Population sampling is the process of selecting a representative subset of individuals from an entire population. This subset, known as a sample, must be sufficiently large to allow for meaningful and statistically valid analysis. Sampling is typically employed because it is often impractical or impossible to test every member of the population due to constraints such as time, cost, and logistical challenges. The primary objective of sampling is to save resources while still obtaining data that accurately reflects the characteristics of the whole population. However, researchers must always keep in mind that the ideal scenario is to assess every individual within the population to achieve results that are as reliable, valid, and precise as possible. Only when testing the entire population is not feasible do researchers resort to sampling techniques, carefully designed to minimize bias and maximize representativeness.
Sampling Distribution Formula
A sampling distribution is defined as the probability distribution of a given statistic based on repeated samples drawn from a population. This distribution enables the calculation (or estimation) of key sample statistics such as the mean, range, standard deviation, and variance.
For a sample size greater than 30, the sampling distribution formulas are as follows:
Where:
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μˉx represents the mean of the sample, which estimates the population mean μ
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σxˉ is the standard deviation of the sampling distribution (also called the standard error of the mean), while σ is the population standard deviation.
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n is the sample size, which is assumed to be greater than 30 in this context.
Many professionals—including analysts, researchers, and statisticians—utilize the concept of sampling distribution for their investigations. When dealing with a large population, this approach allows the selection of a smaller, manageable sample that can be used to estimate population parameters such as the mean and standard deviation.
The process for calculating the sampling distribution typically involves the following steps:
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Identify multiple samples of size nn drawn from the larger population of size NN.
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List these samples and compute the mean of each individual sample.
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Construct a frequency distribution of these sample means obtained in step 2.
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Determine the probability distribution of the sample means based on the frequency distribution, which represents the sampling distribution.
This methodology provides insight into how sample statistics vary from sample to sample and forms the foundation for inferential statistics.
Example
Let’s take the example of taxes paid by vehicles. In California, the average tax paid is $12,225, with a standard deviation of $5,000. Observations were made on a sample of 400 trucks and trailers combined. Help the Department of Transportation determine the sample mean and the sample standard deviation.
Solution
Use the data below to calculate the sampling distribution.
The calculation of the standard deviation for the sample size is as follows:
= $5,000 / √400
The standard deviation of the sample size will be –
σₓ̄ = $250
Therefore, the sample standard deviation, as estimated by the Department of Transportation, is $250, and the sample mean is $12,225.
Sample Size Formula
The sample size formula helps calculate or determine the minimum sample size required to accurately estimate the true proportion of a population, considering the desired confidence level and margin of error.
A sample refers to a subset of the population that allows us to make inferences about the entire population. Therefore, the sample size must be adequate to produce meaningful and statistically significant conclusions. In other words, it is the minimum number of observations needed to estimate the population proportion within a specified margin of error and confidence level. Consequently, determining the appropriate sample size is a common and critical challenge in statistical analysis. This formula is derived using the population size, the critical value from the normal distribution, the sample proportion, and the margin of error.
The formula for sample size n is:
Where:
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N= Population size
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Z= Critical value from the normal distribution corresponding to the desired confidence level
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p = Sample proportion (estimated proportion of the attribute present in the population)
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e = Margin of error (the allowable error tolerance)
As the sample size increases, the sampling distribution approaches a normal distribution. Meanwhile, the standard deviation (standard error) of the sampling distribution decreases as nn increases.
If the sample size is too small, the results may not be reliable. Conversely, an excessively large sample size can lead to unnecessary expenditure of time and resources. Therefore, it is important to choose a reasonable sample size for fields such as market research, healthcare studies, and educational surveys.
How to Calculate Sample Size? (Step-by-step)
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Determine the population size (NN): This is the total number of distinct entities in your population.
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Identify the critical value (ZZ): Find the critical value from the normal distribution associated with your desired confidence level. For example, for a 95% confidence level, the critical value is 1.96.
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Estimate the sample proportion (pp): This can be derived from previous surveys or small pilot studies. Note: If unknown, use 0.5 as a conservative estimate, which will yield the largest required sample size.
