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,
where
 z = critical factor
 ơ = population standard deviation
 n = sample size
The margin of Error Calculation (Step by Step)
 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.
 Step 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.
 Step 3: Next, determine the critical factor or the zscore based on the desired confidence level, and it is denoted by z.
 Step 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
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 zvalue is 1.645 i.e. z = 1.645
Therefore, the error at a 90% confidence level can be made using above the formula as,
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 = 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 zvalue 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 zvalue 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 zvalue 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 = 
 

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.
Recommended Articles
This has been a guide to the Margin of Error and its definition. Here we learn how to calculate the Margin of Error using its formula along with practical examples and a downloadable excel template. You can learn more about financial analysis from the following articles –
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 Random vs. Systematic Error
 Sampling Error Formula
 Formula of Percent Error
 Mortgage Interest Deduction
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