**Sampling Error Formula (Table of Contents)**

## What is the Sampling Error Formula?

Sampling Error can be defined as a statistical error that shall occur when the tester doesn’t select the sample which shall represent the entire data or entire population. The sampling error formula is represented as follows,

**Sampling Error Formula = Z x (σ /**

**√**

**n)**

Where,

- Z is the Z score value based on the confidence interval
- σ is the population standard deviation
- n is the size of the sample

### Explanation of the Sampling Error Formula

The sampling error equation can be explained in the following steps:

**Step1**: Gathered all set of data called the population. Compute the population means and population standard deviation.**Step2**: Now, one needs to determine the size of the sample and further the sample size has to be less than the population and it should not be greater.**Step3**: Determine the confidence level and accordingly one can determine the value of Z score from its table.**Step4**: Now multiply Z score by the population standard deviation and divide the same by the square root of the sample size in order to arrive at a margin of error or sample size error.

### Examples of Sampling Error Formula

Let’s see some simple to advanced examples of the sampling error equation to understand it better.

#### Example #1

**Suppose that the population standard deviation is 0.30 and the size of the sample is 100. What will the sampling error at 95% confidence level?**

**Solution**

Here we are given the population standard deviation as well as the size of the sample, therefore we can use the below formula to calculate the same.

Use the following data for the calculation of the sampling error.

Therefore, the calculation of the sampling error is as follows,

- = 1.96 x 0.30 /√100

**Sampling Error will be –**

**Sampling Error = 0.06**

Hence, the sampling error for this case will be 0.06.

#### Example #2

**Gautam is currently pursuing an accountancy course and he has cleared his entrance exam. He has registered now for an intermediate level and will also be joining a senior accountant as an intern. He will be working in an audit of the manufacturing firms. **

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**One of the firms he was visiting for the first time, was asked to check whether the bills for all the entries for purchases were reasonably available. The sample size he picked was 50 and the population standard deviation for the same was 0.50. **

**Based on available information, you are required to compute sampling error at 95% and 99% confidence interval.**

**Solution**

Here we are given the population standard deviation as well as the size of the sample, therefore we can use below formula to calculate the same.

Z score for 95% confidence level will be 1.96 (available from Z score table)

Use the following data for the calculation of the sampling error.

Therefore, the calculation of the sampling error is as follows,

- = 1.96 x 0.50 /√50

**Sampling Error will be –**

**Sampling Error = 0.14**

Z score for 95% confidence level will be 2.58 (available from Z score table)

Use the following data for the calculation of the sampling error.

Therefore, the calculation of the sampling error is as follows,

- = 2.58 x 0.5 /√50

**Sampling Error will be –**

**Sampling Error** **= 0.18**

As the confidence level increases, the sampling error also increases.

#### Example #3

**In a school, the biometric session was organized so as to check the health of the students. The session was initiated with students of class X standard. In total there are 30 students in B division. Among them 12 students were randomly selected to do detail checkup and rest were, an only basic test was done. The report inferred that the average height of the students in B division is 154.**

**Solution**

The population standard deviation was 9.39. Based on the above information, you are required to compute the sampling error for 90% and 95% confidence interval.

Here we are given the population standard deviation as well as the size of the sample, therefore we can use below formula to compute the same.

Z score for 95% confidence level will be 1.96 (available from Z score table)

Use the following data for the calculation of the sampling error.

Therefore, the calculation of the sampling error is as follows,

- = 1.96 x 0.0939/√12

**Sampling Error will be –**

**Sampling Error = 0.053**

Z score for 90% confidence level will be 1.645 (available from Z score table)

Use the following data for the calculation of the sampling error.

Therefore, the calculation of the sampling error is as follows,

- =1.645 x 0.0939 /√12

**Sampling Error will be –**

**Sampling Error = 0.045**

As the confidence level decreases, the sampling error also decreases.

### Relevance and Uses

This is very much vital to understand this concept as this shall depict how much one can expect that the survey results would, in fact, depict the actual view of the population overall. One needs to keep one thing in mind that a survey is performed using a smaller population called the sample size (also otherwise renowned as the respondents of the survey) to represent a bigger population.

The sampling error formula can be viewed as a way of calculating the effectiveness of the survey. When the sampling margin is higher it shall represent that the survey consequences might stray from the actual total population representation. On the flip side, a sampling error or margin of error is smaller than that shall indicate that the consequences are now closer to the true representation of the population in total and which shall build a higher level of confidence about the survey that is under view.

### Recommended Articles

This has been a guide to the Sampling Error Formula. Here we discuss the formula to calculate the sampling error along with examples and downloadable excel template. You can learn more about accounting from the following articles –

- Formula of Accounting Rate of Return
- Formula of Standard Normal Distribution
- Formula of Midrange
- Central Limit Theorem Calculation
- Formula of Weighted Mean

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