Unit 3: Sampling techniques and descriptive statistics - Subjective Questions

Sampling is the process of selecting a subset (a sample) of individuals, items, or observations from a larger group (the population) in order to estimate the characteristics of the whole population.

Reasons sampling is preferred over a complete census:

• Cost-effective: Studying a sample requires far fewer resources than examining every unit of the population.
• Time-saving: Data collection, processing, and analysis are much faster for a sample.
• Practicality: In many cases the population is infinite or inaccessible, making a full census impossible.
• Reduced effort: Fewer investigators and less equipment are needed.
• Destructive testing: When testing destroys the item (e.g., testing the lifespan of bulbs), only a sample can be tested.
• Greater accuracy of detail: With fewer units, more careful and detailed measurements can be taken, reducing non-sampling errors.

Thus, sampling provides reliable estimates about a population efficiently while conserving resources.

Random (Probability) Sampling and Non-random (Non-probability) Sampling differ mainly in how units are selected.

Example of random sampling: Drawing 50 student roll numbers using a lottery method from a college of 1000 students.

Example of non-random sampling: Interviewing the first 50 shoppers who walk into a mall (convenience sampling).

Conclusion: Random sampling is objective and statistically sound, while non-random sampling is convenient but may introduce bias.

The main types of probability sampling are:

• Simple Random Sampling: Every unit has an equal chance of selection, e.g., using a lottery method or random number tables. Best for homogeneous populations.

• Stratified Random Sampling: The population is divided into homogeneous subgroups (strata) based on a characteristic (e.g., gender, age), and random samples are drawn from each stratum. Ensures representation of all subgroups.

• Systematic Sampling: Every unit is selected after a random start, where (population size / sample size). Simple and quick.

• Cluster Sampling: The population is divided into naturally occurring clusters (e.g., cities, schools); some clusters are randomly chosen and all units within them are studied. Useful for geographically spread populations.

• Multi-stage Sampling: Sampling is carried out in several stages, combining the above methods (e.g., select states → districts → villages → households).

Each method balances accuracy, cost, and feasibility depending on the research context.

The important non-probability sampling methods are:

• Convenience Sampling: Units are selected because they are easily accessible, e.g., surveying friends or nearby respondents. Quick but highly biased.

• Judgment (Purposive) Sampling: The researcher uses expert judgment to select units believed to be representative, e.g., choosing experienced professionals for a study.

• Quota Sampling: The population is divided into categories, and a fixed quota of units is selected from each (non-randomly) to match population proportions.

• Snowball Sampling: Existing subjects refer future subjects; used for hard-to-reach populations (e.g., studying rare disease patients or hidden groups).

Advantages: Low cost, easy, quick, useful for exploratory studies.

Disadvantages: Prone to bias, cannot estimate sampling error, and results are difficult to generalize.

These methods are suitable when probability sampling is impractical or when preliminary insight is required.

Data refers to facts, figures, or information collected for the purpose of analysis and drawing conclusions in research.

Qualitative (Categorical) Data:

• Describes qualities or characteristics that cannot be measured numerically.
• Expressed in categories or labels.
• Examples: Gender (male/female), colour, religion, marital status, satisfaction level (satisfied/unsatisfied).

Quantitative (Numerical) Data:

• Represents measurable quantities expressed as numbers.
• Can be counted or measured.
• Examples: Height, weight, income, age, number of children.

Conclusion: Qualitative data describe attributes, while quantitative data measure amounts, and each requires different analytical techniques.

Data can be classified into four levels of measurement:

• Nominal Scale: Data are classified into categories with no natural order. Only naming/labeling is possible.

  • Example: Gender, blood group, city names.

• Ordinal Scale: Categories have a meaningful order or rank, but differences between ranks are not measurable.

  • Example: Education level (high school < graduate < postgraduate), customer ratings (poor, good, excellent).

• Interval Scale: Ordered data with equal intervals between values, but no true zero point.

  • Example: Temperature in Celsius, IQ scores. ( does not mean 'no temperature').

• Ratio Scale: Has all properties of interval scale plus a true zero, allowing meaningful ratios.

