Population and Sample

Introduction

• In statistical mathematics population can mean a set of observations or objects.

• Population, in statistics and quantitative methodology, can be defined as a collection of data satisfying specific conditions.

• A sample can be defined as a group of observations from a population.

• The sample size is always less than the size of the population.

• Non-probability sampling can further be divided into quota sampling, judgement sampling, and purposive sampling.

• Population and sample are used in market research widely in inferring behaviour of a population.

• Statistical analysis in financial decisions also implements population and sample. In this tutorial we will learn about Population, types of population, sample, type of sample, formulae based on population and sample, such as mean deviation, standard deviation and variance.

Population

Population in statistics and research is a collection of elements that is defined by parameters. Generally, the word “population” is associated with a group of people in a specific area, country or state. For example, the number of car showrooms in Delhi, or the number of grocery stores in a particular region.

Types of Population

Population can be categorized into four types depending on the mathematical characteristics of the data −

• Finite Population

• Infinite Population

• Existent Population

• Hypothetical Population

Finite Population

As the name suggests, finite population is the set of countable populations. In other words, a finite collection of population is defined as a finite population. It is easier to work with a finite population than an infinite population. For example, online consumers of a specific product, people using WI-FI, government employees in a state, etc.

Infinite Population

Infinite population can be described as the set of uncountable populations. In other words, a set having an infinite number of elements is known as an infinite population. For example, the number of bacteria in the air, number of microorganisms in the human body, number of cells in a living organism, etc.

Existent Population

Existent population can be described as a set of elements whose existence is clear or understood and can be described as a number of units. In other words, an existential population is the collection of concrete elements. For example, the number of students in a class, number of computers in an institute, etc.

Hypothetical Population

Hypothetical population can be described as a set of elements whose existence is not clear or not understood and cannot be described as a number of units. In other words, a hypothetical population is the collection of hypothetical observations. Sometimes the elements of a population can be hypothetical, for example, selection of a card from a pile of cards, outcome of a match, etc.

Sample

Sample is a set of elements drawn from a population. It is a subset of the population or represents a population. Samples in a population are derived based on probability. For example, the set of cars manufactured in India in 2022 is a sample of cars manufactured in India since 1990.

Types of Sampling

The sampling is based on two types −

• Probability sampling

• Non-probability sampling

Probability sampling

Probability sampling is the process of selecting a sampling size based on the probability of a certain parameter in a population. The types of probability sampling techniques are given below −

• Simple random sampling

• Cluster sampling

• Stratified Sampling

• Disproportionate sampling

• Proportionate sampling

• Optimum allocation stratified sampling

• Multi-stage sampling

Non-probability sampling

A non-probability sampling can be defined as the process of selecting elements based entirely at the discretion of the sampling person or user. This type of sampling need not have any theoretical basis for the selection of population. Types of non-probability sampling are given below −

• Quota sampling

• Judgement sampling

• Purposive sampling

	Population	Sample
Meaning	Set of objects or observations with similar characteristics.	A subset of the population.
Includes	Represents every element of the group.	Represents a subunit of a population
Characteristics	“Parameter” is used to describe the population	“Statistic” is used to describe the population
Data Collection	It is usually a record of census or enumeration of an entire population	Data is used to do a survey or sampling of a particular characteristic.
Focus on	Collection is used to identify the characteristics of the population.	Sample is used to infer characteristics of a population.

