Theory on sampling

1. Sampling methods
In Statistics, the sampling method or sampling technique is the process of studying the
population by gathering information and analysing that data. It is the basis of the data where
the sample space is enormous. There are several different sampling techniques available, and
they can be subdivided into two groups.All these methods of sampling may involve specifically
targeting hard or approach to reach groups.
Types of Sampling Method
In Statistics, there are different sampling techniques available to get relevant results from the
population. The two different types of sampling methods are:
1. Probability Sampling-
2. Non-probability Sampling.

2. 1. Probability Sampling-
Probability sampling is a sampling technique where a researcher selects a few
criteria and chooses members of a population randomly. All the members have an equal
opportunity to participate in the sample with this selection parameter.
• Simple random sampling:
One of the best probability sampling techniques that helps in saving time and
resources is the Simple Random Sampling method. It is a reliable method of obtaining
information where every single member of a population is chosen randomly, merely by
chance. Each individual has the same probability of being chosen to be a part of a
sample.
For example, in an organization of 500 employees, if the HR team decides on
conducting team-building activities, they would likely prefer picking chits out of a
bowl. In this case, each of the 500 employees has an equal opportunity of being selected.
• Cluster sampling:
Cluster sampling is a method where the researchers divide the entire population
into sections or clusters representing a population. Clusters are identified and included
in a sample based on demographic parameters like age, sex, location, etc. This makes
it very simple for a survey creator to derive effective inferences from the feedback.
For example, suppose the United States government wishes to evaluate the
number of immigrants living in the mainland US. In that case, they can divide it into
clusters based on states such as California, Texas, Florida, Massachusetts, Colorado,
Hawaii, etc. This way of conducting a survey will be more effective as the results will
be organized into states and provide insightful immigration data.
• Systematic sampling:
Researchers use the systematic sampling method to choose the sample members
of a population at regular intervals. It requires selecting a starting point for the sample
and sample size determination that can be repeated at regular intervals. This type of
sampling method has a predefined range; hence, this sampling technique is the least
time-consuming.
For example, a researcher intends to collect a systematic sample of 500 people
in a population of 5000. He/she numbers each element of the population from 1-5000
and will choose every 10th individual to be a part of the sample (Total population/
Sample Size = 5000/500 = 10).

3. • Stratified random sampling:
Stratified random sampling is a method in which the researcher divides the
population into smaller groups that don’t overlap but represent the entire population.
While sampling, these groups can be organized, and then draw a sample from each
group separately.
For example, a researcher looking to analyze the characteristics of people
belonging to different annual income divisions will create strata (groups) according to
the annual family income. E.g. – less than $20,000, $21,000 – $30,000, $31,000 to
$40,000, $41,000 to $50,000, etc. By doing this, the researcher concludes the
characteristics of people belonging to different income groups. Marketers can analyze
which income groups to target and which ones to eliminate to create a roadmap that
would bear fruitful results.
2. Non-probability Sampling.
In non-probability sampling, the researcher randomly chooses members for
research. This sampling method is not a fixed or predefined selection process. This
makes it difficult for all population elements to have equal opportunities to be included
in a sample.
• Convenience Sampling
In a convenience sampling method, the samples are selected from the population
directly because they are conveniently available for the researcher. The samples are
easy to select, and the researcher did not choose the sample that outlines the entire
population.
Example: In researching customer support services in a particular region, we
ask your few customers to complete a survey on the products after the purchase. This
is a convenient way to collect data. Still, as we only surveyed customers taking the same
product. At the same time, the sample is not representative of all the customers in that
area.
• Consecutive Sampling
Consecutive sampling is similar to convenience sampling with a slight
variation. The researcher picks a single person or a group of people for sampling. Then
the researcher researches for a period of time to analyze the result and move to another
group if needed.

