Representative Sample
1.0Introduction
In statistics, a representative sample is one of the most critical concepts. In JEE Mathematics, it arises in the chapter Statistics and Probability, and students will need to understand how the data from a sample can be used to estimate the characteristics of the total population.
A representative sample is a smaller subset of the population that accurately reflects the characteristics of the whole. Instead of studying every element in a population (which could be impossible or impractical), we may study a representative sample and make inferences about the population.
2.0What Is a Representative Sample?
A representative sample is a subset of data that accurately mirrors the distribution, features, and variations of the entire population.
Mathematically, if P is the population with parameters such as mean (μ) and variance (σ2), then a representative sample S with mean xˉ and variance s2 should satisfy: xˉ≈μ, s2≈σ2
3.0Mathematical Foundation of Representative Sampling
In mathematics, representative sampling is connected with probability and statistical estimation.
- Population Mean (μ): μ=N∑i=1nXi
- Sample Mean (xˉ) : xˉ=n∑i=1nxi
- Population Variance (σ2):σ2=N∑i=1n(Xi−μ)2
- Sample Variance (s2):s2=n−1∑i=1n(xi−xˉ)2
4.0Methods of Selecting a Representative Sample
Several methods are used in statistics to select a representative sample:
Simple Random Sampling:
- Each element of the population has an equal chance of selection.
- Example: Drawing names from a hat.
Stratified Sampling:
- Population divided into subgroups (strata).
- Samples taken proportionally from each subgroup.
Systematic Sampling:
- Every kth element is selected from an ordered list.
- Example: Selecting every 10th student from a roll.
Cluster Sampling:
- Entire clusters (groups) are chosen randomly.
- Example: Selecting random schools to survey students.
Multi-Stage Sampling:
- Combination of several sampling techniques applied in stages.
- In JEE-level questions, simple random sampling and stratified sampling are the most relevant.
5.0How It Works (Representative Sample)
A representative sample works by making sure that a small group taken from a population reflects the same characteristics as the whole population.
- Define the Population: Decide the entire group you want to study (e.g., all students in a school).
- Select the Sample: Choose a smaller group using random or systematic methods.
- Calculate Sample Statistics: Find the sample mean, variance, or proportion.
- Compare with Population: If the sample values are close to the actual population values, the sample is representative.
- Use Probability Laws: By the Law of Large Numbers and Central Limit Theorem, larger random samples naturally become more representative of the population.
Example:
If 5% of 10,000 bulbs are defective, we expect 10 defective bulbs in a random sample of 200. If the sample shows 9–11 defective bulbs, it closely represents the population.
6.0Importance of a Representative Sample in Mathematics
- Accurate Predictions: A well-chosen representative sample provides accurate predictions and results that can be generalized to the whole population.
- Reduces Bias: Ensures that the study or experiment is unbiased, giving every group within the population an equal chance of being selected.
- Cost-effective: Surveying or collecting data from an entire population is often impractical or expensive. A representative sample reduces resources spent while maintaining accuracy.
- Improves Reliability: Enhances the credibility of the study, as the results are likely to be valid if the sample truly represents the population
7.0Solved Examples of Representative Samples
Example 1: A class of 100 students has an average score of 70. A sample of 10 students is chosen at random. If the sample mean is 72, is the sample representative?
Solution:
- Population mean = 70.
- Sample mean = 72.
- Since the sample mean is close to the population mean, the sample is reasonably representative (within sampling error).
Example 2: A factory produces 10,000 bulbs, 5% of which are defective. A sample of 200 bulbs is taken. If 12 bulbs are defective in the sample, is it representative?
Solution:
- Expected defective = 5%×200=10.
- Observed defective = 12.
- Since the observed is close to expected, the sample is representative.
Example 3 : If the probability of success in a population is 0.4, find the expected proportion of success in a sample of size 50.
Solution:
- Population probability = p=0.4.
- Expected sample proportion = p=0.4.
- Variance of sample proportion = p(1−p)n=0.4⋅0.650=0.0048.
- Standard deviation = 0.0048≈0.069.
Thus, the sample proportion will likely fall between 0.33 and 0.47, making it representative.
8.0Practice Questions on Representative Sample
- A researcher wants to study the eating habits of college students in a city. Out of 50,000 students, he selects 500 students randomly from different colleges. Is this a representative sample? Why or why not?
- Out of a population of 10,000 households, only 20 households from the same neighborhood are surveyed to understand the average monthly income. Will this sample represent the population fairly? Explain.
- A survey on mobile phone usage is conducted among 1,000 people, with proportional representation of age groups, gender, and income levels as in the population. Is this a good representative sample?
- A political analyst wants to predict election results. He surveys only people from one political party’s rally. Discuss whether this sample is representative or not.
- In a company of 2,000 employees, 200 are selected randomly across departments to study job satisfaction. Identify whether the sample is representative.