Tag: Sampling Techniques

Management Notes

Sampling Techniques (Probability and Non-Probability Sampling Techniques)

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Sampling Techniques refer to the methods used to select individuals, items, or data points from a larger population for research purposes. These techniques ensure that the sample accurately represents the entire population, allowing for valid and reliable conclusions. Sampling techniques are broadly classified into two categories: probability sampling (where every element has an equal chance of being selected) and non-probability sampling (where selection is based on researcher judgment or convenience). Common methods include random sampling, stratified sampling, cluster sampling, convenience sampling, and purposive sampling. Choosing the right sampling technique is crucial because it impacts the quality, accuracy, and generalizability of the research findings. Proper sampling reduces bias and increases research credibility.

Probability Sampling Techniques

Probability sampling techniques are methods where every member of the population has a known and equal chance of being selected for the sample. These techniques aim to eliminate selection bias and ensure that the sample is truly representative of the entire population. Common types of probability sampling include simple random sampling, systematic sampling, stratified sampling, and cluster sampling. Researchers often prefer probability sampling because it allows the use of statistical methods to estimate population parameters and test hypotheses accurately. This approach enhances the validity, reliability, and generalizability of research findings, making it fundamental in scientific studies and decision-making processes.

Types of Probability Sampling Techniques:

  • Simple Random Sampling

Every population member has an equal, independent chance of selection, typically using random number generators or lotteries. This method eliminates selection bias and ensures representativeness, making it ideal for homogeneous populations. However, it requires a complete sampling frame and may miss small subgroups. Despite its simplicity, large sample sizes are often needed for precision. It’s widely used in surveys and experimental research where unbiased representation is critical.

  • Stratified Random Sampling

The population is divided into homogeneous subgroups (strata), and random samples are drawn from each. This ensures representation of key characteristics (e.g., age, gender). It improves precision compared to simple random sampling, especially for heterogeneous populations. Proportionate stratification maintains population ratios, while disproportionate stratification may oversample rare groups. This method is costlier but valuable when subgroup comparisons are needed, such as in clinical or sociological studies.

  • Systematic Sampling

A fixed interval (*k*) is used to select samples from an ordered population list (e.g., every 10th person). The starting point is randomly chosen. This method is simpler than random sampling and ensures even coverage. However, if the list has hidden patterns, bias may occur. It’s efficient for large populations, like quality control in manufacturing or voter surveys, but requires caution to avoid periodicity-related distortions.

  • Cluster Sampling

The population is divided into clusters (e.g., schools, neighborhoods), and entire clusters are randomly selected for study. This reduces logistical costs, especially for geographically dispersed groups. However, clusters may lack internal diversity, increasing sampling error. Two-stage cluster sampling (randomly selecting subjects within chosen clusters) improves accuracy. It’s practical for national health surveys or educational research where individual access is challenging.

  • Multistage Sampling

A hybrid approach combining multiple probability methods (e.g., clustering followed by stratification). Large clusters are selected first, then subdivided for further random sampling. This balances cost and precision, making it useful for large-scale studies like census data collection or market research. While flexible, it requires careful design to minimize cumulative errors and maintain representativeness across stages.

Non-Probability Sampling Techniques:

Non-probability Sampling refers to research methods where samples are selected through subjective criteria rather than random selection, meaning not all population members have an equal chance of participation. These techniques are used when probability sampling is impractical due to time, cost, or population constraints. Common approaches include convenience sampling (easily accessible subjects), purposive sampling (targeted selection of specific characteristics), snowball sampling (participant referrals), and quota sampling (pre-set subgroup representation). While these methods enable faster, cheaper data collection in exploratory or qualitative studies, they carry higher risk of bias and limit result generalizability to broader populations. Researchers employ them when prioritizing practicality over statistical representativeness.

Types of Non-Probability Sampling Techniques:

  • Convenience Sampling

Researchers select participants who are most easily accessible, such as students in a classroom or shoppers at a mall. This method is quick, inexpensive, and requires minimal planning, making it ideal for preliminary research. However, results suffer from significant bias since the sample may not represent the target population. Despite limitations, convenience sampling is widely used in pilot studies, exploratory research, and when time/resources are constrained.

  • Purposive (Judgmental) Sampling

Researchers deliberately select specific individuals who meet predefined criteria relevant to the study. This technique is valuable when studying unique populations or specialized topics requiring expert knowledge. While it allows for targeted data collection, the subjective selection process introduces researcher bias. Purposive sampling is commonly used in qualitative research, case studies, and when investigating rare phenomena where random sampling isn’t feasible.

  • Snowball Sampling

Existing study participants recruit future subjects from their acquaintances, creating a chain referral process. This method is particularly useful for reaching hidden or hard-to-access populations like marginalized communities. While effective for sensitive topics, the sample may become homogeneous as participants share similar networks. Snowball sampling is frequently employed in sociological research, studies of illegal behaviors, and when investigating stigmatized conditions.

  • Quota Sampling

Researchers divide the population into subgroups and non-randomly select participants until predetermined quotas are filled. This ensures representation across key characteristics but lacks the randomness of stratified sampling. Quota sampling is more structured than convenience sampling yet still prone to selection bias. Market researchers often use this method when they need quick, cost-effective results that approximate population demographics.

  • Self-Selection Sampling

Individuals voluntarily choose to participate, typically by responding to open invitations or surveys. This approach yields large sample sizes easily but suffers from volunteer bias, as participants may differ significantly from non-respondents. Common in online surveys and call-in opinion polls, self-selection provides accessible data though results should be interpreted cautiously due to inherent representation issues.

