When conducting quantitative research and intend to collect primary data using survey questionnaires, you’ll be required to determine the minimum sample size for your study. Sample size determination helps ensure that the results & findings you get are representative of the whole population, without wasting time or resources. While there are many sample size formulas you can use to determine the minimum sample size for your study, the most commonly used are: Slovin’s, Cochran’s, and Taro Yamane. In this guide, we’ll explain each of the formulas, when to use them, and how to compute sample size manually, using each of these formulas.
1. Cochran’s Sample Size Formula
Cochran’s formula is commonly used in survey or cross-sectional studies, where the researcher plans to sample data from a large population (infinite population). However, if the target population is known and finite, you need to apply Cochran’s formula and then adjust for the finite population correction.
Cochran’s formula for sample size determination is:
n₀ = Z²·p(1–p)/e²
Where:
- n₀ is the minimum sample size for your study
- Z is the Z-score for your desired confidence level (e.g., 1.96 for 95%)
- p is the estimated proportion of the population (use 0.5 if unknown)
- e is the desired margin of error
Therefore, you should use the Cochran’s sample size formula when you want to determine the minimum sample size for large, unknown, or infinite populations.
Example. Suppose you want to find the minimum sample size for a study involving the proportion of adults in a city with over 1 million inhabitants who plan to vote in an upcoming election. Since the population is very large for sampling purposes, you should use Cochran’s formula.
Now, suppose you know the size of the target population and want to use Cochran’s formula. In this case, you should first compute the minimum sample size (n₀) and then apply the finite population correction (FPC).
The finite population correction formula is n = n₀ / [1 + (n₀–1)/N].
Where:
- n₀ is the sample size from the Cochran’s formula
- N is the known population size
- n is the estimated and adjusted sample size
According to Cochran (1977), applying the FPC when determining the minimum sample size using Cochran’s formula accounts for the population not included in the sample. As a result, it helps in reducing the variance of survey estimates when the sampling fraction is not negligible.
Sample Size Determination Using Cochran’s Formula: Infinite Population Example
Scenario 1. Suppose you are conducting a nationwide survey to estimate how many people support a new education policy. The population is extremely large and unknown. You decide to use a 95% confidence level, a 5% margin of error, and assume maximum variability (p = 0.5). Compute the minimum sample size for this survey.
Solution
To compute the minimum sample size for the survey, we should apply Cochran’s formula:
By definition, Cochran’s formula is: n₀ = Z²·p(1–p)/e²
From the scenario, we know that:
Population proportion, p = 0.5, Margin of error, e = 5% or 0.05. Since we want to be 95% confident, the corresponding z-score is 1.96.
Substituting the values into the formula, we have:
n₀ = 1.96²*0.5(1–0.5)/0.05²
=0.9604/0.0025
=384.16
Rounding up to the nearest whole number, the minimum sample size for the survey is 385 participants.
Sample Size Determination Using Cochran’s Formula: Finite Population Example
Scenario 2: Now, imagine your study targets employees in a company with 10,000 staff members. Assuming all the other parameters remain the same as in scenario 1. That is, 95% confidence, 5% margin of error, and p = 0.5. Compute the minimum sample size for this survey.
Solution
As you can see, in addition to knowing the standard parameters in Cochran’s formula, we have more information about the size of the target population. Specifically, we know that the target population is N = 10,000.
Therefore, we’ll need to adjust for this finite population using the FPC formula.
Since the parameters were the same as in scenario 1, the minimum sample size assuming an infinite population, n₀ = 384.16 (we don’t have to compute this since computation is shown in scenario 1).
Now, we need to adjust for the finite population using the formula
n = n₀ / [1 + (n₀–1)/N].
Substituting the known values in the formula, we have:
n = 384.16 / [1 + (384.16–1)/10000].
Solving the above equation, we get:
n = 384.16/[1+383.16/10000]
=384.16/[1+0.038316]
=384.16/1.038316
= 369.98
Rounding up to the nearest whole number, the minimum sample size is n = 370.
Therefore, the corrected sample size for a finite population of 10,000 is approximately 370 respondents.
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2. Slovin’s Sample Size Formula
Slovin’s formula is used to determine the minimum sample size (n) when the target population is known and finite, and little is known about the population behavior or variance. It is the most appropriate sample size determination formula for quick, small-scale surveys, classroom studies, and initial research where a rough estimate of sample size is acceptable.
The Slovin’s formula is n = N / (1 + Ne²)
Where:
- n is the minimum sample size
- N is the size of the target population
- e is the desired margin of error.
