Sample Size Formula

Sample Size Formula (Table of Contents)

• Formula
• Examples

What is Sample Size Formula?

The term “sample” refers to the portion of the population that enable us to draw inferences about the population and so it is important that the sample size is adequate enough so that meaningful inferences can be made. In other words, it is the minimum sample size that is needed to estimate the true population proportion with the required margin of error and confidence level. As such, the determination of the appropriate sample size is one of the recurrent problems in statistical analysis. The sample size equation can be derived by using population size, the critical value of the normal distribution, sample proportion, and margin of error.

Calculation of Sample Size Formula is represented as follows,

where,

• N = Population size,
• Z = Critical value of the normal distribution at the required confidence level,
• p = Sample proportion,
• e = Margin of error

Explanation of the Sample Size Formula

The calculation of the sample size formula can be done by using the following steps:

Step 1: Firstly, determine the population size which is the total number of distinct entities in your population and it is denoted by N. [Note: In case, the population size is very large but the exact number is not known, then use 100,000 because the sample size doesn’t change much for populations larger than that.]

Step 2: Next, determine the critical value of the normal distribution at the required confidence level. For example, the critical value at 95% confidence level is 1.96.

Step 3: Next, determine the sample proportion which can be used from previous survey results or be collected by running a small pilot survey. [Note: if unsure one can always use 0.5 as a conservative approach and it will give the largest possible sample size.]

Step 4: Next, determine the margin of error which is the range in which the true population is expected to lie. [Note: Smaller the margin of error, more is the precision and hence the exact answer.]

Step 5: Finally, the sample size equation can be derived by using population size (step 1), the critical value of the normal distribution at the required confidence level (step 2), sample proportion (step 3) and margin of error (step 4) as shown below.

Examples of Sample Size Formula (with Excel Template)

Let’s see some simple to advanced examples of sample size equation to understand it better.

Example #1

Let us take the example of a retailer who is interested to know how many of their customers bought an item from them after viewing their website on a certain day. Given that their website has on average 10,000 views per day, determine the sample size of the customers that they have to monitor at a 95% confidence level with a 5% margin of error if:

• They are uncertain of the current conversion rate.
• They know from previous surveys that the conversion rate is 5%.

Given,

• Population size, N = 10,000
• Critical value at 95% confidence level, Z = 1.96
•  Margin of error, e = 5% or 0.05

1 – Since the current conversion rate is unknown, let us assume p = 0.5

Therefore, the sample size can be calculated using the above formula as,

= (10,000 * (1.96 ²)*0.5*(1-0.5)/(0.05 ²)/(10000 – 1+((1.96 ²)* 0.5*(1-0.5)/(0.05 ²))))

Sample Size is calculated as below – 

Sample Size = 370

Therefore, a sample size of 370 customers will be adequate for deriving meaningful inference.

2 – The current conversion rate is p = 5% or 0.05

Therefore, the sample size can be calculated using the above formula as,

= (10,000 * (1.96 ²)*0.05*(1-0.05)/(0.05 ²)/(10000 – 1+((1.96 ²)* 0.05*(1-0.05)/(0.05 ²))))

Sample Size will be –

Sample Size = 72

Therefore, a sample size of 72 customers will be adequate for deriving meaningful inference in this case.

Example #2

Let us take the above example and in this case let us assume that the population size, i.e. daily website view, is between 100,000 and 120,000, but then the exact value is not known. The rest of the values are the same along with a conversion rate of 5%. Determine the sample size for both 100,000 and 120,000.

Given,

• Sample proportion, p = 0.05
• Critical value at 95% confidence level, Z = 1.96
•  Margin of error, e = 0.05

Therefore, the sample size for N = 100,000 can be calculated as,

= (100000 * (1.96 ²)*0.05*(1-0.05)/(0.05 ²)/(100000 – 1+((1.96 ²)* 0.05*(1-0.05)/(0.05 ²))))

Sample Size is calculated as below –

Sample Size = 73

Therefore, the sample size for N = 120,000 can be calculated as,

= (120000 * (1.96 ²)*0.05*(1-0.05)/(0.05 ²)/(120000 – 1+((1.96 ²)* 0.05*(1-0.05)/(0.05 ²))))

Sample Size will be –

Sample Size = 73

Therefore, it is proved that as the population size increases to be very large, it becomes irrelevant in the computation of the sample size.

Relevance and Uses

It is important to understand the concept of appropriate sample size because it is used for the validity of research findings. In case the sample size is too small, it will not yield valid results, while a sample size that is too large may be a waste of both money and time. Statistically, the significant sample size is predominantly used for market research surveys, healthcare surveys, and education surveys.

You can download this Sample Size Formula Excel Template from here – Sample Size Formula Excel Template

Recommended Articles

This has been a guide to Sample Size Formula. Here we discuss the calculation of sample size along with practical examples and downloadable excel template. You can learn more about excel modeling from the following articles –

• Use of Z Test vs T Test
• Sample Standard Deviation Calculation
• F-Test Calculation
• Z Test Calculation
• P-Value Calculation