computational labour and is cost-effective.
• Systematic sampling: Under this method, the units constituting the sample are selected at regular intervals of time, space or order of occurrence after selecting the very first unit at random. If the characteristics under study are independent of the order of arrangement of the units, then a systematic sample is practically equivalent to a random sample.
1. Non-probabilistic or non-random sampling methods

These types of sampling solely depend on the discretion of the investigator or sampler. Some of the important methods are as follows:
• Judgmental sampling: In this method, the enumerator selects the sample based on what he thinks would be appropriate for the study based on certain pre-determined criteria fixed by the sampler.
• Convenience sampling: In this type, the members of the population are chosen based on their relative ease of access. To sample friends, co-workers, or shoppers at a single mall, are all examples of convenience sampling.
• Quota sampling: This is a method used for selecting survey participants. In quota sampling, a population is first segmented into mutually exclusive sub-groups, just as in stratified sampling. Then judgment is used to select the subjects or units from each segment based on a specified proportion.
1. Mixed sampling method

It is a mixture or combination of probability and non-probability sampling i.e., some part of sampling is done probably and some non-probably.