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Set the margin of error (ee): This is the range within which the true population parameter is expected to fall. Note: A smaller margin of error indicates higher precision and requires a larger sample size.
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Apply the formula using the values from steps 1 to 4 to calculate the minimum sample size required.
Additional Practical Note:
When the population size is less than 1,000, an ideal sample size is often around 10% of the population. For example, if the population is 10,000 individuals, a 10% sample corresponds to 1,000 individuals.
Example
Let’s take the example of a retailer who wants to know how many of their customers purchased an item from them after visiting their website on a particular day. Given that their website has, on average, 10,000 views per day, determine the sample size of customers they need to monitor at a 95% confidence level with a 5% margin of error if:
■ They are uncertain about the current conversion rate.
■ They know from previous surveys that the conversion rate is 5%.
Data
■ Population size, N = 10,000
■ Critical value at 95% confidence level, Z = 1.96
■ Margin of error, e = 5% or 0.05
Since the current conversion rate is unknown, assume p = 0.5
Therefore, the sample size can be calculated using the formula:
Substituting the values:
Example 2
Gautam is currently taking accounting courses and has passed his entry exam. He is now enrolled at the intermediate level and will also join a senior accountant as an intern. Additionally, he will be working on an audit of manufacturing companies.
One of the companies he visited for the first time requested verification that invoices for all purchase transactions were reasonably available. Gautam selected a sample size of 50, and the population standard deviation for this audit variable is 0.50.
Based on the information available, you are asked to calculate the sampling error at both 95% and 99% confidence intervals.
Solution
We are given the population standard deviation and the sample size. Therefore, we can use the sampling error formula:
Data for calculation:
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Sample size, n=50n = 50
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Population standard deviation, σ=0.50\sigma = 0.50
At 95% confidence level:
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Z-score value = 1.96
At 99% confidence level:
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Z-score value = 2.576 (from Z-score tables)
Conclusion:
The sampling error is approximately 0.1386 at the 95% confidence level and 0.1820 at the 99% confidence level.
Eliminating Sampling Errors
Understanding the concept of sampling error is essential, as it indicates how accurately the results of a survey are expected to reflect the overall views of the population. It’s important to remember that surveys typically rely on a smaller group, known as the sample size (or survey respondents), to represent a much larger population.
Sampling error can also be viewed as a measure of the effectiveness and precision of the survey. A larger margin of error implies that the survey results may significantly deviate from the true characteristics of the population. Conversely, a smaller margin of error or sampling error indicates that the results more closely approximate the true population values, thereby increasing the reliability and confidence level of the ongoing survey.
Strategies to Reduce Sampling Error:
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Increase the Sample Size
As the sample size increases, the sample more closely resembles the actual population, thereby reducing the risk of deviation from population parameters. For example, the mean of a sample of 10 individuals is likely to fluctuate more than that of a sample of 100 individuals. A larger sample reduces variability and leads to more precise estimates. -
Ensure Representativeness of the Sample
Researchers should take proactive steps to ensure that the sample reflects the diversity and characteristics of the entire population. This includes avoiding over-representation or under-representation of specific subgroups. -
Replicate the Study
One effective way to reduce sampling error is to replicate the research. This can be done by:-
Repeating measurements multiple times.
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Using more than one subject or group in the study.
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Conducting multiple independent studies on the same topic.
Replication helps verify consistency and increases the reliability of the results.
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Use Random Sampling Methods
Random sampling is a powerful technique to minimize sampling errors. It involves using a systematic and unbiased method to select participants from the population. For instance, instead of randomly selecting participants without a plan, a researcher might adopt systematic random sampling by selecting every 10th, 20th, or 30th individual on a list.
Conclusion:
Reducing sampling error is crucial for improving the accuracy and trustworthiness of survey results. Through increasing sample size, ensuring representativeness, replicating studies, and applying random sampling techniques, researchers can minimize bias and better reflect the true characteristics of the population being studied.