Sample Size Calculator

The Core Components of a Sample Size Calculation

Determining the right sample size is a balancing act between resources and statistical rigor. Our Sample Size Calculator simplifies this by handling the complex formulas, but understanding the inputs is crucial for obtaining a meaningful result. Let’s break down the key concepts.

1. Population Size (N)

The population is the entire group you want to draw conclusions about. It could be “all doctors in the United States,” “all users of your app,” or “all cars manufactured last year.”

• When to use it: You should input a population size only when you are dealing with a relatively small and well-defined group.
• When to leave it blank: If your population is very large, unknown, or effectively infinite (like “all potential customers in the world”), you can leave this field blank. The calculator will use a formula for an infinite population. The difference in required sample size becomes negligible once the population exceeds a few tens of thousands.

2. Confidence Level

The confidence level tells you how sure you can be that your results are accurate. It’s expressed as a percentage and represents how often the true percentage of the population who would pick an answer lies within the margin of error. A 95% confidence level is the most common standard in scientific research, meaning if you were to repeat the survey 100 times, 95 of those times the results would match the true population value within your margin of error.

• 90% Confidence: Less certain, requires a smaller sample size.
• 95% Confidence: The industry standard, offering a good balance.
• 99% Confidence: Very certain, requires a much larger sample size.

3. Margin of Error (Confidence Interval)

The margin of error, expressed as a percentage, describes the “plus or minus” range that accompanies your survey results. It tells you how much you can expect your survey results to reflect the views of the overall population. For example, if you find that 60% of your sample prefers a certain product with a 5% margin of error, it means you can be confident that the true percentage in the entire population is between 55% and 65%.

• A smaller margin of error (e.g., ±2%) means your results are more precise, but it requires a larger sample size.
• A larger margin of error (e.g., ±10%) is less precise but allows for a smaller, more affordable sample size.

4. Expected Proportion (or Response Distribution)

This input relates to the expected variance in your results. It’s the proportion of the population you expect to choose a particular answer. If you are unsure, using 50% is the most conservative choice. A proportion of 50% indicates the maximum level of variability in a population (an even split), which requires the largest possible sample size to achieve your desired precision. If you have prior research suggesting the proportion is closer to an extreme (e.g., 90% or 10%), you can use that to get a smaller, more efficient sample size.

Understanding the Formulas Behind the Sample Size Calculator

This calculator uses established statistical formulas to provide its results. Understanding the basis of these calculations can help you appreciate the science behind getting a reliable sample.

1. Cochran’s Sample Size Formula for an Infinite Population

When the population size is large or unknown, the most common method for calculating sample size is Cochran’s formula:

n₀ = (Z² * p * (1-p)) / e²

• n₀ is the initial sample size.
• Z is the Z-score corresponding to your chosen confidence level (e.g., 1.96 for 95% confidence).
• p is the estimated proportion or variance (we use 0.5 for the most conservative estimate).
• e is the margin of error (e.g., 0.05 for ±5%).

2. Finite Population Correction (FPC)

Cochran’s formula assumes the population is infinitely large. However, if your population is small and you are sampling a significant fraction of it, the required sample size can be reduced. The calculator applies the Finite Population Correction formula to adjust the initial sample size:

n = n₀ / (1 + (n₀ – 1) / N)

• n is the adjusted, final sample size.
• n₀ is the initial sample size from Cochran’s formula.
• N is the total population size you entered.

This is why the “Population Adjusted” result is often smaller than the initial required sample size, especially for smaller, well-defined populations.

Beyond Surveys: Sample Size for A/B Testing & Experiments

Calculating sample size for an A/B test or a scientific experiment is different from a simple survey. Here, your goal is not just to estimate a population parameter but to detect a statistically significant difference between two groups (e.g., which webpage design leads to more clicks). This requires two additional concepts.

1. Statistical Power

Statistical power is the probability that your test will correctly detect a true effect if one actually exists. In simpler terms, it’s the probability of avoiding a “false negative” (a Type II error). A power of 80% is a common standard, meaning you have an 80% chance of finding a statistically significant difference if a real difference of your specified magnitude exists.

• Higher Power (e.g., 90%): Reduces the risk of missing a real effect but requires a larger sample size.
• Lower Power (e.g., 70%): Requires a smaller sample size but increases the risk of your test being “underpowered” and failing to detect a real difference.

2. Minimum Detectable Effect (MDE)

This is the smallest difference between your control and variation groups that you want to be able to detect. For example, you might want to know if a new button color increases the conversion rate by at least 2%. This 2% is your MDE.

• A smaller MDE (wanting to detect a very subtle change) requires a much larger sample size.
• A larger MDE (only caring about detecting big changes) allows for a smaller sample size.

Our Sample Size Calculator switches to a power analysis formula when you select the “A/B Test” mode, allowing you to plan experiments that are robust enough to yield conclusive results.

Common Pitfalls to Avoid When Determining Sample Size

Using a sample size calculator is a crucial first step, but it’s not a magic bullet. The quality of your research depends on how you collect your sample. Here are some common mistakes to avoid:

• Using a Convenience Sample: Surveying only people who are easy to reach (like your friends or social media followers) can introduce significant bias. Your sample should be as random as possible to be truly representative of the entire population.
• Ignoring the Non-Response Rate: Not everyone you invite to your survey will respond. You should anticipate a certain non-response rate and aim to collect more responses than the calculated minimum to compensate. If your calculated sample size is 400 and you expect a 50% response rate, you’ll need to send your survey to at least 800 people.
• Choosing a Margin of Error That’s Too Large: While a ±10% margin of error yields a small sample size, the results are often too vague to be actionable. A result of “50% ±10% prefer the new feature” means the true value is somewhere between 40% and 60%, which may not be precise enough to make a business decision.
• Forgetting About Subgroups: If you plan to analyze subgroups within your data (e.g., comparing responses from different age groups), the sample size for each subgroup must also be statistically robust. This may require a much larger overall sample size.