Key differences between Probability and Non-Probability Sampling

Aspect Probability Sampling Non-Probability Sampling
Selection Basis Random Subjective
Bias Risk Low High
Representativeness High Low
Generalizability Strong Limited
Cost High Low
Time Required Long Short
Complexity High Low
Population Knowledge Required Optional
Error Control Measurable Unmeasurable
Use Cases Quantitative Qualitative
Statistical Tests Applicable Limited
Sample Frame Essential Flexible
Precision High Variable
Research Stage Confirmatory Exploratory
Participant Access Challenging Easy
Management Notes

Introduction, Meaning, Definition, Importance and Objective of Research

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Research is a systematic and organized process of collecting, analyzing, and interpreting information to increase understanding of a topic or issue. It aims to discover new facts, verify existing knowledge, or solve specific problems through careful investigation. Research can be theoretical or applied, and it involves forming hypotheses, gathering data, and drawing conclusions. It is essential in academic, scientific, and business fields to make informed decisions and improve practices. A well-conducted research study follows a structured methodology to ensure reliability and validity. Overall, research is a tool for expanding knowledge and contributing to the development of society and industries.

Definition of Research:

  1. Clifford Woody: Research is a careful inquiry or examination to discover new facts or verify old ones.

  2. Creswell: Research is a process of steps used to collect and analyze information to increase our understanding of a topic.

  3. Redman and Mory: Research is a systematized effort to gain new knowledge.

  4. Kerlinger: Research is a systematic, controlled, empirical, and critical investigation of hypothetical propositions.

  5. Lundberg: Research is a systematic activity directed towards the discovery and development of an organized body of knowledge.

Importance of Research:

  • Expansion of Knowledge

Research plays a vital role in expanding human knowledge. It helps us understand concepts, theories, and facts in a deeper and more meaningful way. Through systematic investigation, research uncovers hidden truths and broadens the scope of what is already known. This continuous process of discovery is essential in education, science, and innovation. Without research, the development of new ideas, improvements in technology, and advancements in various fields would come to a standstill.

  • Problem Solving

One of the main purposes of research is to find solutions to problems. In both academic and practical settings, research helps identify the root causes of issues and suggests possible remedies. Whether it’s a social, economic, scientific, or business problem, research provides the tools and frameworks to analyze the situation effectively. It allows decision-makers to make evidence-based choices and implement strategies that are backed by data and analysis, leading to more successful outcomes.

  • Informed Decision Making

Research enables individuals, organizations, and governments to make informed decisions. By analyzing data and studying trends, research provides a factual basis for choosing between alternatives. In business, it helps managers decide on product development, marketing strategies, and investment plans. In public policy, it helps lawmakers craft laws that address real needs. This reduces the risk of failure and ensures that decisions are effective, efficient, and aligned with actual conditions and demands.

  • Economic Development

Research is essential for economic growth and development. It leads to the creation of new products, services, and technologies, which drive industry and generate employment. By improving productivity, reducing costs, and increasing competitiveness, research directly contributes to the success of businesses and national economies. Additionally, research in areas like agriculture, health, and education ensures sustainable development by solving real-world problems and improving the quality of life for individuals and communities.

  • Improvement in Education

Research strengthens the education system by improving teaching methods, learning outcomes, and academic content. It helps educators understand student needs, evaluate curricula, and adopt innovative practices. Research also enables students and teachers to stay updated with the latest knowledge in their field, promoting lifelong learning. Educational research contributes to the development of better textbooks, e-learning tools, and inclusive teaching strategies that cater to diverse learning styles and backgrounds.

  • Policy Formulation

Government and institutional policies must be based on reliable data and analysis, which research provides. Whether in health, education, environment, or public safety, research ensures that policies are relevant, effective, and future-ready. It helps policymakers assess the potential impact of laws and regulations, avoiding guesswork and promoting social welfare. Evidence-based policies are more likely to gain public support and achieve their goals, ultimately benefiting the economy and society as a whole.

  • Innovation and Technology Advancement

Innovation thrives on research. From developing new medical treatments to designing smarter devices, research is the foundation of technological progress. Scientists and engineers rely on research to explore possibilities, test ideas, and turn concepts into real-world applications. Research also encourages creativity and collaboration across disciplines, pushing the boundaries of what’s possible. As technology rapidly evolves, research ensures that innovation continues to meet the needs of people and adapt to changing environments.

  • Social and Cultural Understanding

Research deepens our understanding of social and cultural dynamics. It helps explore human behavior, beliefs, traditions, and societal changes. Through research in fields like sociology, anthropology, and psychology, we gain insights into communities and cultures, fostering tolerance and mutual respect. This understanding is crucial in a globalized world where collaboration and coexistence are key. It also helps in addressing social issues like poverty, gender inequality, and discrimination with informed, data-backed strategies.

Objective of Research:

  • To Gain Familiarity with a Phenomenon

One major objective of research is to explore and understand a phenomenon or concept more clearly. This is often done through exploratory research, especially when little prior knowledge exists. It helps researchers gain insights into new topics, identify trends, and lay the groundwork for future studies. By becoming familiar with unfamiliar issues, researchers can form better hypotheses and research questions. This foundational understanding is critical for developing more in-depth research and creating meaningful contributions to academic and professional fields.

  • To Describe a Phenomenon Accurately

Descriptive research aims to systematically and precisely describe the characteristics of a subject, event, or population. Whether it’s human behavior, market trends, or institutional processes, this type of research collects detailed information to create an accurate picture. The objective is not to determine cause-and-effect but to define “what is” in a clear and factual manner. Such descriptions help researchers, practitioners, and policymakers understand the current state of affairs and serve as a reference point for comparing future changes.

  • To Establish Cause-and-Effect Relationships

Causal or explanatory research seeks to identify and analyze relationships between variables, often using experiments or observational studies. The objective is to determine how and why certain phenomena occur. For instance, a business might study the impact of advertising on sales. Establishing cause-and-effect allows researchers to predict outcomes and design effective interventions. This type of research is essential in fields like science, economics, and medicine, where understanding the effects of one factor on another can lead to critical discoveries and solutions.