4. • Quota Sampling
In the quota sampling method, the researcher forms a sample that involves the
individuals to represent the population based on specific traits or qualities. The
researcher chooses the sample subsets that bring the useful collection of data that
generalizes the entire population.
• Purposive or Judgmental Sampling
In purposive sampling, the samples are selected only based on the researcher’s
knowledge. As their knowledge is instrumental in creating the samples, there are the
chances of obtaining highly accurate answers with a minimum marginal error. It is also
known as judgmental sampling or authoritative sampling.
• Snowball Sampling
Snowball sampling is also known as a chain-referral sampling technique. In this
method, the samples have traits that are difficult to find. So, each identified member of
a population is asked to find the other sampling units. Those sampling units also belong
to the same targeted population.
Sampling Distribution
Sampling distribution in statistics refers to studying many random samples collected
from a given population based on a specific attribute. The results obtained provide a clear
picture of variations in the probability of the outcomes derived. As a result, the analysts remain
aware of the results beforehand, and hence, they can make preparations to take action
accordingly.
Sampling
Distribution
Sampling Distribution
of Mean
Sampling Distribution
of Proportion
T-Distribution

5. • Sampling Distribution of Mean
The Sampling Distribution of the Mean is the mean of the population from
where the items are sampled. If the population distribution is normal, then the sampling
distribution of the mean is likely to be normal for the samples of all sizes.
Following are the main properties of the sampling distribution of the mean:
 Its mean is equal to the population mean, thus,
𝜇𝜇𝑥𝑥̅ = 𝜇𝜇
(𝜇𝜇𝑥𝑥̅ =sample mean and 𝜇𝜇 Population mean)
 The population standard deviation divided by the square root of the sample size
is equal to the standard deviation of the sampling distribution of the mean, thus:
𝜎𝜎𝑥𝑥̅ =
𝜎𝜎
√𝑛𝑛
(σ = population standard deviation, n = sample size)
 The sampling distribution of the mean is normally distributed.
• Sampling distribution of proportion
This method involves choosing a sample set from the overall population to get the
proportion of the sample.
The mean of the proportions ends up becoming the proportion of the larger group.
Mean:
𝜇𝜇𝑃𝑃 = 𝑝𝑝
𝜇𝜇𝑃𝑃 – Mean of population
p- Sample
Deviation:
𝜎𝜎𝑃𝑃 = �
𝑝𝑝𝑝𝑝
𝑛𝑛
𝜎𝜎𝑃𝑃- Deviation of population
𝑝𝑝− Population proportion
𝑞𝑞 = 1 −p
𝑛𝑛 − Sample size

6. • T- Distribution-
T-Distribution is a continuous probability distribution. It is used when sample sizes are smaller
than the normal distribution, say less than 30. This method identifies the disparity between the
sample and population means when the population standard deviation is unknown.
𝑡𝑡 =
𝑥𝑥̅ − 𝜇𝜇
𝑆𝑆
√𝑛𝑛
Where, 𝑥𝑥̅ is the mean of first sample.
𝜇𝜇- Mean of second sample
𝑆𝑆
√𝑛𝑛
- the estimate of the standard error of difference between the means.
Properties of T Distribution
• It ranges from −∞ to +∞.
• It has a bell-shaped curve and symmetry similar to normal distribution.
• The variance of the t-distribution is always greater than ‘1’.
Sampling Error
A sampling error is a statistical error that occurs when an analyst does not select a sample that
represents the entire population of data. As a result, the results found in the sample do not
represent the results that would be obtained from the entire population.
Sampling Error = 𝑧𝑧 ×
𝜎𝜎
√𝑛𝑛
z= Z score value based on the confidence interval (approx.=1.96)
σ=Population standard deviation
n=Size of the sample
Types of Sampling Errors
There are different categories of sampling errors.
• Population-Specific Error
A population-specific error occurs when a researcher doesn't understand who to survey.

7. • Selection Error
Selection error occurs when the survey is self-selected, or when only those participants who
are interested in the survey respond to the questions. Researchers can attempt to overcome
selection error by finding ways to encourage participation.
• Sample Frame Error
A sample frame error occurs when a sample is selected from the wrong population data.
• Non-response Error
A non-response error occurs when a useful response is not obtained from the surveys because
researchers were unable to contact potential respondents (or potential respondents refused to
respond).