How to Use the Yamane Formula Calculator
This Yamane’s sample size calculator will help you calculate the minimum sample size for a survey when you already know the size of a finite population.
To calculate the sample size:
- Enter the total population size, N.
- Enter the desired margin of error, e, as a decimal.
- Click Calculate.
The calculator will instantly return the minimum sample size and show you exactly how the value was computed, step-by-step.
Note. The calculator rounds up the value to make sure the minimum sample size is met.
What Is Yamane’s Formula?
Yamane’s formula is a simplified method for estimating the sample size required to study a known and finite population. It is most commonly used when a researcher wants to estimate a population proportion but does not have a previous estimate of that proportion.
The method is associated with Taro Yamane’s 1967 book, Statistics: An Introductory Analysis. It is popular in surveys, dissertations, theses, and other academic research because it only requires two parameters: population size (N) and the desired margin of error (e).
Note. Yamane’s formula only provides the minimum number of participants you should sample from the target population. However, it does not account for people who decline to participate, fail to complete the questionnaire, or provide unusable responses.
Yamane Sample Size Formula
Taro Yamane’s formula is n = N / (1 + Ne2)
Where:
- n is the estimated minimum sample size
- N is the total population size (target population)
- e is the desired level of precision (margin of error), expressed as a decimal
Tip. The value of e is often described as the margin of error. A smaller margin of error produces a larger sample because greater precision requires information from more members of the population.
How to Calculate Sample Size Using Yamane’s Formula
A public health researcher wants to study the dietary habits of undergraduate students living on a university campus. The total university student population living on campus is exactly 4,000 students. Due to limited budget and time constraints, the researcher cannot survey everyone and decides to use a sample. She chooses to accept a 5% margin of error for her findings. Calculate the minimum required sample size for this study using Taro Yamane’s formula, showing all the steps. Also, explain what would happen to the required sample size if the researcher decided to change the margin of error to 2%.
Solution
From the question, we know that:
- The target population, N = 4000
- The desired margin of error, e = 5%
We can compute the minimum sample size as follows:
By definition, Yamane’s sample size formula is: n = N / (1 + Ne²)
Substituting the values into the formula, we get:
n = 4000/ (1 + 4000(0.05)2)
= 4000/11
= 363.64
Since the sample size must be a whole number, we should round up the value. Therefore, the minimum sample size for a study with population size N = 4000 and margin of error, e = 0.05, is 364.
Want to verify the result using the Yamane formula calculator? Follow these steps:
- Enter 4000 as the population size.
- Enter 0.05 as the margin of error.
- Click Calculate.
The calculator will return a minimum sample size of 364. It will also provide a clear, step-by-step solution showing you how the value was obtained.
What happens when the margin of error changes to 2%?
Dropping the margin of error from 5% to 2% increases the required sample because a smaller margin of error requires a much larger, more precise representation of the population to ensure the same level of confidence.
When to Use Yamane’s Formula
Yamane’s sample size formula is only suitable when:
- The target population size is known and finite. If the population size is unknown, Cochran’s formula may be useful
- A simple random sampling technique was used during data collection to ensure every member in the target population has an equal chance of being selected.
- You don’t have a prior estimate for the population proportion or variability. Yamane formula assumes a conservative 50% proportion to maximize the sample size.
- You only know the population size and margin of error and want a quick estimate of the sample size
Adjusting the Sample Size for Nonresponse
The minimum sample size from Yamane’s formula represents the minimum number of usable responses for your study and not the actual number of participants you should survey. In a real-world setting, it is very unlikely to get all the surveys filled out, and this is why it is important to account for non-response.
Therefore, to know the number of participants to survey, you should divide the minimum sample size by the expected response rate.
Example. Suppose the required sample is 385 and the researcher expects an 80% response rate. Calculate the adjusted sample size to account for non-response.
Solution
From the example, we know that:
- n = 385
- response rate = 0.80 (80%)
Therefore, the adjusted sample size is: nadj = n/response rate
=385/0.80
=481.25
Rounding up gives 482.
Therefore, as a researcher, you may need to invite about 482 participants to obtain approximately 385 completed responses if the expected response rate is achieved.
Yamane Formula vs Slovin Formula
Many students and researchers wonder whether the Yamane formula and Slovin’s formula are the same. Mathematically, both of these sample size formulas are identical. The primary difference lies in their academic attribution, origin, and specific usage context in research.
- Origin and attribution. While Slovin’s formula was first popularized in the early 1960s, its true original author remains obscure or undocumented in standard statistical literature. However, Yamane’s formula is said to have been introduced by statistician Taro Yamane in 1967 and is a formally established, widely accepted formula because it is backed by a distinct publication and clear academic traceability
- Research context and usage. Yamane’s formula is generally preferred in formal academic writing, including theses and dissertations, because researchers can accurately cite Yamane’s 1967 book. On the other hand, Slovin’s formula is frequently used in practical or undergraduate social science surveys, especially in student projects, simply due to its ease of use.
Frequently Asked Questions
Yamane’s formula is a simplified equation for estimating the number of observations required from a known finite population. The formula is n = N/(1 + Ne²), where N is the target population size and e is the margin of error.
The method is associated with Taro Yamane’s 1967 statistics text, Statistics: An Introductory Analysis.
No. It requires a known finite population size. When the population is unknown or effectively infinite, use Cochran’s formula or another suitable method
As the population becomes large, the effect of N decreases, and the sample approaches 1/e². At a 5% margin of error, the limiting sample size is approximately 400 under the formula’s assumptions.