  • To Test Hypotheses

Another key objective of research is hypothesis testing, where assumptions or predictions made before a study are examined for accuracy. Researchers design experiments or surveys to gather data that supports or refutes their hypotheses. The goal is to provide empirical evidence for or against theoretical statements. This process sharpens theories, confirms findings, and promotes scientific accuracy. Testing hypotheses is particularly important in quantitative research, as it relies on statistical techniques to validate conclusions and ensure objectivity.

  • To Develop New Theories and Concepts

Research often leads to the creation or refinement of theories and models that explain how the world works. The objective here is to go beyond existing knowledge and offer new perspectives or conceptual frameworks. Through in-depth analysis, researchers can challenge outdated views and propose innovative explanations. These new theories guide future research, inform policy, and influence practice across disciplines. In academic fields, theoretical research forms the basis for scholarly progress and intellectual advancement.

  • To Find Solutions to Practical Problems

Applied research is conducted with the specific objective of solving real-world problems. Whether it’s improving product design, enhancing public health, or increasing workplace efficiency, the goal is to apply scientific methods to practical challenges. This kind of research is widely used in industries, education, and government. It not only addresses current issues but also anticipates future needs. By developing effective strategies and solutions, applied research makes a direct contribution to societal well-being and economic development.

  • To Predict Future Trends

Research aims to forecast what may happen in the future based on current and past data. Predictive research uses statistical tools and modeling techniques to identify patterns and trends that inform future outcomes. For example, businesses use market research to predict consumer behavior, and climate scientists use data to forecast environmental changes. These predictions guide planning and strategic decisions. Accurate forecasting is essential for minimizing risk, improving preparedness, and making proactive decisions in dynamic environments.

  • To Enhance Understanding and Clarify Doubts

Research helps deepen our understanding of complex topics and clarifies uncertainties that may exist in previous studies or beliefs. By investigating issues from multiple angles, using various methods, and verifying results, research ensures greater clarity and accuracy. This objective is crucial in academia and science, where incomplete or conflicting information often leads to confusion. Ongoing research contributes to refinement, resolution of debates, and filling knowledge gaps, ensuring a more complete and reliable understanding of any subject.

Management Notes

Sampling and Sampling Distribution

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Sample design is the framework, or road map, that serves as the basis for the selection of a survey sample and affects many other important aspects of a survey as well. In a broad context, survey researchers are interested in obtaining some type of information through a survey for some population, or universe, of interest. One must define a sampling frame that represents the population of interest, from which a sample is to be drawn. The sampling frame may be identical to the population, or it may be only part of it and is therefore subject to some under coverage, or it may have an indirect relationship to the population.

Sampling is the process of selecting a subset of individuals, items, or observations from a larger population to analyze and draw conclusions about the entire group. It is essential in statistics when studying the entire population is impractical, time-consuming, or costly. Sampling can be done using various methods, such as random, stratified, cluster, or systematic sampling. The main objectives of sampling are to ensure representativeness, reduce costs, and provide timely insights. Proper sampling techniques enhance the reliability and validity of statistical analysis and decision-making processes.

Steps in Sample Design

While developing a sampling design, the researcher must pay attention to the following points:

  • Type of Universe:

The first step in developing any sample design is to clearly define the set of objects, technically called the Universe, to be studied. The universe can be finite or infinite. In finite universe the number of items is certain, but in case of an infinite universe the number of items is infinite, i.e., we cannot have any idea about the total number of items. The population of a city, the number of workers in a factory and the like are examples of finite universes, whereas the number of stars in the sky, listeners of a specific radio programme, throwing of a dice etc. are examples of infinite universes.

  • Sampling unit:

A decision has to be taken concerning a sampling unit before selecting sample. Sampling unit may be a geographical one such as state, district, village, etc., or a construction unit such as house, flat, etc., or it may be a social unit such as family, club, school, etc., or it may be an individual. The researcher will have to decide one or more of such units that he has to select for his study.

  • Source list:

It is also known as ‘sampling frame’ from which sample is to be drawn. It contains the names of all items of a universe (in case of finite universe only). If source list is not available, researcher has to prepare it. Such a list should be comprehensive, correct, reliable and appropriate. It is extremely important for the source list to be as representative of the population as possible.

  • Size of Sample:

This refers to the number of items to be selected from the universe to constitute a sample. This a major problem before a researcher. The size of sample should neither be excessively large, nor too small. It should be optimum. An optimum sample is one which fulfills the requirements of efficiency, representativeness, reliability and flexibility. While deciding the size of sample, researcher must determine the desired precision as also an acceptable confidence level for the estimate. The size of population variance needs to be considered as in case of larger variance usually a bigger sample is needed. The size of population must be kept in view for this also limits the sample size. The parameters of interest in a research study must be kept in view, while deciding the size of the sample. Costs too dictate the size of sample that we can draw. As such, budgetary constraint must invariably be taken into consideration when we decide the sample size.

  • Parameters of interest:

In determining the sample design, one must consider the question of the specific population parameters which are of interest. For instance, we may be interested in estimating the proportion of persons with some characteristic in the population, or we may be interested in knowing some average or the other measure concerning the population. There may also be important sub-groups in the population about whom we would like to make estimates. All this has a strong impact upon the sample design we would accept.

  • Budgetary constraint:

Cost considerations, from practical point of view, have a major impact upon decisions relating to not only the size of the sample but also to the type of sample. This fact can even lead to the use of a non-probability sample.

  • Sampling procedure:

Finally, the researcher must decide the type of sample he will use i.e., he must decide about the technique to be used in selecting the items for the sample. In fact, this technique or procedure stands for the sample design itself. There are several sample designs (explained in the pages that follow) out of which the researcher must choose one for his study. Obviously, he must select that design which, for a given sample size and for a given cost, has a smaller sampling error.

Types of Samples

  • Probability Sampling (Representative samples)

Probability samples are selected in such a way as to be representative of the population. They provide the most valid or credible results because they reflect the characteristics of the population from which they are selected (e.g., residents of a particular community, students at an elementary school, etc.). There are two types of probability samples: random and stratified.

