What Is Margin of Error? Definition, Formula, & Examples

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When reading survey results, especially in published work, you might come across a percentage followed by a ± value. For instance, results from a poll may show that 58% ± 4% of the respondents support a certain policy. In this case, the ±4% is known as the margin of error (MOE). It means the true population percentage is likely to be about 4 percentage points above or below the reported survey estimate.

Therefore, if you’re working with surveys, opinion polls, market research, academic studies, or even public research, you’ll be required to use the margin of error while reporting the findings. In this guide, you’ll learn what the margin of error is, its formula, how to calculate it manually, and how to interpret and present the results in your research work.

Need a quick answer? Use our free Margin of Error Calculator to get instant results, with a clear, step-by-step explanation.
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What is Margin of Error?

The margin of error is a statistical value that measures how much a sample survey’s results are expected to differ from the true population result. It is mainly denoted by a plus/minus symbol (±) followed by a percentage. The margin of error helps quantify uncertainty, with a smaller margin suggesting higher precision and a larger margin indicating lower confidence.

Why Margin of Error Matters

Instead of collecting data from every individual in the target population, researchers often prefer randomly selecting individuals from this population. Although the resulting sample is representative of the population, there is always some level of uncertainty.

In other words, even if you’re keen when drawing the sample, the results will not always match the exact population values. This is where the margin of error becomes useful.

It helps you answer practical questions such as:

How accurate is this survey result?
How much could the estimate vary?
Is the difference between two percentages meaningful?
Can I trust a close result?

Margin of Error Formula

The margin of error formula depends on whether you are working with a very large population (infinite population) or a finite population. For most surveys and opinion polls, it is calculated for a sample proportion, such as the percentage of respondents who support a policy, prefer a product, or agree with a statement. This formula is closely related to Cochran’s sample size formula. However, instead of solving for the minimum sample size, we use the formula to calculate the margin of error for a given sample size.

Therefore, for a large unknown population (infinite population), the margin of error formula for a sample proportion is:

MOE = Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}

Where:

MOE is the margin of error
Z is the critical value from the Z tables
p is the sample proportion
n is the sample size

However, for a known and finite population, we need to adjust for this population size using the finite population correction. Therefore, the margin of error formula for a finite population is:

MOE = Z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}} \times\sqrt{\frac{N-n}{N-1}}

Where N is the known population size.

Want to understand every symbol present in both formulas? See the table below.

Symbol	Meaning
E	Margin of error
z	Critical value from the Z tables
p	Sample proportion
n	Sample size
N	Population size

How to Calculate Margin of Error

To manually find the margin of error for a sample proportion, you should be able to determine whether the population is unknown (infinite) or known (finite). If the population is finite, you should always adjust the formula using the finite population correction.

While the formula may seem complex, we’ve simplified the process for you. Just follow these steps to find the correct MOE for your sample proportion:

Identify the parameters (confidence level, sample proportion, sample size, and population size if known).
Find the z critical value (z-score) corresponding to the confidence level
Select the correct formula
Substitute the values into the formula and solve
Convert the final answer to a percentage

Still unclear? Let’s walk through two examples to help you learn how to find margin of error for finite and infinite populations.

Example 1. Large or Unknown Population

Suppose a survey finds that 60% of respondents support a proposal. The sample size is 400, and the confidence level is 95%. Find the margin of error.

Solution

Step 1. Identify the parameters

From the question, we know that:

The confidence level is 95%.
The sample proportion is 60%. That’s p = 0.6
The sample size, n = 400

Step 2. Find the z critical value for the confidence level

The confidence level is 95%. To find the corresponding z critical value, first calculate the significance level:

We use the formula: α=1−0.95

=0.05

Since a margin of error uses a two-tailed confidence interval, we divide the significance level by 2.

Hence, α/2 = 0.025

Using positive z tables, we look up the z score value corresponding to probability 1-0.025 = 0.975. This gives a z-score value of 1.96. Therefore, the z critical value for a 95% confidence level is 1.96.

Want to learn how to find these z-score values from standard normal tables? Check out our complete guide on how to find a critical value from the Z tables. However, if you want a quicker method, you can either use our free z critical value calculator or read our guide on finding z critical values using Excel.

Step 3. Select the correct formula

Since the population size is unknown, we assume an infinite population. As such, the correct MOE formula to use in this example is:

MOE = Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}

Step 4. Substitute the values into the formula and solve

Substituting the values into the formula, we have:

MOE = 1.96\times\sqrt{\frac{0.6(1-0.6)}{400}}

= 1.96\times\sqrt{\frac{0.6(0.4)}{400}}

= 1.96\times\sqrt{0.0006}

Thus, MOE = 0.0480

Step 5. Convert the final answer to a percentage

Converting the answer to a percentage gives MOE = 4.80%. This implies that the estimated population proportion is likely to fall within 60% ± 4.80%, or between 55.2% and 64.8%, at the 95% confidence level.

