Margin of Error Formula (Table of Contents)
 Margin of Error Formula
 Examples of Margin of Error Formula (With Excel Template)
 Margin of Error Formula Calculator
Margin of Error Formula
In statistic, we calculate the confidence interval to see where the value of the data of sample statistic will fall. The range of values which are below and above the sample statistic in a confidence interval is known as Margin of Error. In other words, it is basically the degree of error in the sample statistic. Higher the margin of error, lesser will the confidence in the results because the degree of deviation in these results is very high. As its name suggests, the margin of error is a range of values above and below the actual results. For example, if we get a response in a survey wherein 70% people have responded “good” and margin of error is 5%, this means that in general, 65% to 75% of the population think that the answer is “good”.
The formula for Margin of Error –
Where:
 Z – Z score
 S – Standard Deviation of a population
 n – Sample Size
Another formula for calculating the margin of error is:
Where:
 p – Sample Proportion (fraction of sample which is a success)
Now to find the desired z score, you need to know the confidence interval of the sample because the Z score is dependent on that. Below table is given to see the relation of a confidence interval and z score:
Confidence Interval  Z – Score 
80%  1.28 
85%  1.44 
90%  1.65 
95%  1.96 
99%  2.58 
Once you know the confidence interval, you can use the corresponding z value and calculate the margin of error from there.
Examples of Margin of Error Formula (With Excel Template)
Let’s take an example to understand the calculation of Margin of Error in a better manner.
Margin of Error Formula – Example #1
Let’s say we are conducting a survey to see what the grade score which university students are getting is. We have selected 500 students randomly and asked their grade score. The average of that is 2.4 out of 4 and the standard deviation is say 30%. Assume the confidence interval is 99%. Calculate the margin of error.
Solution:
Margin of Error is calculated using the formula given below
Margin of Error = Z * S / √n
 Margin of Error = 2.58 * 30% / √(500)
 Margin of Error = 3.46%
This means that with 99% confidence, the average grade of students is 2.4 plus or minus 3.46%.
Margin of Error Formula – Example #2
Let say you are launching a new health product in the market but you are confused which flavor will people like. You are confused between banana flavor and vanilla flavor and have decided to conduct a survey. Your population for that is 500,000 which is your target market and out of that you decided to ask the opinion of 1000 people and that will the sample. Assume that a confidence interval is 90%. Calculate the margin of error.
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Solution:
Once the survey is done, you came to know that 470 people liked the banana flavor and 530 has asked for vanilla flavor.
Margin of Error is calculated using the formula given below
Margin of Error = Z * √((p * (1 – p)) / n)
 Margin of Error = 1.65 * √((0.47 * (1 – 0.47)) / 1000)
 Margin of Error = 2.60%
So we can say that with 90% confidence that 47% of all people liked banana flavor plus or minus 2.60%.
Explanation
As discussed above, the margin of error helps us understand whether the sample size of your survey is appropriate or not. In case the margin error is too big, it can be the case that our sample size is too small and we need to increase that so that sample results match more closely with population results.
There are some scenarios where the margin of error will not be of much use and will not help us in tracking the error:
 If the questions of the survey are not designed and do not help in getting the required answer
 If the people who are responding to the survey have some bias regarding the product for which the survey is being done, then also, the result not be very accurate
 If the sample chosen itself is the proper representative of the population, in that case also, results will be way off.
Also, one big assumption here is that the population is normally distributed. So if the sample size is too small and the population distribution is not normal, z score cannot be calculated and we will not be able to find the margin of error.
Relevance and Uses of Margin of Error Formula
Whenever we use sample data to find some relevant answer for the population set, there is some uncertainty and chances that the result might deviate from the actual result. The margin of error will tell us that what is the level of deviation is there is the sample output. We need to minimize the margin of error so that our sample results depict the actual story of population data. So lower the margin of error, the better will be the results. The margin of error complements and completes the statistical information which we have. For example, if a survey finds that 48% of the people prefer spending time at home during the weekend, we cannot be so precise and there are some missing elements in that information. When we introduced a margin of error here, say, 5%, then the outcome will be interpreted as 4353% people liked the idea of being at home during the weekend, which makes complete sense.
Margin of Error Formula Calculator
You can use the following Margin of Error Calculator
Z  
S  
√n  
Margin of Error
 
Margin of Error  = 


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