  • Example: Height, weight, income, age. (A weight of 0 kg means no weight, and 40 kg is twice 20 kg.)

Summary: The scales increase in mathematical strength from nominal → ordinal → interval → ratio, allowing progressively more advanced statistical analysis.

Measures of central tendency are statistical values that represent the centre or typical value of a data set, summarizing the entire distribution with a single representative figure.

Main measures:

• Arithmetic Mean: The sum of all observations divided by their number.
  

• Median: The middle value when data are arranged in ascending order.

• Mode: The value that occurs most frequently.

Other measures: Geometric Mean, Harmonic Mean, Weighted Mean.

Purpose/Importance:

• Provides a single representative value of the data.
• Facilitates comparison between different groups.
• Helps in decision-making and further statistical analysis.
• Summarizes large data sets concisely.

Conclusion: Central tendency measures give a concise, meaningful summary that describes the general nature of the data.

The Arithmetic Mean is the most common average, obtained by dividing the sum of all observations by the total number of observations.

Formula (ungrouped data):

Formula (grouped data):

where is the frequency and the class midpoint.

Merits:

• Simple to understand and easy to calculate.
• Based on all observations.
• Rigidly defined and suitable for further algebraic treatment.
• Least affected by sampling fluctuations.

Demerits:

• Highly affected by extreme values (outliers).
• Cannot be determined for open-ended class intervals.
• May give a value not present in the data.
• Not suitable for qualitative data.

Example: For data :

The Median is the value that divides a data set into two equal halves when the observations are arranged in ascending or descending order. It is a positional average.

Ungrouped data:

• Arrange the data in ascending order.
• If is odd: Median = value of the item.
• If is even: Median = average of the and items.

Grouped data:

where = lower boundary of median class, = total frequency, = cumulative frequency before median class, = frequency of median class, = class width.

Importance of Median:

• Not affected by extreme values, making it ideal for skewed data.
• Can be computed for open-ended distributions.
• Useful for qualitative/ordinal data (e.g., ranking).
• Easy to understand and locate graphically (ogive).

Conclusion: The median is a robust average, especially valuable when data contain outliers.

An ideal average should satisfy the following characteristics laid down by statisticians (notably Yule and Kendall):

• Rigidly defined: It should have a clear, unambiguous mathematical definition so that everyone obtains the same value.

• Easy to understand and calculate: It should be simple enough for a layman to grasp and compute.

• Based on all observations: A good average should consider every item in the data set.

• Suitable for further algebraic treatment: It should be usable in advanced statistical computations (e.g., combining means of groups).

• Least affected by sampling fluctuations: Its value should remain stable across different samples from the same population.

• Not unduly affected by extreme values: Outliers should not distort the average significantly.

• Capable of being determined for all types of data: It should be computable even for open-ended distributions where possible.

Conclusion: No single average satisfies all these conditions perfectly; the arithmetic mean meets most criteria but is affected by extreme values, whereas the median resists outliers but ignores their magnitude.

Mean, Median, and Mode are the three principal measures of central tendency.

When to use each:

• Mean: When data are symmetrical, quantitative, and free from extreme outliers (e.g., average marks).
• Median: When data are skewed or contain outliers, or are ordinal (e.g., median income).
• Mode: When the most typical/common value is needed, especially for categorical data (e.g., most popular shoe size).

Empirical relationship (moderately skewed data):

Given data: (already in ascending order), .

Arithmetic Mean:

Median:
Since (odd), Median = value of the item.
The item = 18.

Mode:
The value occurring most frequently is 15 (appears twice), while all others appear once.

Summary:

• Mean
• Median
• Mode

The three measures differ slightly, indicating a mildly skewed distribution.

Correlation is a statistical technique that measures the degree and direction of the relationship between two or more variables. If a change in one variable is accompanied by a change in another, the variables are said to be correlated.

Types of Correlation:

• Positive Correlation: Both variables move in the same direction (increase or decrease together).

  • Example: Height and weight; more study hours and higher marks.

• Negative Correlation: Variables move in opposite directions; one increases as the other decreases.

  • Example: Price and demand; speed and travel time.

