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 $({\sigma }^2)$, then a representative sample S with mean $\bar{x}$ and variance $s^2$ should satisfy: $\bar{x}\approx \mu ,s^2\approx {\sigma }^2$

3.0Mathematical Foundation of Representative Sampling

In mathematics, representative sampling is connected with probability and statistical estimation.

1. Population Mean (μ): $\mu =\frac{{\sum }_{i=1}^n{X}_i}{N}$
2. Sample Mean $(\bar{x})$ : $\bar{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}$
3. Population Variance $({\sigma }^2):{\sigma }^2=\frac{{\sum }_{i=1}^n(X_i-\mu {)}^2}{N}$
4. Sample Variance $(s^2):s^2=\frac{{\sum }_{i=1}^n(x_i-\bar{x}{)}^2}{n-1}$

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.

1. Define the Population: Decide the entire group you want to study (e.g., all students in a school).
2. Select the Sample: Choose a smaller group using random or systematic methods.
3. Calculate Sample Statistics: Find the sample mean, variance, or proportion.
4. Compare with Population: If the sample values are close to the actual population values, the sample is representative.
5. 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

1. 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?
2. 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.
3. 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?
4. 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.
5. In a company of 2,000 employees, 200 are selected randomly across departments to study job satisfaction. Identify whether the sample is representative.