How to find sample size in statistics

Q: How do you calculate the sample size of a population?

X = Zα/22 *p*(1-p) / MOE2, and Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, p is the sample proportion, and N is the population size.

Formula to Determine Sample Size of Population

The sample size formula helps calculate or determine the minimum sample size, which is required to know the adequate or correct proportion of the population, along with the confidence level and the margin of error.
Table of Contents Show

Formula to Determine Sample Size of Population
How to Calculate Sample Size? (Step by Step)
Relevance and Uses
Recommended Articles
How do you determine sample size?
What is the size of a sample in statistics?
How do you calculate the sample size of a population?
What are the 4 ways to determine the sample size?

The term “sample” refers to the portion of the population that enables us to draw inferences about the population. So, the sample size must be adequate to make meaningful inferences. In other words, it is the minimum size needed to estimate the true population proportion with the required margin of error and confidence level. As such, determining the appropriate sample size is one of the recurrent problems in statistical analysis. Its equation can derive using population size, the critical value of the normal distribution, sample proportion, and margin of errorThe margin of error is a statistical expression to determine the percentage point the result arrived at will differ from the actual value. Standard deviation divided by the sample size, multiplying the resultant figure with the critical factor. Margin of Error = Z * ơ / √nread more.

Sample Size n = N * [Z2 * p * (1-p)/e2] / [N – 1 + (Z2 * p * (1-p)/e2]

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For eg:
Source: Sample Size Formula (wallstreetmojo.com)

where,

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

How to Calculate Sample Size? (Step by Step)

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 distributionNormal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays a key role in assets return calculation and in risk management strategy.read more 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, the range in which the true population expects 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 derive 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

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, one can calculate the sample size using the formula,

= (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))))

Therefore, 370 customers will be adequate for deriving meaningful inferences.

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

Therefore, one can calculate the sample size using the above formula,

= (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))))

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

Example #2

Let us take the above example. Let us assume that the population size, i.e., daily website view, is between 100,000 and 120,000, but the exact value is unknown. However, the rest of the values are the same, with a conversion rate of 5%. Therefore, calculate 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, one can calculate the sample size for N = 100,000 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))))

Therefore, one can calculate the sample size for N = 120,000 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))))

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

Relevance and Uses

Sample size calculation is important to understand the concept of the appropriate sample size because one may use it to validate research findings. In case it is too small, it will not yield valid results, while a sample that is too large may be a waste of both money and time. Therefore, one should use a considerable sample size for market research, healthcare, and education surveys.

How do you determine sample size?

Five steps to finding your sample size.

Define population size or number of people..

Designate your margin of error..

Determine your confidence level..

Predict expected variance..

Finalize your sample size..

What is the size of a sample in statistics?

Sample size refers to the number of participants or observations included in a study. This number is usually represented by n. The size of a sample influences two statistical properties: 1) the precision of our estimates and 2) the power of the study to draw conclusions.

How do you calculate the sample size of a population?

X = Z_α_/₂² *p*(1-p) / MOE², and Z_α_/₂ is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, p is the sample proportion, and N is the population size.

What are the 4 ways to determine the sample size?

There are several approaches to determining the sample size. These include using a census for small populations, imitating a sample size of similar studies, using published tables, and applying formulas to calculate a sample size.