Difference Between Population and Sample with Examples
The concept of population and sample plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding these terms helps us collect, analyze, and interpret data correctly in statistics and other subjects. Knowing when to use a population and when to use a sample is especially important for exams, projects, and daily reasoning.
What Is Population and Sample?
A population in statistics means the entire group you want to study or get information about. A sample is a smaller part or subset taken from that population. You’ll find this concept applied in data collection, survey analysis, research projects, and all forms of statistics. For example, if we want to know the average marks of all students in a school, the school’s students are the population. If we check only 50 students, those 50 form the sample.
Key Differences Between Population and Sample
| Population | Sample |
|---|---|
| Entire group under study (e.g., all students) | Part of the population (e.g., selected students) |
| Described by ‘parameters’ | Described by ‘statistics’ |
| Often large or infinite | Small and manageable |
| Hard to measure directly | Easy to survey or experiment |
This table makes it clear how populations and samples differ. Remember: populations are usually bigger, samples are easier to work with.
Examples of Population and Sample
- All the plants in a garden (population); 10 randomly chosen plants (sample).
- Every book in a library (population); 100 books checked for missing pages (sample).
- All voters in a city (population); 500 questioned in a survey (sample).
In math questions, always read carefully to identify what is the population and which is the sample.
Formulas: Population and Sample Mean & Variance
Here are the standard formulas used in most statistics questions:
| Statistic | Population Formula | Sample Formula |
|---|---|---|
| Mean | \(\mu = \frac{1}{N} \sum_{i=1}^N x_i\) | \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) |
| Variance | \(\sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i-\mu)^2\) | \(s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i-\bar{x})^2\) |
Notice that for samples, we divide by n-1 (not n). This corrects for estimation error and is very important for exams!
Step-by-Step Illustration: Sample Mean Calculation
1. Suppose a sample of 5 students scores: 72, 75, 68, 80, 852. Add all the scores: 72 + 75 + 68 + 80 + 85 = 380
3. Divide by the sample size (n = 5):
\( \bar{x} = \frac{380}{5} = 76 \)
4. Final answer: The sample mean is 76.
When to Use Population vs Sample?
Use a population when you have access to every member in the group. Use a sample when the group is too large, and you want to work faster or save effort.
- Population: Census of an entire country, all students in a school.
- Sample: Polling a few voters before elections, testing a batch of products from a factory.
Try These Yourself
- Identify the population and the sample: Out of 200 mangoes, 30 are picked to check for sweetness.
- Calculate the sample mean for 4 observation values: 90, 95, 100, 85.
- Find variance for the values: 5, 5, 7, 7, 7.
- Decide: when would you use a sample instead of a full population?
Frequent Errors and Misunderstandings
- Mixing up if the question is about a population or sample
- Using the formula for population variance on sample data (remember n-1!)
Relation to Other Concepts
The idea of population and sample connects closely with concepts like mean, variance, and sampling methods. A strong understanding also helps with more complex statistics topics like probability.
Classroom Tip
A quick way to remember: Population = Whole group, Sample = Small part. Draw circles to show a large set and shade inside for the sample. Vedantu’s teachers often use such visuals and worksheet practice for better memory in live classes and class notes.
We explored population and sample—from definition, formula, examples, mistakes, and connections to other maths topics. Continue practicing with Vedantu to become confident in statistics and make fewer mistakes in exams and daily problem-solving!
Useful Links:
- Mean: Learn how averages are used for both populations and samples.
- Variance: See how data spreads around the mean in statistics.
- Types of Data in Statistics: Know the difference between quantitative, qualitative, and how this links to sampling.
- Mean Absolute Deviation: Related measure of dispersion.
FAQs on Understanding Population and Sample in Statistics
1. What is population in statistics?
In statistics, a population is the complete set of all individuals, items, or observations that you want to study. It includes every possible data point relevant to a research question.
- It can be finite (e.g., 500 students in a school).
- It can be infinite (e.g., all possible coin toss results).
- Population data describes the entire group, not just a part of it.
2. What is a sample in statistics?
A sample is a subset of the population selected for analysis. It represents a smaller group chosen to draw conclusions about the whole population.
- It saves time and cost compared to studying the entire population.
- The quality of conclusions depends on how well the sample represents the population.
- Example: Surveying 50 students out of 500 is taking a sample.
3. What is the difference between population and sample?
The main difference is that a population includes all members of a group, while a sample includes only a part of that group.
- Population size is usually denoted by N.
- Sample size is usually denoted by n.
- Population parameters (like μ) describe the whole group.
- Sample statistics (like x̄) estimate population values.
4. What is a population parameter?
A population parameter is a numerical value that describes a characteristic of the entire population. It is usually fixed but often unknown.
- Population mean: μ
- Population standard deviation: σ
- Population proportion: p
5. What is a sample statistic?
A sample statistic is a numerical measure calculated from a sample to estimate a population parameter.
- Sample mean: x̄ = (Σx) / n
- Sample standard deviation: s
- Sample proportion: p̂
6. How do you calculate the population mean and sample mean?
The population mean is calculated using μ = (ΣX) / N, and the sample mean is calculated using x̄ = (Σx) / n.
- ΣX = sum of all population values
- N = total number of population values
- Σx = sum of sample values
- n = number of sample values
7. Why do we use a sample instead of the whole population?
We use a sample because studying the entire population is often costly, time-consuming, or impractical. Sampling makes statistical analysis more efficient.
- Reduces time and expense.
- Useful when population is very large or infinite.
- Allows quicker decision-making.
8. What are the types of sampling methods?
The main types of sampling methods are probability and non-probability sampling. These methods determine how a sample is selected from a population.
- Simple random sampling
- Stratified sampling
- Systematic sampling
- Cluster sampling
9. Can you give an example of population and sample?
An example of population and sample is when all 1,000 voters in a town form the population, and 100 selected voters form the sample.
- Population: All 1,000 voters.
- Sample: 100 voters surveyed.
- Goal: Use sample data to estimate opinions of all voters.
10. What are common mistakes when distinguishing population and sample?
A common mistake is confusing sample statistics with population parameters. These terms represent different concepts in statistics.
- Using x̄ (sample mean) as if it were μ (population mean).
- Assuming a small sample perfectly represents the population.
- Ignoring sampling bias.
