Sample Size Formula

The sample size formula helps us find the accurate sample size through the difference between the population and the sample. To recall, the number of observations in a given sample population is known as sample size. Since it not possible to survey the whole population, we take a sample from the population and then conduct a survey or research. The sample size is denoted by “n” or “N”. Here, it is written as “SS”. Let us learn the sample size formula along with a few solved examples.

What Is Sample Size Formula? 

The sample size formula is determined in two steps. First, we calculate the sample size for the infinite population and second we adjust the sample size to the required population. The sample size formula can be given as:

Formula 1: Sample size for infinite population

S= Z² × P ×  \(\dfrac{(1-P)}{M^2}\)

Formula 2: Adjusted sample size

Adjusted Sample Size = \(\dfrac{(S)}{1 + \dfrac{(S-1)}{\text{Population}}}\)

where,

• S = sample size for infinite population
• Z = Z score
• P = population proportion ( Assumed as 50% or 0.5)
• M = Margin of error

Note: Z score is determined based on the confidence level. 

Confidence Level: Probability that the value of a parameter falls within a specified range of values. For example, for 95% confidence level Z score is 1.960. The margin of error: It is defined as a small amount that is allowed for in case of miscalculation or change of circumstances. Generally, the margin of error is taken as 5% or 0.05.
 

How to Apply Sample Size Formula?

In order to calculate the required sample size, we need to find several other sets of values and then substitute them into an appropriate formula. Let's look at the steps to be followed to calculate the sample size.

Step 1: Determining Key Values
One of the key values to be determined is the population size which refers to the total number of people within the required demographic. For much larger studies, we can consider using an approximated value instead of using a precise number.

• When you are working with a smaller group, precision plays a major role in having a greater statistical impact. For example, if you are performing a survey among the employees of a very small business, then you need to make sure that the population size is accurate within a number of a dozen or so people.
• When you are working on larger surveys, there could be deviance with respect to the actual population. For instance, if the demographic chosen includes everyone who is living in Canada, then the size can be estimated roughly to 30 million people, although the actual size could vary by some hundreds of thousands.

Step 2: Determining the margin of error or confidence interval

The margin of error is considered to be the amount of error that can be allowed in the study. The margin of error is actually a percentage that shows how close the sample results will be with respect to the true value of the overall population that is considered in the study.

• Usually, you can obtain more accurate answers with a smaller margin of error, but if a small margin of error is chosen, then you may require a larger sample.
• The margin of error usually is represented with a minus or a plus percentage when the results of a survey are presented.

For instance, 35% of people choose option B, with a margin of error of +/- 5%". In this particular example, the margin of error actually indicates that, if the question was asked to the entire population, then you are confident that between 30% (35 - 5) and 40% (35 + 5) of the people will agree with option B.

Step 3: Setting the confidence level.

The confidence level is pretty closely related to the margin of error or confidence interval. This value is used to measure the degree of certainty about how well a sample actually represents the entire population within the margin of error chosen for the study.

• When the confidence level is chosen as 95%, then this means that you can be 95% certain that the results will accurately fall within the margin of error chosen by you.
• When a larger confidence level is chosen, it shows a greater degree of accuracy provided that the sample size is larger. Some of the most common confidence levels used in studies are 99% confident, 90% confident, and 95% confident.
• When the confidence level is set to 95%, then it shows that you are 95% confident that 30% to 40% of the total chosen population would definitely agree with option B of the survey.

Step 4: Specifying the standard of deviation.

The standard of deviation shows how much variation can be expected from the responses of the study.

• Compared to the moderate results, you can expect extreme answers to be more accurate.
• Consider an example where 1% of the survey responses says "No", and then 99% answer "Yes", then it means that the sample actually represents the overall population in an accurate manner.
• In another case, if 55% answer "No" and 45% answer "Yes," then this means that there could be a greater chance of error.

Since this value is difficult to be calculated in an actual survey, most people choose to use 0.5 (50%) as the value which is actually the worst-case scenario percentage. Thus, using this value will actually guarantee that the calculated sample size is huge enough to show the overall population within the confidence level and the confidence interval in an accurate manner.

Step 5: Finding the Z-score.

The Z-score can be considered as a constant value that is set automatically depending on the confidence level. Z-score shows the number of standard deviations or the standard normal score between the average/mean of the population and any selected value.

Z-score is very easy to calculate that one can do it with their hand, or find an online calculator.

Due to the fact that the confidence levels are all standardized, most researchers actually memorize the required z-score for most of the commonly used confidence levels:

Using the Standard Formula

Lets' look at the below-given equation. If the size of the population is small to moderate, then it is easier to know all the key values and thus the standard formula can be used. The standard formula for calculating the sample size is:

Sample Size Formula = [z² * p(1-p)] / e² / 1 + [z² * p(1-p)] / e² * N]

Where,

• N is the population size
• z is the z-score
• e is the margin of error
• p is the standard of deviation

Let's provide the required values specific to the survey in this standard formula. Consider the example of determining the ideal survey size for a population size of 425 people. Let's use these:

• Confidence level = 99%
• Standard Deviation = 50%
• Margin of error = 5%
• The z-score wil be 2.58 if the confidence level is 99%
• This means that: N = 425, z = 2.58, e = 0.05 and p = 0.5

Let us see how to use the sample size formula in the following solved examples section.

Examples Using Sample Size Formula

Example 1: Calculate the sample size for a population of 100000. Take confidence level as 95% and margin of error as 5%.

Solution:

To find: Sample size for 100000 population.
We will calculate the sample size first by calculating it for infinite size and then adjusting it to the required size.

Given: Z = 1.960, P = 0.5, M = 0.05 

Using sample size formula,

S= Z² × P ×  \(\dfrac{(1-P)}{M^2}\)

S= (1.960)² × 0.5 ×  \(\dfrac{(1-0.5)}{0.05^2}\)

= 3.8416 × 0.25 / 0.0025
S = 384.16

Answer: The sample size for the infinite population is 384.16

Example 2 : Using the sample size formula, adjust the sample size for the required population in solved example 1.

Solution:

To Find: Adjusted sample size.

Given: Z = 1.960, P =0.5, M = 0.05

Using sample size formula for adjusted sample size ,

Adjusted Sample Size = \(\dfrac{(S)}{1 + \dfrac{(S-1)}{\text{Population}}}\)

= \(\dfrac{(384.16)}{1 + \dfrac{(384.16 -1)}{100000}}\)

= 382.69 or 
= 383 approx.

Answer: The required sample size for a population of 100000 is 383.

Example 3: Using the Sample Size Formula, find the sample size for a survey where confidence level = 95%, standard deviation = .5, and margin of error = +/- 5%.

Solution:

The Sample Size can be calculated as = (Z-score)² * SD*(1-SD) / (margin of error)²
= ((1.96)² x .5(.5)) / (.05)²

= (3.8416 x .25) / .0025

= .9604 / .0025

= 384.16

Thus, you will be needing 385 respondents for this survey.