Sample Size Calculator

Hypothesis Testing and Statistical Inference

Find the ideal sample size to ensure your survey or study results are statistically significant. Adjust the confidence level, margin of error, and population details to get your required sample size.

Practical Examples

See how the Sample Size Calculator works with real-world scenarios.

Political Poll

example1

A political pollster wants to gauge the approval rating of a candidate in a city of 500,000 people. They want to be 95% confident in their results, with a margin of error of 3%.

Conf: 95%, Margin: 3%

Prop: 50%, Pop: 500000

Market Research

example2

A company is launching a new product and wants to survey potential customers. They are unsure of the market size (assume it's very large) and want a 99% confidence level with a 5% margin of error.

Conf: 99%, Margin: 5%

Prop: 50%, Pop: N/A

Educational Study

example3

A researcher is studying the reading habits of 1,200 high school students. They expect that about 60% of students read for pleasure and need a 90% confidence level with a 4% margin of error.

Conf: 90%, Margin: 4%

Prop: 60%, Pop: 1200

Website A/B Test

example4

A developer wants to determine if a new button color increases click-through rate. They want to be extremely confident (99.9%) with a tight margin of error (1%) before making a change.

Conf: 99.9%, Margin: 1%

Prop: 50%, Pop: N/A

Other Titles
Understanding the Sample Size Calculator: A Comprehensive Guide
Dive deep into the principles of sample size calculation to ensure your research is robust, reliable, and statistically significant. Learn how to balance various factors to get the right sample size for your needs.

What is Sample Size?

  • The Role of a Sample in Research
  • Why Sample Size Matters
  • The Goal of Sample Size Calculation
Sample size refers to the number of participants or observations included in a study. In most research scenarios, studying an entire population is impractical, expensive, and time-consuming. Instead, we select a smaller, manageable subset of the population, known as a sample. The goal is to use this sample to make inferences about the entire population. The accuracy and reliability of these inferences depend heavily on how well the sample represents the population, and a key factor in this representation is the sample size.
Why is Determining the Right Sample Size Crucial?
Choosing the correct sample size is a critical step in the design of a study. A sample that is too small may lead to inconclusive results, as it may not have enough statistical power to detect a true effect or difference. On the other hand, a sample that is too large wastes resources, time, and can be ethically questionable, as it exposes more participants than necessary to potential risks. Therefore, a sample size calculation helps strike a balance, ensuring the study is both scientifically valid and efficient.

Step-by-Step Guide to Using the Sample Size Calculator

  • Inputting Confidence Level
  • Setting the Margin of Error
  • Estimating Population Proportion and Size
Our calculator simplifies the process of determining your sample size. Here's a breakdown of each input and how to use it effectively:
1. Confidence Level
This reflects how confident you want to be that your results fall within a certain range. It is expressed as a percentage (e.g., 95%). A 95% confidence level means that if you were to repeat the survey 100 times, you would get the same results within the margin of error 95 of those times. Common choices are 90%, 95%, and 99%. Higher confidence requires a larger sample.
2. Margin of Error
Also known as the confidence interval, the margin of error is a percentage that describes how much you can expect your survey results to deviate from the true population value. A 5% margin of error means your result could be 5% higher or lower than the actual population figure. A smaller margin of error provides more precision but requires a larger sample size.
3. Population Proportion
This is the fraction of the population that has a specific characteristic you are interested in. If you have preliminary data (e.g., from a previous study), you can use that value. If you are completely unsure, use 50%. A proportion of 50% (or 0.5) maximizes the variance, which results in the most conservative (i.e., largest possible) sample size, ensuring you have enough power.
4. Population Size (Optional)
This is the total number of individuals in the group you're studying. If the population is small, you can use a formula to adjust the sample size downwards. This is known as the 'finite population correction'. If your population is very large (e.g., over 100,000) or you don't know the exact size, you can leave this field blank, and the calculator will assume an infinite population.

Real-World Applications of Sample Size Calculation

  • Market Research and Surveys
  • Medical and Clinical Trials
  • Quality Control in Manufacturing
Political Polling
Before an election, pollsters need to know how many voters to survey to confidently predict the outcome within a certain margin of error. For example, a national poll might calculate the sample size needed to represent all eligible voters in a country.
Customer Satisfaction Surveys
A large e-commerce company wants to measure customer satisfaction. Instead of surveying all of its millions of customers, it calculates the sample size needed to get a reliable picture of overall satisfaction.
A/B Testing
A tech company wants to test a new website layout. A sample size calculation determines how many users need to see each version (A and B) to reliably detect a statistically significant difference in conversion rates.

Mathematical Derivation and Formulas

  • Cochran's Formula for Infinite Populations
  • Finite Population Correction
  • Understanding the Z-score
The calculation for sample size is based on established statistical formulas. The primary formula used for an unknown or very large population is Cochran's formula.
Formula for Infinite Population
The formula to calculate the sample size (n) is: n = (Z² p (1-p)) / E²
Where: Z is the Z-score corresponding to the chosen confidence level (e.g., 1.96 for 95% confidence). p is the estimated population proportion (as a decimal). E is the margin of error (as a decimal).
Formula with Finite Population Correction
When the population size (N) is known and relatively small, the initial sample size (n) can be adjusted using the finite population correction formula to get the final sample size (n'): n' = n / (1 + (n - 1) / N)
This correction factor reduces the required sample size, as a sample that is a significant fraction of a finite population provides more information than one from an infinite population.

Common Z-scores:

  • For a 95% confidence level, the Z-score is 1.96.
  • For a 99% confidence level, the Z-score is 2.576.

Common Misconceptions and Best Practices

  • Sample Size vs. Response Rate
  • The Myth of the 'Magic Number'
  • Using 50% for Population Proportion
Misconception: A Sample Size is the Same as the Number of People to Contact
The calculated sample size is the number of completed responses you need. It does not account for non-responses. You must estimate your response rate and contact more people to achieve your target sample size. For example, if you need 400 responses and expect a 10% response rate, you'll need to contact 400 / 0.10 = 4,000 people.
Misconception: A Percentage of the Population is a Good Rule of Thumb
The required sample size does not scale linearly with the population size. For large populations (e.g., >100,000), the sample size needed for a given confidence level and margin of error tends to plateau. A sample of 1,000 people can be just as informative for a population of 1 million as it is for 10 million.
Best Practice: Using 50% for Proportion
When in doubt about the population proportion (p), using 50% (or 0.5) is the most conservative choice. The term p * (1-p) in the formula is maximized when p is 0.5. This ensures your sample size is large enough to meet your precision requirements, regardless of the true underlying proportion.