# Sampling Error Formula

## Formula to Calculate Sampling Error

Sampling Error Formula refers to the formula that is used in order to calculate statistical error that occurs in the situation where person conducting the test doesn’t select sample that represents the whole population under consideration and as per the formula Sampling Error is calculated by dividing the standard deviation of the population by the square root of the size of sample and then multiplying the resultant with the Z score value which is based on confidence interval.

Sampling Error = Z x (σ /n)

For eg:
Source: Sampling Error Formula (wallstreetmojo.com)

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

### Step by Step Calculation of Sampling Error

The following are the steps to calculate sampling error.

1. Gathered all set of data called the population. Compute the population means and population standard deviation.

2. 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.

3. Determine the confidence level, and accordingly, one can determine the value of Z score from its table.

4. 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 error.

### Examples

You can download this Sampling Error Formula Excel Template here – Sampling Error Formula Excel Template

#### 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 have 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.

• Z Factor Value: 1.96
• Population of Standard Deviation: 0.3
• Sample Size: 100

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

Sampling Error will be –

#### 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.

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 calculate 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 the 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.

• Z Factor Value: 1.96
• Population of Standard Deviation: 0.50
• Sample Size: 50

Therefore, the calculation is as follows,

Sampling Error will be –

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

Use the following data for the calculation.

Therefore, the calculation is as follows,

Sampling Error will be –

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 the B division. Among them, 12 students were randomly selected to do a detailed checkup, and the rest was only a 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 calculate 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 the 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.

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

Sampling Error will be –

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

Use the following data for the calculation.

Therefore, the calculation is as follows,

Sampling Error will be –

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.

It 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 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.

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