Example 2. Known/Finite Population

Suppose a school has 2,000 students. A survey of 300 students finds that 55% prefer online revision quizzes. Use a 95% confidence level to calculate the margin of error for the survey results.

Solution

Step 1. Identify the parameters

From the question, we know that:

The population size, N = 2000
The sample size, n = 300
The confidence level is 95%.
The sample proportion, p = 0.55

Step 2. Find the Z critical value

Since the confidence level is the same as in example 1 above (95%), the corresponding z-critical value is 1.96

Step 3. Select the appropriate formula

Since the population is known and finite, N = 2000, we need to use the formula that adjusts for this using the finite population correction. Thus, the correct formula is:

MOE = Z_{\alpha/2} \sqrt{\frac{p(1-p)}{n}} \times\sqrt{\frac{N-n}{N-1}}

Step 4. Substitute the values into the formula and solve

Substituting the values into the formula, we get:

MOE = 1.96\times \sqrt{\frac{0.55(1-0.55)}{300}} \times \sqrt{\frac{2000-300}{2000-1}}

= 1.96\times \sqrt{\frac{0.55(0.45)}{300}} \times \sqrt{\frac{1700}{1999}}

= 1.96\times \sqrt{0.000825} \times\sqrt{0.850425}

= 0.0519

Step 5. Convert the answer to a percentage

Converting to a percentage, we get MOE = 5.19%. This implies that the true population proportion is likely to be within 55% ± 5.19%, or between 49.81% and 60.19%, at the 95% confidence level.

Common Confidence Levels and Z-Scores

If you’re a statistics student or a math student, you’ll need to master how to find z-score values from confidence levels. However, if you’re a researcher, you only need to master the z-scores for the most common confidence levels. The table below provides a summary of z-scores corresponding to confidence levels you’ll come across in the research process:

Confidence Level	Z-Score
80%	1.282
85%	1.440
90%	1.645
95%	1.960
99%	2.576

Conditions and Assumptions

For the MOE results to be correct, several assumptions must be met. These include:

The sample should be randomly selected from the target population. If the sample is biased and not random, the result will be inaccurate.
The responses should be independent. In other words, the response from one participant should not influence the responses from other participants.
The sample size should be sufficiently large and satisfy these conditions: np≥10 and n(1−p)≥10
The target population should have the characteristics of interest.

What Affects the Margin of Error?

The size of the margin of error mainly depends on three things.

Sample size. Sample size has a very strong effect on the MOE. In general, larger samples reduce the MOE, while smaller samples increase it. That is why increasing the number of respondents is one of the best ways to improve precision.
Confidence level. The more confident you want to be in the result, the larger the MOE. For example, a 90% confidence level gives a smaller MOE than 95%, while 99% gives an even larger one. This happens because higher confidence requires a wider interval.
Sample proportion. The margin of error is usually largest when the sample proportion is close to 50% and smaller when the proportion is closer to 0% or 100%. This is why many sample size and MOE formulas use 50% as the safest assumption when the true proportion is unknown.

Common Mistakes to Avoid

When working with the margin of error, people often make a few common mistakes. Some of these mistakes include:

Treating the estimate as exact. A survey result is still an estimate. The margin of error reminds you that a range is usually more realistic than a single fixed number.
Ignoring the confidence level. A margin of error always depends on the selected confidence level. If the confidence level changes, the MOE changes too.
Forgetting the finite population correction. If the population is known and the sample makes up a meaningful share of it, ignoring the finite population correction can make the result less precise than it should be.
Assuming a small margin of error guarantees no bias. A small margin of error only reflects sampling error. It does not fix problems caused by poor sampling, low response quality, bad question wording, or nonresponse bias.
Comparing close results without checking overlap. Two results that look different at first may not be meaningfully different once the margin of error is taken into account.

Frequently Asked Questions

What is a margin of error?

A margin of error is a statistical value that shows how much a sample estimate may differ from the true population value.

What is the margin of error formula?

For an infinite population, the margin of error formula is $E = Z_{\alpha/2}\sqrt{\frac{p(1-p)}{n}}$ . However, if you’re working with a known but finite population, you should apply the finite population correction. In this case, the formula becomes: $E = Z \sqrt{\frac{p(1-p)}{n}} \sqrt{\frac{N-n}{N-1}}$ .

How do I reduce the margin of error?

The most common way is to increase the sample size.

Does population size affect the margin of error?

Yes, but mainly when the sample is large relative to the population. In that case, the finite population correction reduces the margin of error.

Is the margin of error the same as the standard error?

No. Standard error measures sampling variability, while MOE combines that variability with a confidence level to create a practical range.

What does a 5% margin of error mean?

It means the true population value is likely to be within 5 percentage points above or below the sample estimate at the chosen confidence level.