Margin of Error

The margin of error is an important measure in statistics. The degree of error in random sampling surveys is known as the margin of error. It expresses random sampling error in the results of a survey. It depicts the confidence that the researchers should have in the data obtained from the survey. It tells the variance from the actual population in terms of the percentage points. A 95% confidence interval with a 4% margin of error, for instance, indicates that your statistic will, 95% of the time, be within 4% of the true population figure. A wider margin of error in statistics denotes a reduced chance of relying on a survey’s or poll’s findings, meaning that there will be less trust in the results’ ability to accurately reflect a community.

What is a Margin of Error?

A margin of error tells you how many percentages points our results differ from the real value. The margin of error is defined as the range of values below and above the sample statistic in a confidence interval. The confidence interval is a way to show what the uncertainty is with a certain statistic. 

Importance of Margin of Error

The importance of Magin of Error is discussed below in this article:

• The margin of error indicates how close the results would be if the whole population was surveyed instead of a sample population.
• The higher the margin of error, the lesser the chances of relying upon the results of the survey.
• If the margin of error is low, it means that the results obtained from the sample are highly reliable and will be very close to the ones obtained when surveying the complete population.
• The margin of error is a very important measure from the market viewpoint.
• It is a crucial tool for market research since it shows the amount of assurance that should be placed in survey data by the researchers.
• It is used to show how confident a statistician is in determining whether or not the outcomes of an online survey or online poll are representative of the full population.

Calculation of Margin of Error

Each person in a population has a chance of being chosen to be a part of the sample when using probability sampling. By using this technique, statisticians and researchers may choose participants who have expertise in their particular field of study, ensuring the lowest possible margin of error for the data drawn from these samples. Non-probability sampling excludes some segments of the population due to the selection technique used to create the samples, which is based on convenience or cost-effectiveness rather than applicability. Only by separating respondents into groups based on their interests and relevance to the poll at hand will surveys be effective.

In terms of sample size, the formula is given by, 

M.O.E. = 1 / √(n)

where,
M.O.E. denotes the margin of error
n is the sample size

The formula is not widely used as it does not give accurate results it only takes the sample size into consideration and not any other parameters. But in terms of z-index and sample size, the formula is given by,

M.O.E.= z × σ / √(n)

where,
z denotes the z-index
n is the sample size
σ is the population standard deviation

Thus, the margin of error is the product of the critical value and the standard deviation. In the above formula, the z score is calculated using the confidence level given in the question. Some widely used z scores and confidence levels are shown below:

Confidence level	Z-score
90%	1.645
95%	1.96
99%	2.575

How to Calculate Margin of Error?

Follow the steps discussed below to calculate the Magin of Error.

Step 1: Find the critical value i.e. either a t-score or a z-score. In general, for small sample sizes (under 30) use a t-score otherwise, use a z-score.

Step 2: Find the Standard Deviation or the Standard Error.

Step 3: Multiply the critical value from Step 1 by the standard deviation or standard error from Step 2. 

Example: If your critical value is 1.645 and your standard error is 0.26, then:

Margin of Error = 1.645 × 0.26 
                         = 0.4277

Solved Examples on Margin of Error

Example 1: A random sample of 50 students has average pocket money of 2500 and a standard deviation of 567. Find the margin of error if c = 0.95.

Solution:

Given, n = 50

σ = 567

For c = 0.95 i.e. 95% confidence level, z = 1.96

M.O.E = 1.96 × 567 / √(50)

M.O.E. = 157.16

Example 2: 600 students were surveyed, and it was found that the average CGPA of the students was 8.7, with a population standard deviation of 0.2. Calculate the margin of error for the 90% confidence level.

Solution:

Given, n = 600

σ = 0.2

For confidence level of 90%, z = 1.645

M.O.E = 1.645 × 0.2 / √(600)

M.O.E. = 0.013

Example 3: In a survey, it was found that the average salary of 5000 employees was 350000, with a population standard deviation of 0.3. What is the margin of error at a 99% confidence level?

Solution:

Given, n = 5000

σ = 0.3

For 99% confidence level, z = 2.575

M.O.E = 2.575 × 0.3 / √(5000)

M.O.E. = 0.0109

Example 4: Find the margin of error when a population of 400 people is surveyed.

Solution:

Given, sample size = 400

∴M.O.E. = 1 / √(400)

               = 1/20

               = 0.05

FAQs on Margin of Error

Question 1: What do you mean by the margin of error?

Solution:

The margin of error shows how similar the findings would be if the entire population had been polled as opposed to just a sample.

Question 2: How is the z-value determined in statistics?

Solution:

Z-score is a metric that quantifies how closely a value relates to the mean of a set of values. Standard deviations of data from their mean are used to calculate Z-score.

Question 3: Can the margin of error have a negative value?

Solution:

The level of accuracy needed in a survey is called the margin of error. For example, a 8% margin of error indicates that the actual results’ variation might be either negative or positive eight points.

Question 4: How can we figure out the margin of error with an 85% level of confidence?

Solution:

It is known that the error value is 0.0219 or z-value, is 1.96 for a 95% confidence level. As a result, the method above may be used to calculate the margin of error at a 85% confidence level.

Related Resources

• Statistics Formulas
• Variance and Standard Deviation
• Frequency Distributions