Sample Size Determination Using Slovin’s Formula: Example
To learn how to determine the minimum sample size for your study using Slovin’s formula, let’s walk through an example.
Scenario. Imagine you’re conducting a survey among college students in a university with 5,000 students. You don’t have information about how opinions vary among students. Thus, you decide to use Slovin’s Formula to determine a suitable sample size. Assuming the margin of error (e) is 5%, compute the minimum sample size (n) for the survey.
Solution
From the scenario, we know that:
The size of the target population, N = 5000, and the desired margin of error, e = 5% or 0.05
To compute the required sample size, we apply Slovin’s formula.
By definition, Slovin’s formula is n = N / (1 + Ne²)
Substituting the values in the formula, we have:
n = 5000 / (1 + 5000*0.05²)
=5000/(1+12.5)
=5000/13.5
= 370.37
Rounding up to the nearest whole number, the minimum sample size for the survey is n = 371 respondents.
Need a quick way to find your sample size using Slovin’s formula? Try our Slovin’s Formula Calculator for instant results with step-by-step explanation.
3. Taro Yamane’s Formula
The Taro Yamane sample size formula is one of the simplest and most widely used methods for determining sample size when the population size is known. It is best suited when the population size is small to moderate, and the total population is known.
Although the formula is identical to Slovin’s formula, it has a stronger statistical foundation because it was derived from Cochran’s formula under the assumptions of a 95% confidence level and maximum variability (p = 0.5). As such, you’re more likely to find Yamane’s formula cited in most academic work, such as dissertations, theses, or published papers.
The Taro Yamane formula is n = N / (1 + Ne²)
Where:
- n is the minimum sample size
- N is the size of the target population
- e is the desired margin of error.
Sample Size Determination Using Yamane’s Formula: Example
Scenario: Assume you are surveying all households in a small town with N = 2,000 households. You want a 5% margin of error (e = 0.05). Calculate the minimum sample size for this survey using Taro Yamane’s Formula.
Solution
From the scenario, we know that:
Target population, N = 2000
Margin of error, e = 0.05
By definition, Yamane’s formula is n = N / (1 + Ne²)
Substituting the known values in the formula, we have:
n = 2000 / (1 + 2000 * 0.05²)
=2000/(1+2000*0.0025)
= 2000/(1+5)
= 2000/6
n = 333.33
Rounding up to the nearest whole number, we have:
n = 334.
Therefore, the minimum sample size for the survey using Taro Yamane’s formula is 334 households.
Want a quick way to determine the minimum sample size for your survey using Yamane’s formula? Try our Taro Yamane Sample Size Calculator for instant results with a step-by-step solution.
Key Takeaways
- Sample size determination ensures the results are reliable, representative of the population, and not a waste of time or resources.
- Cochran’s sample size formula is best suited for large, unknown, or infinite populations, especially in survey and cross-sectional studies involving proportions. However, when the population size is known and finite, you should adjust the formula using the finite population correction (FPC) to reduce sampling variance.
- Slovin’s formula is a quick and simple method used when the population size is known, but population variability is unknown. It is commonly applied in small-scale surveys, classroom research, and preliminary studies where an approximate sample size is acceptable.
- Taro Yamane’s formula is ideal for finite populations with known size and is widely used in academic research, including theses and dissertations.
- The choice of sample size formula depends on three main factors:
- Whether the population is finite or infinite
- Whether population variability is known or unknown
- The level of precision required (margin of error and confidence level)
Frequently Asked Questions (FAQs)
Sample size determination is the process of calculating how many participants, items, or observations are needed in a study. It helps researchers ensure that their findings are reliable, valid, and generalizable. A well-determined sample size increases the accuracy of conclusions while minimizing time and resources.
The three most common formulas are:
– Cochran’s Formula for large or infinite populations.
– Slovin’s Formula for quick estimation when population variability is unknown.
– Taro Yamane’s Formula for finite populations with known size.
You can use any of these depending on your study design and population size.
Key factors include:
– Population size (N) – the total number of units in your target group.
– Confidence level – how sure you want to be about your results (e.g., 95%).
– Margin of error (e) – how much error you can tolerate.
– Population variability (p) – how diverse your population responses are.
– Expected non-response rate – the percentage of participants who might not respond
To adjust for expected non-response, divide your calculated sample size by the anticipated response rate.
For example, if you expect 80% of people to respond and your sample size is 400
n_adjusted = 400/0.8
= 500
This means that you should target 500 people to get 400 responses.