  • Random Sample

The term random has a very precise meaning. Each individual in the population of interest has an equal likelihood of selection. This is a very strict meaning you can’t just collect responses on the street and have a random sample.

The assumption of an equal chance of selection means that sources such as a telephone book or voter registration lists are not adequate for providing a random sample of a community. In both these cases there will be a number of residents whose names are not listed. Telephone surveys get around this problem by random-digit dialling but that assumes that everyone in the population has a telephone. The key to random selection is that there is no bias involved in the selection of the sample. Any variation between the sample characteristics and the population characteristics is only a matter of chance.

  • Stratified Sample

A stratified sample is a mini-reproduction of the population. Before sampling, the population is divided into characteristics of importance for the research. For example, by gender, social class, education level, religion, etc. Then the population is randomly sampled within each category or stratum. If 38% of the population is college-educated, then 38% of the sample is randomly selected from the college-educated population.

Stratified samples are as good as or better than random samples, but they require fairly detailed advance knowledge of the population characteristics, and therefore are more difficult to construct.

  • Non-probability Samples (Non-representative samples)

As they are not truly representative, non-probability samples are less desirable than probability samples. However, a researcher may not be able to obtain a random or stratified sample, or it may be too expensive. A researcher may not care about generalizing to a larger population. The validity of non-probability samples can be increased by trying to approximate random selection, and by eliminating as many sources of bias as possible.

  • Quota Sample

The defining characteristic of a quota sample is that the researcher deliberately sets the proportions of levels or strata within the sample. This is generally done to insure the inclusion of a particular segment of the population. The proportions may or may not differ dramatically from the actual proportion in the population. The researcher sets a quota, independent of population characteristics.

Example: A researcher is interested in the attitudes of members of different religions towards the death penalty. In Iowa a random sample might miss Muslims (because there are not many in that state). To be sure of their inclusion, a researcher could set a quota of 3% Muslim for the sample. However, the sample will no longer be representative of the actual proportions in the population. This may limit generalizing to the state population. But the quota will guarantee that the views of Muslims are represented in the survey.

  • Purposive Sample

A purposive sample is a non-representative subset of some larger population, and is constructed to serve a very specific need or purpose. A researcher may have a specific group in mind, such as high level business executives. It may not be possible to specify the population they would not all be known, and access will be difficult. The researcher will attempt to zero in on the target group, interviewing whoever is available.

  • Convenience Sample

A convenience sample is a matter of taking what you can get. It is an accidental sample. Although selection may be unguided, it probably is not random, using the correct definition of everyone in the population having an equal chance of being selected. Volunteers would constitute a convenience sample.

Non-probability samples are limited with regard to generalization. Because they do not truly represent a population, we cannot make valid inferences about the larger group from which they are drawn. Validity can be increased by approximating random selection as much as possible, and making every attempt to avoid introducing bias into sample selection.

Sampling Distribution

Sampling Distribution is a statistical concept that describes the probability distribution of a given statistic (e.g., mean, variance, or proportion) derived from repeated random samples of a specific size taken from a population. It plays a crucial role in inferential statistics, providing the foundation for making predictions and drawing conclusions about a population based on sample data.

Concepts of Sampling Distribution

A sampling distribution is the distribution of a statistic (not raw data) over all possible samples of the same size from a population. Commonly used statistics include the sample mean (Xˉ\bar{X}), sample variance, and sample proportion.

Purpose:

It allows statisticians to estimate population parameters, test hypotheses, and calculate probabilities for statistical inference.

Shape and Characteristics:

    • The shape of the sampling distribution depends on the population distribution and the sample size.
    • For large sample sizes, the Central Limit Theorem states that the sampling distribution of the mean will be approximately normal, regardless of the population’s distribution.

Importance of Sampling Distribution

  • Facilitates Statistical Inference:

Sampling distributions are used to construct confidence intervals and perform hypothesis tests, helping to infer population characteristics.

  • Standard Error:

The standard deviation of the sampling distribution, called the standard error, quantifies the variability of the sample statistic. Smaller standard errors indicate more reliable estimates.

  • Links Population and Samples:

It provides a theoretical framework that connects sample statistics to population parameters.

Types of Sampling Distributions

  • Distribution of Sample Means:

Shows the distribution of means from all possible samples of a population.

  • Distribution of Sample Proportions:

Represents the proportion of a certain outcome in samples, used in binomial settings.

  • Distribution of Sample Variances:

Explains the variability in sample data.

Example

Consider a population of students’ test scores with a mean of 70 and a standard deviation of 10. If we repeatedly draw random samples of size 30 and calculate the sample mean, the distribution of those means forms the sampling distribution. This distribution will have a mean close to 70 and a reduced standard deviation (standard error).

Management Notes

Range and co-efficient of Range

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The range is a measure of dispersion that represents the difference between the highest and lowest values in a dataset. It provides a simple way to understand the spread of data. While easy to calculate, the range is sensitive to outliers and does not provide information about the distribution of values between the extremes.

Range of a distribution gives a measure of the width (or the spread) of the data values of the corresponding random variable. For example, if there are two random variables X and Y such that X corresponds to the age of human beings and Y corresponds to the age of turtles, we know from our general knowledge that the variable corresponding to the age of turtles should be larger.

Since the average age of humans is 50-60 years, while that of turtles is about 150-200 years; the values taken by the random variable Y are indeed spread out from 0 to at least 250 and above; while those of X will have a smaller range. Thus, qualitatively you’ve already understood what the Range of a distribution means. The mathematical formula for the same is given as:

Range = L – S

where

L: The Largets/maximum value attained by the random variable under consideration

S: The smallest/minimum value.

Properties

  • The Range of a given distribution has the same units as the data points.
  • If a random variable is transformed into a new random variable by a change of scale and a shift of origin as:

Y = aX + b

where

Y: the new random variable

X: the original random variable

a,b: constants.

Then the ranges of X and Y can be related as:

RY = |a|RX

Clearly, the shift in origin doesn’t affect the shape of the distribution, and therefore its spread (or the width) remains unchanged. Only the scaling factor is important.

