What is the Difference Between Population and Sample in Statistics?
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 Population and Sample Explained in Maths
1. What are population and sample in the context of Maths and Statistics?
In statistics, a population refers to the entire, complete set of individuals, items, or data points that share a common characteristic of interest for a study. For example, all the students in a particular school. A sample, on the other hand, is a smaller, manageable subset selected from the population. For instance, a group of 100 students chosen from that same school would be a sample. Researchers study samples to make inferences or draw conclusions about the entire population.
2. What is the main difference between a population and a sample?
The primary difference lies in their scope and the measurements derived from them. Here’s a quick comparison:
- Scope: A population includes every single member of a defined group, while a sample includes only a portion of it.
- Characteristic Measurement: A measurable characteristic of a population is called a parameter (e.g., population mean, μ). The corresponding characteristic measured from a sample is called a statistic (e.g., sample mean, x̄).
- Purpose: The goal of collecting sample data is to use the statistic to estimate the population parameter.
3. Can you give a real-world example of a population and a corresponding sample?
Certainly. Imagine a mobile phone company wants to know the average battery life of its latest smartphone model.
- The population would be every single smartphone of that model ever manufactured. It is impractical to test them all.
- A sample would be a batch of, say, 500 phones randomly selected from the production line to be tested. The average battery life of these 500 phones (a statistic) is then used to estimate the average battery life for all phones (the parameter).
4. Why do statisticians use samples instead of the whole population?
Using a sample is often more practical than studying an entire population for several key reasons:
- Cost-Effectiveness: Collecting data from a whole population (a process called a census) can be extremely expensive.
- Time-Saving: It is much faster to collect and analyse data from a smaller sample.
- Practicality: Sometimes, the population is too large or inaccessible to study completely. In some cases, like product testing, the process is destructive (e.g., crash-testing cars), making a census impossible.
- Accuracy: A well-chosen, representative sample can provide highly accurate and reliable results that reflect the population.
5. What is the difference between population variance (σ²) and sample variance (s²)?
Both measure the spread or dispersion of data points around the mean, but they are calculated differently. Population variance (σ²) is the average of the squared differences from the population mean (μ), calculated for every individual in the population. The formula divides by the total population size, N. In contrast, sample variance (s²) is an estimate of the population variance based on a sample. Its formula divides the sum of squared differences by the sample size minus one (n-1), known as Bessel's correction, which provides a more accurate, unbiased estimate of the true population variance.
6. What happens if a sample is not representative of the population?
If a sample is not representative, it is considered a biased sample. This is a major issue in statistics because any conclusions drawn from a biased sample will be flawed and cannot be reliably generalised to the entire population. For example, if you survey only the students in the front row of a class to gauge understanding, your results will likely be skewed and not reflect the understanding of the whole class. This leads to inaccurate predictions and poor decision-making.
7. How does sampling error occur and how can it be minimised?
Sampling error is the natural difference between a sample statistic (like the sample mean) and the actual population parameter it is estimating. It occurs simply because a sample is only a part of the population and cannot perfectly mirror it. This is not a mistake, but an inherent aspect of sampling. The most effective way to minimise sampling error is to increase the sample size. A larger sample is more likely to be representative of the population, thus reducing the discrepancy between the sample statistic and the population parameter.
8. Are populations in statistics always finite?
No, populations can be either finite or infinite.
- A finite population has a limited, countable number of individuals. For example, the number of cars registered in Delhi or the number of words in a specific book.
- An infinite population has an unlimited, uncountable number of individuals. For example, the set of all possible outcomes from flipping a coin indefinitely, or all the real numbers between 0 and 1. The type of population influences the statistical methods used.
9. In what situations would studying the entire population (a census) be better than sampling?
While sampling is common, a census is preferred in specific situations. It is the better choice when the population is small and easily accessible. For example, if a teacher wants to find the average test score for their class of 30 students, it makes sense to use the scores of all 30 students. A census is also necessary when complete accuracy is critical and resources (time and money) are not a constraint, such as in a national population census conducted by a government.
10. What are some common methods for selecting a representative sample?
To ensure a sample is representative and free from bias, statisticians use specific sampling techniques. Some common methods include:
- Simple Random Sampling: Every member of the population has an equal chance of being selected, like drawing names out of a hat.
- Stratified Sampling: The population is divided into subgroups (strata) based on a shared characteristic (e.g., age groups), and a random sample is taken from each subgroup.
- Cluster Sampling: The population is divided into clusters (e.g., geographic areas), and entire clusters are randomly selected to be part of the sample.
- Systematic Sampling: Members are selected from a larger population according to a random starting point but with a fixed, periodic interval (e.g., every 10th person on a list).
