📊 Universe, Finite Population, and Sample: A Complete Guide to Understanding Sample Size

When conducting research, surveys, or market studies, it is essential to understand three key concepts: universe, finite population, and sample. These elements are closely related and form the foundation of any statistical study.

🌍 What is the Universe?

The universe is:

The total set of elements, real or potential, about which information is desired in a study.

👉 It represents the broadest scope of the research.

🧠 Example:

• Universe: all people in a country

The universe can be very large or even conceptual, making it impractical to study directly.

🧠 What is a Finite Population?

A finite population is:

A subset of the universe that has a limited and known (or estimable) size (N).

👉 It is the specific, real, and measurable group you can study.

📌 Key Characteristics

• Has a defined size (N)
• Is limited (not infinite)
• Can be counted or estimated
• Comes from real data sources

📊 Examples

• 500,000 residents of a city
• 2,000 university students
• 350 employees in a company

🔎 How is it obtained?

❗ It is not calculated using formulas

👉 It comes from real sources such as:

• Census data
• Administrative records
• Databases
• Official lists

Example:

A company has 500 employees → N = 500

🧪 What is a Sample?

A sample is:

A subset of the population selected to represent it.

👉 It is the portion you actually study.

🎯 What is a Sample used for?

A sample allows you to:

• Save time
• Reduce costs
• Conduct studies efficiently
• Draw conclusions without analyzing the entire population

📊 Relationship between the Concepts

👉 The structure is hierarchical:

Universe ⟶ Finite Population ⟶ Sample

Level	Description
Universe	Everything of interest
Finite Population	Specific and measurable group
Sample	Representative subset

🎯 Full Example

Study: Consumer behavior

• 🌍 Universe: all people in a country
• 👥 Finite population: residents of a city (500,000)
• 🧪 Sample: 384 individuals

🔢 What is Sample Size?

It is the number of elements selected from the population:

👉 How many individuals need to be studied?

📐 Sample Size Formula (Finite Population)

$$𝑛=\frac{{𝑍}^2\cdot 𝑝\cdot (1-𝑝)}{{𝑒}^2}÷(1+\frac{{𝑍}^2\cdot 𝑝\cdot (1-𝑝)}{{𝑒}^2\cdot 𝑁})$$

n=e2Z2⋅p⋅(1−p)​÷(1+e2⋅NZ2⋅p⋅(1−p)​)

Where:

• n = sample size
• N = finite population
• Z = confidence level (e.g., 1.96 for 95%)
• p = expected proportion (usually 0.5)
• e = margin of error

📊 Practical Example

Given:

• Population: 500,000
• Confidence level: 95%
• Margin of error: 5%
• Expected proportion: 50%

👉 Result:

n ≈ 384

🧠 Why this result?

• Using p = 50% is the most conservative scenario
• A 95% confidence level is standard
• For large populations, the finite population correction has minimal effect

📉 Effect of Population Size

Population (N)	Sample Size
500	~218
5,000	~357
500,000	~384

👉 As population size increases, the adjustment becomes less significant.

⚖️ Finite vs Infinite Population

Type	Characteristic
Finite	Countable
Infinite	No clear limit

🎯 Conclusion

• The universe defines the total scope of the study
• The finite population defines the real and measurable group
• The sample allows efficient analysis
• The sample size determines how many elements to study

👉 In summary:

The universe is ideal, the population is real, and the sample is practical.

💡 Final Recommendation

If the true proportion is unknown:

👉 Use p = 0.5 (50%)

This ensures a more reliable and conservative sample size.

📚 References (APA 7 format)

• Hernández Sampieri Roberto, R., Fernández Collado, C., & Baptista Lucio, M. P. (2014). Research methodology (6th ed.). McGraw-Hill.
• Levin Richard I., R. I., & Rubin, D. S. (2010). Statistics for management and economics (7th ed.). Pearson.
• Daniel Wayne W., W. W., & Cross, C. L. (2013). Biostatistics: A foundation for analysis in the health sciences (10th ed.). Wiley.
• Calculator.net Sample Size Calculator. (n.d.). Sample Size Calculator. Retrieved from https://www.calculator.net/sample-size-calculator.html

Sample Calculatior:

https://www.calculator.net/sample-size-calculator.html?type=1&cl=95&ci=5&pp=50&ps=500000&x=Calculate