  • For a grouped class distribution, the Range is defined as the difference between the two extreme class boundaries.
  • A better measure of the spread of a distribution is the Coefficient of Range, given by:

Coefficient of Range (expressed as a percentage) = L – SL + S × 100

Clearly, we need to take the ratio between the Range and the total (combined) extent of the distribution. Besides, since it is a ratio, it is dimensionless, and can, therefore, one can use it to compare the spreads of two or more different distributions as well.

  • The range is an absolute measure of Dispersion of a distribution while the Coefficient of Range is a relative measure of dispersion.

Due to the consideration of only the end-points of a distribution, the Range never gives us any information about the shape of the distribution curve between the extreme points. Thus, we must move on to better measures of dispersion. One such quantity is Mean Deviation which is we are going to discuss now.

Interquartile range (IQR)

The interquartile range is the middle half of the data. To visualize it, think about the median value that splits the dataset in half. Similarly, you can divide the data into quarters. Statisticians refer to these quarters as quartiles and denote them from low to high as Q1, Q2, Q3, and Q4. The lowest quartile (Q1) contains the quarter of the dataset with the smallest values. The upper quartile (Q4) contains the quarter of the dataset with the highest values. The interquartile range is the middle half of the data that is in between the upper and lower quartiles. In other words, the interquartile range includes the 50% of data points that fall in Q2 and

The IQR is the red area in the graph below.

The interquartile range is a robust measure of variability in a similar manner that the median is a robust measure of central tendency. Neither measure is influenced dramatically by outliers because they don’t depend on every value. Additionally, the interquartile range is excellent for skewed distributions, just like the median. As you’ll learn, when you have a normal distribution, the standard deviation tells you the percentage of observations that fall specific distances from the mean. However, this doesn’t work for skewed distributions, and the IQR is a great alternative.

I’ve divided the dataset below into quartiles. The interquartile range (IQR) extends from the low end of Q2 to the upper limit of Q3. For this dataset, the range is 21 – 39.

Management Notes

Karl Pearson and Rank co-relation

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Karl Pearson Coefficient of Correlation (also called the Pearson correlation coefficient or Pearson’s r) is a measure of the strength and direction of the linear relationship between two variables. It ranges from -1 to +1, where +1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 indicates no linear relationship. The formula for Pearson’s r is calculated by dividing the covariance of the two variables by the product of their standard deviations. It is widely used in statistics to analyze the degree of correlation between paired data.

The following are the main properties of correlation.

  1. Coefficient of Correlation lies between -1 and +1:

The coefficient of correlation cannot take value less than -1 or more than one +1. Symbolically,

-1<=r<= + 1 or | r | <1.

  1. Coefficients of Correlation are independent of Change of Origin:

This property reveals that if we subtract any constant from all the values of X and Y, it will not affect the coefficient of correlation.

  1. Coefficients of Correlation possess the property of symmetry:

The degree of relationship between two variables is symmetric as shown below:

  1. Coefficient of Correlation is independent of Change of Scale:

This property reveals that if we divide or multiply all the values of X and Y, it will not affect the coefficient of correlation.

  1. Co-efficient of correlation measures only linear correlation between X and Y.
  2. If two variables X and Y are independent, coefficient of correlation between them will be zero.

Karl Pearson’s Coefficient of Correlation is widely used mathematical method wherein the numerical expression is used to calculate the degree and direction of the relationship between linear related variables.

Pearson’s method, popularly known as a Pearsonian Coefficient of Correlation, is the most extensively used quantitative methods in practice. The coefficient of correlation is denoted by “r”.

If the relationship between two variables X and Y is to be ascertained, then the following formula is used:

Properties of Coefficient of Correlation

  • The value of the coefficient of correlation (r) always lies between±1. Such as:

    r=+1, perfect positive correlation

    r=-1, perfect negative correlation

    r=0, no correlation

  • The coefficient of correlation is independent of the origin and scale.By origin, it means subtracting any non-zero constant from the given value of X and Y the vale of “r” remains unchanged. By scale it means, there is no effect on the value of “r” if the value of X and Y is divided or multiplied by any constant.
  • The coefficient of correlation is a geometric mean of two regression coefficient. Symbolically it is represented as:
  • The coefficient of correlation is “ zero” when the variables X and Y are independent. But, however, the converse is not true.

Assumptions of Karl Pearson’s Coefficient of Correlation

  1. The relationship between the variables is “Linear”, which means when the two variables are plotted, a straight line is formed by the points plotted.
  2. There are a large number of independent causes that affect the variables under study so as to form a Normal Distribution. Such as, variables like price, demand, supply, etc. are affected by such factors that the normal distribution is formed.
  3. The variables are independent of each other.                                     

Note: The coefficient of correlation measures not only the magnitude of correlation but also tells the direction. Such as, r = -0.67, which shows correlation is negative because the sign is “-“ and the magnitude is 0.67.

Spearman Rank Correlation

Spearman rank correlation is a non-parametric test that is used to measure the degree of association between two variables.  The Spearman rank correlation test does not carry any assumptions about the distribution of the data and is the appropriate correlation analysis when the variables are measured on a scale that is at least ordinal.

The Spearman correlation between two variables is equal to the Pearson correlation between the rank values of those two variables; while Pearson’s correlation assesses linear relationships, Spearman’s correlation assesses monotonic relationships (whether linear or not). If there are no repeated data values, a perfect Spearman correlation of +1 or −1 occurs when each of the variables is a perfect monotone function of the other.

Intuitively, the Spearman correlation between two variables will be high when observations have a similar (or identical for a correlation of 1) rank (i.e. relative position label of the observations within the variable: 1st, 2nd, 3rd, etc.) between the two variables, and low when observations have a dissimilar (or fully opposed for a correlation of −1) rank between the two variables.

