Non-Representative Sample 

1.0What Is a Non-Representative Sample?

A non-representative sample (also “unrepresentative sample”) is a sample drawn from a population in such a way that the sample does not accurately reflect the characteristics of the whole population with respect to one or more relevant parameters. In mathematical/statistical terms, if you compute a statistic (mean, proportion, variance, etc.) based on the sample, that statistic may not be a good estimator of the corresponding population parameter.

• The sample may omit some subgroups, overrepresent others, or be collected by a biased method.
• The key issue is sampling bias (selection bias): the method of selecting units causes the sample to deviate systematically from the population.

2.0Types of Non-Representative Sampling Errors

1. Selection Bias: Selection bias arises when certain groups within the population have a higher or lower chance of being included in the sample than others. This results in a sample that is skewed towards specific characteristics, making it non-representative.

• Example: If a survey about study habits is conducted only among students who attend morning classes, students who prefer evening classes are excluded, leading to selection bias.

2. Non-response Bias: Non-response bias occurs when some selected individuals do not participate in the survey or study. If the non-respondents differ significantly from the respondents in terms of the study variable, the sample becomes biased.

• Example: In a survey about exam stress, if students who are highly stressed choose not to respond, the sample will not represent the proper distribution of stress levels among all students.

3. Undercoverage Bias: Undercoverage bias happens when some members of the population are inadequately represented in the sample. This often occurs if the sampling frame (the list from which the sample is drawn) does not cover the entire population.

• Example: Conducting a survey using a school’s email list will exclude students without access to email, leading to undercoverage.

4. Overcoverage Bias: Overcoverage bias occurs when the sampling frame includes elements that should not be part of the population. This leads to over-representation of certain groups.

• Example: If a list of students from previous years is used to sample current JEE aspirants, students who are no longer eligible may be included, causing overcoverage.

5. Voluntary Response Bias: Voluntary response bias arises when participants self-select into the study. Those with strong opinions or specific interests are more likely to participate, skewing the results.

• Example: An online poll about JEE preparation methods may attract only those students who are highly engaged or have strong preferences, not the average aspirant.

6. Convenience Bias: Convenience bias results from choosing a sample that is easy to access rather than a random one. This method often excludes significant sections of the population.

• Example: Surveying only friends or classmates about JEE preparation overlooks the diversity of the entire student population.

3.0Non-Representative Sample: Advantages vs. Disadvantages

4.0Causes of Non-Representative Samples

Some common causes (especially relevant for contest or exam style problems):

• Convenience sampling: Taking samples that are easy to access (friends, volunteers, the closest, etc.), not randomly chosen.
• Frame bias: The sampling frame does not include all of the population (e.g. sampling via phone excludes people with no phones).
• Non-response bias: Some from the selected sample refuse/respond late, etc., so the final sample differs.
• Undercoverage / Overcoverage: Some subgroups are underrepresented (or overrepresented) in the sample.
• Sampling without randomisation: If selection is not random, one cannot guarantee equal probabilities among population units.

5.0Examples of Non-Representative Samples 

Example 1: A school has 800 boys and 200 girls. A survey is conducted with 50 students, but 45 boys and only 5 girls are chosen.

• Population proportion (boys : girls) = 80:20.
• Sample proportion = 90:10.

Since proportions differ significantly, the sample is non-representative.

Example 2: In a factory of 500 bulbs, 50 are defective. A sample of 20 bulbs is taken, all from one batch that had more defects. If 6 defective bulbs are found in the sample:

• Population defect rate = $\frac{50}{500}=0.1$
• Sample defect rate = $\frac{6}{20}=0.3$

The sample clearly overestimates defects and is non-representative.

6.0Practice Questions on Non-Representative Sample

1. Define a non-representative sample with an example.
2. A city has 60% voters in favor of a candidate. A sample of 100 voters contains 80 supporters. Is the sample representative?
3. Explain why biased sampling methods usually lead to non-representative samples.
4. A population has variance ${\sigma }^2=25$. A sample of 10 elements gives variance 50. Is the sample representative? Why or why not?
5. Distinguish between representative and non-representative samples in terms of mean and variance.