# Margin of Error  ## What is the Margin of Error?

Margin of Error is a statistical expression that is used in order to determine the percentage point by which the result arrived will differ from value of the real population and it is calculated by dividing the standard deviation of the population by the sample size and lastly multiplying the resultant with the critical factor.

A higher error indicates a high chance that the result of the sample reported may not be the true reflection of the whole population.

### The margin of Error Formula

The formula for the margin of error is calculated by multiplying a critical factor (for a certain confidence level) with the population standard deviation, and then the result is divided by the square root of the number of observations in the sample.

Mathematically, it is represented as,

Margin of Error = Z * ơ / √n

For eg:
Source: Margin of Error (wallstreetmojo.com)

where

• z = critical factor
• ơ = population standard deviation
• n = sample size

### The margin of Error Calculation (Step by Step)

1. Firstly, gather the statistical observations to form a data set called the population. Now, calculate the mean of the population. Next, compute the population standard deviation based on each observation, the population means, and the number of observations of the population, as shown below. 2. Next, determine the number of observations in the sample, denoted by n. Remember that sample size is less than equal to the total population, i.e., n ≤ N.

3. Next, determine the critical factor or the z-score based on the desired confidence level, and it is denoted by z.

4. Next, finally, the margin error is calculated by multiplying the critical factor for the desired confidence level and the population standard deviation, and then the result is divided by the square root of the sample size as shown above.

### Example

You can download this Margin of Error Formula Excel Template here – Margin of Error Formula Excel Template

Let us take the example of 900 students who were part of a survey, and it was found that the average GPA of the population was 2.7, with a population standard deviation of 0.4. Calculate the margin of error for

• 90% confidence level
• 95% confidence level
• 98% confidence level
• 99% confidence level

We are going to use the following data for the calculation.

For a 90% Confidence Level

For a 90% confidence level, the critical factor or z-value is 1.645 i.e. z = 1.645

Therefore, the error at a 90% confidence level can be made using above the formula as,

• = 1.645 * 0.4 / √900

Margin Error at 90% confidence level will be-

• Error = 0.0219

For a 95% Confidence Level

For a 95% confidence level, the critical factor or z-value is 1.96 i.e. z = 1.96

Therefore, the calculation of margin of error at a 95% confidence level can be done using the above the formula as,

• = 1.96 * 0.4 / √900

Margin Error at 95% confidence level will be-

• Error = 0.0261

For a 98% confidence level

For a 98% confidence level, the critical factor or z-value is 2.33 i.e. z = 2.33

Therefore, the calculation of margin of error at a 98% confidence level can be done using the above the formula as,

• = 2.33 * 0.4 / √900

Margin Error at 98% confidence level will be-

• Error = 0.0311

Therefore, the error for the sample at 98% confidence level is 0.0311.

For a 99% confidence level

For a 99% confidence level, the critical factor or z-value is 2.58 i.e. z = 2.58

Therefore, the calculation of margin at a 99% confidence level can be done using the above the formula as,

• = 2.58 * 0.4 / √900

Margin Error at 99% confidence level will be-

• Error = 0.0344

Consequently, it can be seen that the error of a sample increases with the increase in confidence level.

### The margin of Error Calculator

You can use the following calculator.

 z σ n Margin of Error Formula =

Margin of Error Formula =
 z * σ = √n
 0 * 0 = 0 √0

### Relevance and Uses

It is essential to understand this concept because it indicates how much one can expect that survey results actually reflect the true view of the overall population. It should be kept in mind that a survey is done using a smaller group of people (also known as survey respondents) to represent a much larger population (also known as the target market). The margin of error equation can be seen as a way of measuring the effectiveness of the survey. A higher margin indicates that the survey results may stray from the actual views of the total population. On the other hand, a smaller margin indicates that the results are close to the true reflection of the total population, which builds more confidence about the survey.

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