The following formula is used to calculate the Spearman rank correlation:

ρ = Spearman rank correlation

di = the difference between the ranks of corresponding variables

n = number of observations

Assumptions

The assumptions of the Spearman correlation are that data must be at least ordinal and the scores on one variable must be monotonically related to the other variable.

Management Notes

Methods of Primary Data Collection: Observation, Interview, Questionnaire, and Survey

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Primary Data is information collected firsthand by a researcher for a specific research purpose. It is original, fresh, and tailored directly to the research question or objective. Methods such as surveys, interviews, experiments, and observations are commonly used to gather primary data. Since it is collected directly from the source, primary data is highly relevant, specific, and accurate. However, it often requires more time, effort, and resources compared to using existing information. It is essential for studies needing updated or detailed insights.

Methods of Primary Data Collection:

  • Observation

Observation involves systematically watching and recording behaviors, events, or phenomena as they occur naturally or in a controlled setting. It allows researchers to gather real-time, unbiased data without influencing the subject’s behavior. Observations can be structured (following a predefined checklist) or unstructured (open-ended). It is especially useful when participants are unwilling or unable to provide accurate verbal responses. Researchers may act as participants (participant observation) or as non-intrusive observers. Observation is widely used in fields like anthropology, psychology, and marketing to understand behaviors, workflows, or consumer interactions. It provides deep insights but may sometimes lack the ability to explain the reasons behind certain actions, requiring combination with other methods like interviews for richer analysis.

  • Interview

An interview is a direct, face-to-face, telephonic, or video-based conversation between the researcher and the participant aimed at gathering detailed information. Interviews can be structured (fixed questions), semi-structured (guided by a framework but flexible), or unstructured (open conversation). This method allows for in-depth exploration of opinions, emotions, experiences, and motivations. Interviews can be personal or group-based, depending on research needs. They are commonly used in qualitative research to gain comprehensive understanding and context behind responses. Although interviews provide rich, detailed data, they can be time-consuming and may introduce biases if not conducted carefully. Proper interviewer skills are essential for encouraging honest and open communication from participants.

  • Questionnaire

Questionnaire is a set of written or digital questions designed to collect information from respondents. It can include closed-ended questions (like multiple-choice) or open-ended questions (where respondents write answers in their own words). Questionnaires are often used for surveys and research studies where standardized information is needed from a large audience. They are cost-effective, easy to distribute, and efficient in data collection. Responses are easy to quantify for statistical analysis. However, the design of the questionnaire is crucial — poorly framed questions can lead to misunderstandings and unreliable data. Questionnaires are widely used in education, social science, market research, and customer satisfaction studies.

  • Survey

Survey is a research method involving the systematic collection of information from a sample of individuals, usually through questionnaires or interviews. Surveys can be conducted in-person, via phone, online, or by mail. They are useful for gathering quantitative as well as qualitative data about behaviors, attitudes, preferences, or demographics. Surveys are popular because they can cover large populations at relatively low cost and produce statistically significant results if designed properly. However, their effectiveness depends on clear question framing, respondent honesty, and sampling methods. Surveys are widely used in fields like business, healthcare, political science, and social research for decision-making and trend analysis.

Management Notes

Data Tabulation

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Tabulation is the systematic arrangement of the statistical data in columns or rows. It involves the orderly and systematic presentation of numerical data in a form designed to explain the problem under consideration. Tabulation helps in drawing the inference from the statistical figures.

Tabulation prepares the ground for analysis and interpretation. Therefore a suitable method must be decided carefully taking into account the scope and objects of the investigation, because it is very important part of the statistical methods.

Types of Tabulation

In general, the tabulation is classified in two parts, that is a simple tabulation, and a complex tabulation.

Simple tabulation, gives information regarding one or more independent questions. Complex tabulation gives information regarding two mutually dependent questions.

  • Two-Way Table

These types of table give information regarding two mutually dependent questions. For example, question is, how many millions of the persons are in the Divisions; the One-Way Table will give the answer. But if we want to know that in the population number, who are in the majority, male, or female. The Two-Way Tables will answer the question by giving the column for female and male. Thus the table showing the real picture of divisions sex wise is as under:

  • Three-Way Table

Three-Way Table gives information regarding three mutually dependent and inter-related questions.

For example, from one-way table, we get information about population, and from two-way table, we get information about the number of male and female available in various divisions. Now we can extend the same table to a three way table, by putting a question, “How many male and female are literate?” Thus the collected statistical data will show the following, three mutually dependent and inter-related questions:

  1. Population in various division.
  2. Their sex-wise distribution.
  3. Their position of literacy.

Presentation of Data

Presentation of data is of utter importance nowadays. Afterall everything that’s pleasing to our eyes never fails to grab our attention. Presentation of data refers to an exhibition or putting up data in an attractive and useful manner such that it can be easily interpreted. The three main forms of presentation of data are:

  1. Textual presentation
  2. Data tables
  3. Diagrammatic presentation

Textual Presentation

The discussion about the presentation of data starts off with it’s most raw and vague form which is the textual presentation. In such form of presentation, data is simply mentioned as mere text, that is generally in a paragraph. This is commonly used when the data is not very large.

This kind of representation is useful when we are looking to supplement qualitative statements with some data. For this purpose, the data should not be voluminously represented in tables or diagrams. It just has to be a statement that serves as a fitting evidence to our qualitative evidence and helps the reader to get an idea of the scale of a phenomenon.

For example, “the 2002 earthquake proved to be a mass murderer of humans. As many as 10,000 citizens have been reported dead”. The textual representation of data simply requires some intensive reading. This is because the quantitative statement just serves as an evidence of the qualitative statements and one has to go through the entire text before concluding anything.

Further, if the data under consideration is large then the text matter increases substantially. As a result, the reading process becomes more intensive, time-consuming and cumbersome.

Data Tables or Tabular Presentation

A table facilitates representation of even large amounts of data in an attractive, easy to read and organized manner. The data is organized in rows and columns. This is one of the most widely used forms of presentation of data since data tables are easy to construct and read.