Formulae

Let us discuss some of the formulae associated with population and sample. For a population of size “n” where “n-1” is the sample size, the formula for mean absolute deviation (MAD), variance and standard deviation are given below −

Population mean absolute deviation$\mathrm{=\frac{1}{n} ∑_{i=1}^n |x_i-x̄ |}$

Sample mean absolute deviation$\mathrm{=\frac{1}{n-1} ∑_{i=1}^n |x_i-x̄ |}$

Population variance $\mathrm{(σx)^2=\frac{1}{n} \sum_{i=1}^n (x_i-x̄)^2}$

Sample variance $\mathrm{(Sx)^2=\frac{1}{(n-1)} \sum_{i=1}^n (x_i-x̄)^2}$

Population standard deviation $\mathrm{σx=\sqrt{\frac{1}{n} \sum_{i=1}^n (x_i-x̄)^2}}$

Sample standard deviation $\mathrm{Sx=\sqrt{\frac{1}{(n-1)} \sum_{i=1}^n (x_i-x̄)^2}}$

Solved Examples

1.Find the population variance for the set of observations {12,15,14,18,6}.

Solution: From the data given, we can determine

$$\mathrm{n=5}$$

$$\mathrm{x̄=(12+15+14+18+6)/5}$$

$$\mathrm{x̄=1}$$

Population variance $\mathrm{=\frac{1}{n} \sum_{i=1}^n (x_i-x̄)^2}$

$$\mathrm{=\frac{1}{5}[(12-13)^2+(15-13)^2+(14-13)^2+(18-13)^2+(6-13)^2]}$$

$$\mathrm{=\frac{1}{5}[(-1)^2+(2)^2+(1)^2+(5)^2+(7)^2]}$$

$$\mathrm{=\frac{1}{5}[1+4+1+25+49]}$$

$$\mathrm{=\frac{1}{5}[80]}$$

Population variance=16

2.Calculate the sample standard deviation for the dataset {6,15,18,11,20}.

Solution: From the data given, we can determine

$$\mathrm{n=5}$$

$$\mathrm{x̄=(6+15+18+11+20)/5}$$

$$\mathrm{x̄=14}$$

Sample standard deviation $\mathrm{Sx=\sqrt{\frac{1}{(n-1)} \sum_{i=1}^n (x_i-x̄)^2}}$

$\mathrm{Sx=\sqrt{\frac{1}{(n-1)} \sum_{i=1}^n (x_i-x̄)^2}}$

$\mathrm{Sx=\sqrt{\frac{1}{4}[(6-14)^2+(15-14)^2+(18-14)^2+(11-14)^2+(20-14)^2]}}$

$\mathrm{Sx=\sqrt{\frac{1}{4}[(-8)^2+(1)^2+(4)^2+(-3)^2+(6)^2]}}$

$\mathrm{Sx=\sqrt{\frac{1}{4}[64+1+16+9+36]}}$

$\mathrm{Sx=\sqrt{\frac{1}{4}[126]}}$

Sample standard deviation $\mathrm{=\sqrt{\frac{1}{4}[126]}}$

Conclusion

Population in statistics and research is a collection of elements that is defined by parameters. For example, the number of car showrooms in Delhi, or the number of grocery stores in a particular region. A sample can be defined as a group of observations from a population, for example students studying in class 9 in a school. Population can be categorized into four types depending on the mathematical characteristics: finite population, infinite population, existent population, and hypothetical population. Sample can be divided into two types: probability sampling and non-probability sampling. Probability sampling can further be divided into Simple random sampling, cluster sampling, stratified sampling, disproportionate sampling, proportionate sampling, optimum allocation stratified sampling, and multi-stage sampling. Non-probability sampling can further be divided into quota sampling, judgement sampling, and purposive sampling. Population and sample are used in market research widely in inferring behaviour of a population. Statistical analysis in financial decisions also implements population and sample.

FAQs

1.What is a statistic in reference to samples?

A statistic is a measure on which the sample is taken from a population.

2.What is a parameter in reference to population?

A parameter is a tool that describes a population.

3.What is a sampling error?

Difference between population parameter and the sample statistic is called the sampling error.

4.What are the types of population?

There are four types of population depending on the mathematical characteristics: finite population, infinite population, existent population, and hypothetical population.

5.What are the types of probability sample?

Types of probability sampling: Simple random, cluster, stratified, disproportionate, proportionate, optimum allocation stratified, and multi-stage sampling.