Random Sampling Explained: Methods, Formulas & Practice Questions

The concept of random sampling plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Random sampling ensures unbiased data collection and helps simplify statistical analysis. It’s a must-know for students preparing for board exams, olympiads, and entrance tests.

What Is Random Sampling?

A random sampling is a method in which every member of a population has an equal and fair chance of being chosen for a sample. In maths, random sampling in statistics and probability helps create unbiased and representative groups for surveys, experiments, and research. You'll find this concept applied in Probability, data handling, and statistical analysis.

Types of Random Sampling

• Simple Random Sampling: Every member is equally likely to be included. Example: Drawing 10 random student names from a class list.
• Systematic Random Sampling: Selects members at regular intervals after a random starting point. Example: Choosing every 5th person in a waiting line, after picking the 3rd as starting point.
• Stratified Random Sampling: The population is divided into subgroups (strata) and random samples are taken from each. Example: Picking boys and girls separately from a class, in proportion to their numbers.
• Cluster Random Sampling: The population is split into clusters (groups) and whole clusters are randomly selected. Example: Picking 2 random schools from a city and surveying every student in those schools.

Key Formula for Random Sampling

Here’s the standard formula for simple random sampling probability:
\( P = \frac{n}{N} \)
where:

To estimate required sample size (for larger populations), a common formula is:
\( n = \frac{N \cdot X}{X + (N - 1)} \)

Step-by-Step Illustration

Let's solve a typical random sampling example:

Example: Out of 5000 students, you want to randomly select 100 for a survey. What is the probability that any one student is chosen?

1. Sample size, n = 100

2. Population size, N = 5000

3. Use the formula \( P = \frac{n}{N} \)

4. \( P = \frac{100}{5000} = 0.02 \) (or 2%)

5. Final Answer: Each student has a 2% chance of being selected.

Random Sampling in Statistics & Research

Random sampling is a core idea in collecting survey data, conducting research studies, and ensuring findings are valid. It minimizes bias and makes sure results are representative of the actual population. For example, surveys use random sampling to gauge election preferences or product feedback fairly.

Sample MCQ:
Which method ensures every member of a population has an equal chance of selection?
A) Random Sampling    B) Purposive Sampling    C) Convenience Sampling
Answer: A) Random Sampling

Random vs Non-Random Sampling

Feature	Random Sampling	Non-Random Sampling
Definition	Every member has an equal probability of selection.	Selection based on choice, convenience, or judgment.
Bias	Low (unbiased)	High (can be biased)
Use Case	Surveys, research, statistics exams	Quick opinions, pilot studies, limited access
Examples	Lottery, random draw, dice roll	Taking responses from nearby friends

Sample Practice Problems (with Answers)

1. Out of 1000 tickets, 50 are to be chosen at random for a prize. What is the probability for a ticket to be chosen?

1. n = 50, N = 1000

2. \( P = \frac{n}{N} = \frac{50}{1000} = 0.05 \) (5%)

3. Final Answer: 5% chance for each ticket.

2. A college with 1200 students wants to select a sample of 60 for a feedback survey. Which sampling method will ensure fairness?

1. Fairness → every student has equal chance

2. Final Answer: Simple random sampling.

3. In systematic random sampling, a list of 300 names is available. You want a sample of 30. What interval should you use?

1. Population (N) = 300, Sample (n) = 30

2. Interval (k) = N/n = 300/30 = 10

3. Final Answer: Pick every 10th name after a random start.

Key Takeaways & Exam Tips

• Random sampling means each member has equal chance; reduces bias.
• Know the four types: simple, systematic, stratified, cluster.
• Use the probability formula: \( P = \frac{n}{N} \).
• Random ≠ Non-random sampling. Don’t confuse them on exams!
• Write steps and show sample calculations in all answers.
• Watch for shortcuts for quick MCQ solving—practice with Vedantu resources!

Further Reading & Related Topics

• Types of Sampling Methods
• Data Collection and Handling
• Statistics
• Cluster Analysis
• Population and Sample
• Data Handling
• Probability

We explored random sampling—definitions, formulas, types, solved examples, and key differences with non-random techniques. Mastering this concept will make you more confident in statistics and help you score better in school and competitive exams. Keep practicing with Vedantu for more easy explanations and exam strategies!