Components of Data Tables

  • Table Number: Each table should have a specific table number for ease of access and locating. This number can be readily mentioned anywhere which serves as a reference and leads us directly to the data mentioned in that particular table.
  • Title: A table must contain a title that clearly tells the readers about the data it contains, time period of study, place of study and the nature of classification of data.
  • Headnotes: A headnote further aids in the purpose of a title and displays more information about the table. Generally, headnotes present the units of data in brackets at the end of a table title.
  • Stubs: These are titles of the rows in a table. Thus a stub display information about the data contained in a particular row.
  • Caption: A caption is the title of a column in the data table. In fact, it is a counterpart if a stub and indicates the information contained in a column.
  • Body or field: The body of a table is the content of a table in its entirety. Each item in a body is known as a ‘cell’.
  • Footnotes: Footnotes are rarely used. In effect, they supplement the title of a table if required.
  • Source: When using data obtained from a secondary source, this source has to be mentioned below the footnote.

Construction of Data Tables

There are many ways for construction of a good table. However, some basic ideas are:

  • The title should be in accordance with the objective of study: The title of a table should provide a quick insight into the table.
  • Comparison: If there might arise a need to compare any two rows or columns then these might be kept close to each other.
  • Alternative location of stubs: If the rows in a data table are lengthy, then the stubs can be placed on the right-hand side of the table.
  • Headings: Headings should be written in a singular form. For example, ‘good’ must be used instead of ‘goods’.
  • Footnote: A footnote should be given only if needed.
  • Size of columns: Size of columns must be uniform and symmetrical.
  • Use of abbreviations: Headings and sub-headings should be free of abbreviations.
  • Units: There should be a clear specification of units above the columns.

Advantages of Tabular Presentation:

  • Ease of representation: A large amount of data can be easily confined in a data table. Evidently, it is the simplest form of data presentation.
  • Ease of analysis: Data tables are frequently used for statistical analysis like calculation of central tendency, dispersion etc.
  • Helps in comparison: In a data table, the rows and columns which are required to be compared can be placed next to each other. To point out, this facilitates comparison as it becomes easy to compare each value.
  • Economical: Construction of a data table is fairly easy and presents the data in a manner which is really easy on the eyes of a reader. Moreover, it saves time as well as space.

Classification of Data and Tabular Presentation

Qualitative Classification

In this classification, data in a table is classified on the basis of qualitative attributes. In other words, if the data contained attributes that cannot be quantified like rural-urban, boys-girls etc. it can be identified as a qualitative classification of data.

Sex Urban Rural
Boys 200 390
Girls 167 100

Quantitative Classification

In quantitative classification, data is classified on basis of quantitative attributes.

Marks No. of Students
0-50 29
51-100 64

Temporal Classification

Here data is classified according to time. Thus when data is mentioned with respect to different time frames, we term such a classification as temporal.

Year Sales
2016 10,000
2017 12,500

Spatial Classification

When data is classified according to a location, it becomes a spatial classification.

Country No. of Teachers
India 139,000
Russia 43,000

Advantages of Tabulation

  1. The large mass of confusing data is easily reduced to reasonable form that is understandable to kind.
  2. The data once arranged in a suitable form, gives the condition of the situation at a glance, or gives a bird eye view.
  3. From the table it is easy to draw some reasonable conclusion or inferences.
  4. Tables gave grounds for analysis of the data.
  5. Errors, and omission if any are always detected in tabulation.
Management Notes

Mean (AM, Weighted, Combined)

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Arithmetic Mean

The arithmetic mean,’ mean or average is calculated by summ­ing all the individual observations or items of a sample and divid­ing this sum by the number of items in the sample. For example, as the result of a gas analysis in a respirometer an investigator obtains the following four readings of oxygen percentages:

14.9
10.8
12.3
23.3
Sum = 61.3

He calculates the mean oxygen percentage as the sum of the four items divided by the number of items here, by four. Thus, the average oxygen percentage is

Mean = 61.3 / 4 =15.325%

Calculating a mean presents us with the opportunity for learning statistical symbolism. An individual observation is symbo­lized by Yi, which stands for the ith observation in the sample. Four observations could be written symbolically as Yi, Y2, Y3, Y4.

We shall define n, the sample size, as the number of items in a sample. In this particular instance, the sample size n is 4. Thus, in a large sample, we can symbolize the array from the first to the nth item as follows: Y1, Y2…, Yn. When we wish to sum items, we use the following notation:

The capital Greek sigma, Ʃ, simply means the sum of items indica­ted. The i = 1 means that the items should be summed, starting with the first one, and ending with the nth one as indicated by the i = n above the Ʃ. The subscript and superscript are necessary to indicate how many items should be summed. Below are seen increasing simplifications of the complete notation shown at the extreme left:

Properties of Arithmetic Mean:

  1. The sum of deviations of the items from the arithmetic mean is always zero i.e.

∑(X–X) =0.

  1. The Sum of the squared deviations of the items from A.M. is minimum, which is less than the sum of the squared deviations of the items from any other values.
  2. If each item in the series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.                       

Merits of A.M:

  1. It is simple to understand and easy to calculate.
  2. It is affected by the value of every item in the series.
  3. It is rigidly defined.
  4. It is capable of further algebraic treatment.
  5. It is calculated value and not based on the position in the series.

Demerits of A.M:

  1. It is affected by extreme items i.e., very small and very large items.
  2. It can hardly be located by inspection.
  3. In some cases A.M. does not represent the actual item. For example, average patients admitted in a hospital is 10.7 per day.
  4. M. is not suitable in extremely asymmetrical distributions.

Weighted Mean

In some cases, you might want a number to have more weight. In that case, you’ll want to find the weighted mean. To find the weighted mean:

  1. Multiply the numbers in your data set by the weights.
  2. Add the results up.

For that set of number above with equal weights (1/5 for each number), the math to find the weighted mean would be:
1(*1/5) + 3(*1/5) + 5(*1/5) + 7(*1/5) + 10(*1/5) = 5.2.