• Zero (No) Correlation: No relationship exists between the variables.

  • Example: Shoe size and intelligence.

Based on number of variables:

• Simple correlation (two variables), Partial and Multiple correlation (more than two variables).

Based on linearity:

• Linear correlation (constant ratio of change) and Non-linear/curvilinear correlation.

Conclusion: Correlation indicates association but does not imply causation.

Karl Pearson's Coefficient of Correlation () is a numerical measure of the linear relationship between two variables. It is also called the product-moment correlation coefficient.

Formula:

Alternative (direct) formula:

Properties:

• The value of always lies between and , i.e., .
• → perfect positive correlation; → perfect negative correlation; → no linear correlation.
• It is independent of the change of origin and scale.
• It is a pure number without units.

Interpretation:

• to : weak; to : moderate; to : strong correlation.

Assumptions: Linear relationship, normally distributed variables, and cause-effect need not be implied.

Step 1: Prepare the calculation table ().

Step 2: Apply the formula:

Numerator:

Denominator:

Step 3: Result:

Interpretation: indicates a strong positive linear correlation between and .

Inferential Statistics is the branch of statistics that uses data from a sample to make generalizations, predictions, or inferences about a larger population. It relies on probability theory to draw conclusions under uncertainty.

Key components of inferential statistics:

• Estimation: Point estimates and confidence intervals for population parameters.
• Hypothesis Testing: Testing claims about population (e.g., t-test, z-test, chi-square test, ANOVA).
• Prediction/Regression: Predicting outcomes based on relationships.

Difference between Descriptive and Inferential Statistics:

Conclusion: Descriptive statistics organizes data, while inferential statistics extends findings from samples to populations.

Sampling Error:

• The error that arises because only a sample, rather than the entire population, is studied.
• It reflects the difference between the sample estimate and the true population parameter.
• It is unavoidable in sampling but can be measured statistically.

Causes: Small sample size, faulty sampling method, natural variability.

Non-Sampling Error:

• Errors that occur due to factors other than sampling, present in both censuses and sample surveys.
• Causes: Faulty questionnaire design, measurement mistakes, data entry errors, non-response, respondent bias.

Minimizing Sampling Error:

• Increase the sample size.
• Use appropriate probability sampling methods.
• Apply stratification to reduce variability.

Minimizing Non-Sampling Error:

• Careful questionnaire design and pre-testing.
• Proper training of investigators.
• Reducing non-response through follow-ups.
• Accurate data processing and validation.

Conclusion: While sampling error decreases with larger samples, non-sampling error requires careful survey design and execution.

The sampling process involves systematic steps to select an appropriate sample:

• Step 1 – Define the Target Population: Clearly specify the group about which conclusions are to be drawn (e.g., all college students in a city).

• Step 2 – Determine the Sampling Frame: Prepare a complete list of all units in the population from which the sample is drawn (e.g., voter list, student register).

• Step 3 – Select the Sampling Technique: Decide between probability (random) and non-probability (non-random) methods based on objectives, cost, and time.

• Step 4 – Determine the Sample Size: Decide the number of units () to select, balancing accuracy and resources.

• Step 5 – Execute the Sampling Plan: Actually select the units according to the chosen method.

• Step 6 – Collect Data and Validate: Gather information from the selected units and check for representativeness.

Conclusion: A well-designed sampling process ensures the sample is representative, reducing bias and improving the reliability of research findings.

Both are probability sampling techniques that divide the population into groups, but they differ fundamentally.

When Stratified Sampling is preferred:

• When the population has distinct subgroups (e.g., age, income) and each must be represented.
• When high precision is required.

When Cluster Sampling is preferred:

• When the population is large and geographically dispersed.
• When a complete sampling frame is unavailable but natural groups (villages, schools) exist.
• When cost and time are major constraints.

Conclusion: Stratified sampling maximizes accuracy; cluster sampling maximizes economy and convenience.

Step 1: Find class midpoints () and compute .

Step 2: Apply the formula for grouped data:

Result: The Arithmetic Mean of the marks is 24.5.

Interpretation: On average, students scored marks, which lies in the class — the modal region of the distribution.