Guide to margin of error (with examples)

The margin of error (MOE) is a statistic that tells you how much your results are likely to differ from the true value in the population. A survey with a ±3% margin of error at a 95% confidence level means that if you ran the same survey repeatedly, the results would land within 3 percentage points of the reported figure 95% of the time.

Unless you survey every member of a population, your results are an estimate—and the margin of error quantifies how precise that estimate is. Here’s how it works, what affects it, and how to calculate it for your own surveys.

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What is the margin of error in a survey?

The margin of error is the range within which the actual value of a survey parameter falls at a given confidence level. It quantifies how much the results from a sample might differ from the value you’d get if you studied the whole population.

The is the range between the upper and lower bounds of an estimate. The narrower the interval, the more precise your results.

For instance, suppose a random sample has a ±2% margin of error at a 95% confidence level. If you repeated the same survey 100 times, you’d expect the percentage of people who gave a particular answer to fall within 2% of the reported result in 95 of those surveys.

The margin of error usually shrinks as the random grows. More responses mean more confidence that your results are reliable.

When can you use a margin of error?

You use a margin of error when you have a random sample—a set of randomly selected respondents from the population you’re studying. A random sample is also known as a probability sample, since every member of your population has a known probability of being included.

For example, say your company wants to know whether workers would prefer an extra leave day or bonus pay. You can randomly select employees from your workforce and ask them to choose. A margin of error helps decision-makers understand how accurate those results are.

What factors affect the margin of error?

The size of the margin of error depends on several factors:

Sample size: the more respondents who complete your study, the smaller the margin of error
Confidence level (CL): increasing the confidence level widens the margin of error
Population variance: the higher the variance, the larger the margin of error
Poll design: the exact wording of your questions can influence how people answer, affecting the MOE
: a lower response rate increases the MOE. It can also increase if respondents don’t closely resemble the larger population.
Non-sampling errors: errors from sources such as coding and measurement can affect the margin of error

What is an acceptable margin of error for a quantitative survey?

Most quantitative surveys aim for a margin of error between 4% and 8% at a 95% confidence level. The right target depends on how high-stakes the decision is—the more it matters, the tighter you want the margin.

For example, if you were surveying a population of 1,000 people at a 90% confidence level, surveying at least 250 of them would give you an MOE of roughly 4%. With that margin, you could be reasonably confident the results reflect the audience you’re surveying.

How to calculate the margin of error

Whenever you run a statistical survey, you should calculate the MOE. Before you do, define your population.

A population is the group of elements you want to survey and gather data from. Properly defining the population and selecting an appropriate sample size helps reduce the MOE.

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Here’s the formula:

MOE= z-value √ (p(1-p)/n)

Where:

n is the sample size
p is the sample percentage
z-value is the critical value corresponding to your confidence level

Here are standard confidence levels with their corresponding z-values:

90%: 1.64
95%: 1.96
99%: 2.58

Finding the maximum margin of error formula

For the maximum MOE, use 0.5 as p and the z-value for 95% (the standard confidence level), which is 1.96. The formula becomes a direct transformation of the sample size:

Maximum 95% MOE = 1.96√0.5 (1 - 0.5)/n =0.98/√n

Plug your sample size into the formula and you’ll get the maximum margin of error for your survey. The figure is unique for a given sample size.

For example, the maximum MOE for a sample size of 100 would be 0.98/√100 = 0.098

How to calculate the margin of error with your survey data

Once you have the formula, the calculation takes four steps:

Take n, the sample size, and p, the sample percentage.
Calculate p(1 - p) and divide the result by n.
Find the square root of the value.
Multiply the figure you got in Step 3 by the z-value.

An example of calculating the margin of error

Suppose you surveyed 1,000 respondents about their views on volunteering, and 500 agreed that volunteering is an excellent part of life. What’s the margin of error at a 95% confidence level?

MOE= z√(p (1 - p))/n

P = 500/1,000 = 50%

z-score = 1.96 (at 95% confidence level)

Therefore, MOE= 1.96√ (0.5(1 - 0.5))/1,000

=0.031=3.1%

So the sample’s MOE is ±3%. If you repeated the survey several times, the number of people who love volunteering would fall within 3% of the sample percentage (50%) 95% of the time.

How sample size affects the margin of error

Sample size is a key factor in your margin of error. To increase your study’s precision, interview more respondents and make sure they complete your survey.

Increasing sample size reduces the standard deviation (a measure of your estimate’s variability). It narrows the range of possible values, leading to more precise estimates and a lower MOE.

For example, if you’re estimating the average income of your existing customers, a sample of two people will produce a wide range of figures, so the margin of error will be high. Increase the sample to 1,000 people and the MOE narrows significantly.

That makes MOE critical to judging whether your sample size is appropriate. If the margin is too wide, you’ll need to survey more people to capture your population’s attitudes accurately.

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How to increase your data’s reliability

To boost your survey’s reliability, minimize the MOE. Here are three tried-and-true ways to generate more accurate results:

1. Minimize the variables

A high number of variables can introduce more errors to your survey. Variables push your standard deviation up, increasing the MOE. The fix is to change how you collect data—make the process rigorous and measure your variables accurately.

2. Increase the sample size

This is often the easiest option. Statistically, the more people who complete your study, the better your chances of a representative response, because the confidence interval decreases. The result is a lower MOE. Just make sure you have the resources and time to reach a larger sample.

3. Lower your confidence level

A lower confidence level produces a narrower MOE. But be careful—a lower level means decision-makers will be less confident in the results.

Only reduce the CL if the benefits of a smaller MOE outweigh the downsides of a lower CL. For example, if increasing the sample size is unfeasible due to cost, you can reduce the CL to achieve a narrower interval.