# Sampling error

Sampling error |
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**Sampling error** is the difference between the results of a sample survey and the results of the same survey if it had been conducted on the entire population. It is caused by randomly selecting a sample of individuals from a population and measuring the responses from that sample. In management, sampling error can be a risk when drawing conclusions from the data collected. It can lead to inaccurate forecasts and conclusions, resulting in decisions that might not be based on the true characteristics of the population. To reduce sampling error, researchers should use a representative sample of the population, accurately measure all variables, and use a larger sample size.

## Example of sampling error

- A poll that indicates that a certain political party will win a certain percentage of the vote in an upcoming election could be subject to sampling error. If the sample size is not adequately representative of the population, the results of the poll could not accurately reflect the true preferences of the general population.
- In a survey of customer satisfaction, sampling error could occur if the sample size is too small or the sample is not representative of the population. For example, if the survey only includes people who have recently purchased a certain product, the survey results may not accurately reflect the satisfaction of all customers who have purchased the product.
- In a medical study, sampling error can occur if the sample size is too small or the sample is not representative of the population. For example, if a study only includes people of a certain age or gender, the study results may not accurately reflect the medical condition of all people in the population.

## Formula of sampling error

The formula for sampling error is as follows:

Sampling Error = $$\sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n(n-1)}}$$

where $$x_i$$ is a single observation, $$\bar{x}$$ is the sample mean, and n is the sample size.

This formula is used to calculate the standard error of the estimate, which is the standard deviation of the population estimated from a sample. It is a measure of how well the sample represents the population and is used to quantify the variability of the estimate. The lower the sampling error, the more accurate the estimate.

The formula works by summing the squared differences between each observation and the sample mean (x_i - \bar{x})^2 and then dividing by the number of observations that were used to calculate the sample mean (n(n-1)). This gives the sum of squared errors, which is then divided by the sample size (n) to give the standard error of the estimate.

## When to use sampling error

Sampling error is an important concept to understand when conducting research. It is important to consider sampling error whenever a sample survey is conducted as it can lead to inaccurate results and conclusions. Sampling error can be used in a variety of ways, including:

- To ensure that survey results accurately represent a population by using a representative sample.
- To inform decisions by providing insight into the potential errors associated with a sample survey.
- To identify potential sources of bias in a sample survey.
- To help determine the most appropriate sample size for a given survey.
- To estimate the confidence level of survey results.

## Types of sampling error

Sampling error is the difference between the results of a sample survey and the results of the same survey if it had been conducted on the entire population. There are several types of sampling error, including:

**Selection bias**: this occurs when the sample of individuals chosen for the survey does not accurately represent the population.**Non-response bias**: this occurs when individuals selected for the survey do not respond or do not provide accurate information.**Undercoverage**: this occurs when the sample does not include all of the population groups that should be represented.**Measurement error**: this occurs when variables are inaccurately measured or recorded.**Sampling variability**: this occurs when the sample population used is too small and not representative of the population as a whole.

## Steps of sampling error

The following are the steps to reduce sampling error:

- Use a representative sample of the population. This means that the sample should accurately reflect the characteristics of the population, such as age, gender, location, and so on.
- Accurately measure all variables. This includes selecting the right type of data collection instruments and ensuring that they accurately capture the required information.
- Use a larger sample size. A larger sample size will reduce the impact of any errors in the sample.
- Use random sampling methods. This will ensure that the sample is representative of the population and that it is not biased in any way.
- Eliminate any sources of bias. This may include removing any outliers or items that are not representative of the population.
- Follow up with a pilot study. This can be used to identify any issues that may have been missed in the initial sample.

## Limitations of sampling error

Sampling error can limit the accuracy of survey results and lead to inaccurate conclusions. Some of the limitations of sampling error include:

**Sample size**: A sample size that is too small may not be representative of the target population, resulting in sampling error.**Sample selection**: Sampling error can occur when a sample is not chosen randomly or when certain groups are over- or under-represented in the sample.**Measurement error**: Measurement errors, such as incorrectly gathering information or using an invalid survey instrument, can lead to sampling error.**Non-response bias**: Non-response bias can occur when certain groups of people do not respond to the survey, leading to sampling error.**Response bias**: Response bias can occur when survey respondents are not truthful or do not answer questions accurately, leading to sampling error.

## Suggested literature

- Lin, L. (2018).
*Bias caused by sampling error in meta-analysis with small sample sizes*. PloS one, 13(9), e0204056. - McNemar, Q. (1947).
*Note on the sampling error of the difference between correlated proportions or percentages*. Psychometrika, 12(2), 153-157. - Altman, D. G., & Bland, J. M. (2014).
*Uncertainty and sampling error*. Bmj, 349.