Sample problem: You take three 100-point exams in your statistics class and score 80, 80 and 95. The last exam is much easier than the first two, so your professor has given it less weight. The weights for the three exams are:

  • Exam 1: 40 % of your grade. (Note: 40% as a decimal is .4.)
  • Exam 2: 40 % of your grade.
  • Exam 3: 20 % of your grade.

What is your final weighted average for the class?

  1. Multiply the numbers in your data set by the weights:

    .4(80) = 32

    .4(80) = 32

    .2(95) = 19

  2. Add the numbers up. 32 + 32 + 19 = 83.

The percent weight given to each exam is called a weighting factor.

Weighted Mean Formula

The weighted mean is relatively easy to find. But in some cases the weights might not add up to 1. In those cases, you’ll need to use the weighted mean formula. The only difference between the formula and the steps above is that you divide by the sum of all the weights.

The image above is the technical formula for the weighted mean. In simple terms, the formula can be written as:

Weighted mean = Σwx / Σw

Σ = the sum of (in other words…add them up!).
w = the weights.
x = the value.

To use the formula:

  1. Multiply the numbers in your data set by the weights.
  2. Add the numbers in Step 1 up. Set this number aside for a moment.
  3. Add up all of the weights.
  4. Divide the numbers you found in Step 2 by the number you found in Step 3.

In the sample grades problem above, all of the weights add up to 1 (.4 + .4 + .2) so you would divide your answer (83) by 1:
83 / 1 = 83.

However, let’s say your weighted means added up to 1.2 instead of 1. You’d divide 83 by 1.2 to get:
83 / 1.2 = 69.17.

Combined Mean

A combined mean is a mean of two or more separate groups, and is found by:

  1. Calculating the mean of each group,
  2. Combining the results.

Combined Mean Formula

More formally, a combined mean for two sets can be calculated by the formula :

Where:

  • xa = the mean of the first set,
  • m = the number of items in the first set,
  • xb = the mean of the second set,
  • n = the number of items in the second set,
  • xc the combined mean.

A combined mean is simply a weighted mean, where the weights are the size of each group.

Management Notes

Baye’s Theorem

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Bayes’ Theorem is a way to figure out conditional probability. Conditional probability is the probability of an event happening, given that it has some relationship to one or more other events. For example, your probability of getting a parking space is connected to the time of day you park, where you park, and what conventions are going on at any time. Bayes’ theorem is slightly more nuanced. In a nutshell, it gives you the actual probability of an event given information about tests.

“Events” Are different from “tests.” For example, there is a test for liver disease, but that’s separate from the event of actually having liver disease.

Tests are flawed:

Just because you have a positive test does not mean you actually have the disease. Many tests have a high false positive rate. Rare events tend to have higher false positive rates than more common events. We’re not just talking about medical tests here. For example, spam filtering can have high false positive rates. Bayes’ theorem takes the test results and calculates your real probability that the test has identified the event.

Bayes’ Theorem (also known as Bayes’ rule) is a deceptively simple formula used to calculate conditional probability. The Theorem was named after English mathematician Thomas Bayes (1701-1761). The formal definition for the rule is:

In most cases, you can’t just plug numbers into an equation; You have to figure out what your “tests” and “events” are first. For two events, A and B, Bayes’ theorem allows you to figure out p(A|B) (the probability that event A happened, given that test B was positive) from p(B|A) (the probability that test B happened, given that event A happened). It can be a little tricky to wrap your head around as technically you’re working backwards; you may have to switch your tests and events around, which can get confusing. An example should clarify what I mean by “switch the tests and events around.”

Bayes’ Theorem Example

You might be interested in finding out a patient’s probability of having liver disease if they are an alcoholic. “Being an alcoholic” is the test (kind of like a litmus test) for liver disease.

A could mean the event “Patient has liver disease.” Past data tells you that 10% of patients entering your clinic have liver disease. P(A) = 0.10.

B could mean the litmus test that “Patient is an alcoholic.” Five percent of the clinic’s patients are alcoholics. P(B) = 0.05.

You might also know that among those patients diagnosed with liver disease, 7% are alcoholics. This is your B|A: the probability that a patient is alcoholic, given that they have liver disease, is 7%.

Bayes’ theorem tells you:

P(A|B) = (0.07 * 0.1)/0.05 = 0.14

In other words, if the patient is an alcoholic, their chances of having liver disease is 0.14 (14%). This is a large increase from the 10% suggested by past data. But it’s still unlikely that any particular patient has liver disease.

Management Notes

Conditional Probability

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Conditional probability refers to the probability of an event occurring, given that another event has already occurred. It quantifies the likelihood of one event under the condition that the related event is known.

The probability of the occurrence of an event A given that an event B has already occurred is called the conditional probability of A given B:

The same is explained in Figure 2.15 using the sample spaces related to the events A and B, assuming that there are few sample points common to these two events. Part 1 of the figure shows the total sample space related to the experiment as in the form of rectangle and the sample space related to the event A as a circle. Similarly part 2 of the figure shows the total sample space and the sample space related to event B. As explained earlier in conditional probability the total sample space is restrained to the sample space that is related to event B (which has already occurred). The same is shown in part 3 of Figure 2.15. Now the sample space for event A (B is the total sample space available) is nothing but the sample points related to event A and falling in the sample space. This is nothing but the intersection of the events A and B and is shown in part 3 of the figure as the hatched area.  

Figure 2.15: Representation of conditional probability using the Venn diagrams

For example, there are 100 trips per day between two places X and Y. Out of these 100 trips 50 are made by car, 25 are made by bus and the other 25 are by local train. Probabilities associated to these modes are 0.5, 0.25, and 0.25, respectively. In transportation engineering both the bus and the local train are considered as public transport so the event space associated to this is the summation of the event spaces associated to bus and local train. Probability of choosing public transportation is 0.5. Now if one is interested in finding the probability of choosing bus given public transportation is chosen the conditional probability is useful in finding that.

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