Cochran’s Sample Size Calculator

When planning research, it is important to choose the right sample size. A sample that is too small may not give reliable results, while a sample that is too large can waste time and resources. To help solve this problem, researchers often use Cochran’s sample size formula, one of the most trusted methods for calculating the correct sample size. In this article, we explain how Cochran’s formula works in simple terms. We also provide a free Cochran’s sample size calculator so you can quickly and easily find the right sample size for your study.

What is Cochran’s Formula?

Cochran’s formula is a method used to calculate the ideal sample size for a research study. It was introduced by the statistician William G. Cochran in 1977 to help researchers determine how many people (or items) they should include in a survey or experiment. The main purpose of this formula is to make sure the sample is large enough to give reliable results, without being unnecessarily large and costly.

Researchers often use Cochran’s sample size formula when working with large populations or when the population size is unknown. It is especially useful in fields like social sciences, health research, and market surveys, where collecting data from the entire population is not practical.

Cochran’s Sample Size Formula Explained

The Cochran’s formula is written as:

$$n_0 = \frac{Z^2 \cdot p \cdot q}{e^2}$$

Where:

• n₀ is the sample size you wish to determine
• Z is the Z-score, which corresponds to your chosen confidence level. For example, 95% confidence uses a Z of 1.96, and 99% confidence uses 2.576.
• p is the estimated proportion of the population with the characteristic of interest. If you don’t know, researchers often use 0.5 (50%) because it gives the most conservative (largest) sample size.
• q = 1 – p. This is the proportion of the population without the characteristic.
• e is the margin of error (also called the level of precision). For example, 0.05 means your results will be within ±5% of the true population value.

In simple terms, the formula multiplies the confidence level (Z²) by the variation in the population (p × q), then divides by how precise you want the results to be (e²). The result is the minimum number of people or items you should include in your sample to get reliable results.

Example 1: Using Cochran’s Formula (Without Finite Population Correction)

Let’s say you want to determine the sample size for a survey with the following conditions:

• Confidence level = 95% (Z = 1.96).
• Proportion (p) = 0.5
• Margin of error (e) = 0.05

NOTE: The Z score of 1.96 is obtained from the z score table by looking from inside of the positive z score table. Specifically, you look for (1-0.05/2) = 0.975 from the positive z table, which gives 1.96

Step 1: Apply the formula:

$$n_0 = \frac{Z^2 \cdot p \cdot q}{e^2}$$

Here,

• Z = 1.96
• p = 0.5
• q = 1 – p = 0.5
• e = 0.05

Step 2: Substitute the values

$$n_0 = \frac{1.96^2 \cdot 0.5 \cdot 0.5}{0.05^2}$$

$$ = \frac{3.8416 \cdot 0.25}{0.0025}$$

$$ = \frac{0.9604}{0.0025}$$

$$n_0= 384.16$$

Step 3: Round up the result

The required sample size is about 385 respondents.

Adjusting Cochran’s Formula for Finite Population

Cochran’s formula is most accurate when the population is very large or unknown. But when the total population is small or finite, the initial result (n₀) often gives a sample size that is larger than necessary. To correct this, researchers use the finite population correction (FPC).

Thus, the adjusted formula becomes:

$$n = \frac{n_0}{1 + \frac{n_0 – 1}{N}}$$

Where:

• n = adjusted sample size
• n₀ = initial sample size from Cochran’s formula
• N = total population size

Example 2: Using Cochran’s Formula (Without Finite Population Correction)

Suppose Cochran’s formula gives an initial sample size of 384 for a large population (using 95% confidence, p = 0.5, margin of error = 5%). But if your actual population size is only 1,000 people, the corrected sample size can be calculated as follows:

$$n = \frac{384}{1 + \frac{384 – 1}{1000}} \approx 278$$

Therefore, instead of surveying 384 people, you only need about 278 respondents when the population is 1,000.

Assumptions and Conditions for Using Cochran’s Formula

Cochran’s formula works best when the population is very large or unknown and when random sampling is used. It assumes that every individual in the population has an equal chance of being selected. However, for smaller or finite populations, the formula may overestimate the sample size. In such cases, researchers should apply the finite population correction (FPC) to adjust the result. This ensures the sample size is accurate and not larger than necessary.

In summary, use Cochran’s formula for large-scale surveys and make adjustments if your population is relatively small.

Frequently Asked Questions