A margin of error calculator tells you how accurately your survey results reflect the wider population. capture the views of a sample, not the whole group, so the results carry what’s called “random sampling error.” The margin of error quantifies that gap—the range within which the true population value is likely to fall.
Used well, the calculation reduces guesswork by showing how closely your sample data represents the population it’s drawn from. Use the free calculator below, or follow the formula to work it out yourself.
What is the margin of error in a survey?
Surveying a limited sample is usually the only feasible way to learn something about the whole group. The margin of error tells you how accurately those sample results reflect the population, using a confidence interval to express the likely range.
A confidence level is a percentage showing how statistically reliable a sample measurement likely is. Together, the and show how well survey results reflect the wider population.
Mathematically speaking, the margin of error calculation uses a confidence level to determine how closely a sample’s observed score matches the true score of the population it was taken from.
Unless you survey the entire population, you can’t obtain the true score. The closest you’ll get is to run your survey results through a margin of error calculator, which shows the lower and/or upper limits of a measurement (e.g., ±2%) based on a predetermined confidence level (e.g., 95%).
How do you calculate the margin of error?
As powerful as it can be, only a few measurements and variables go into a margin of error calculator.
Physics, medicine, and other scientific fields use the margin of error when they don’t have empirical data but still need to know how closely their limited data approximates the real thing. In some cases, researchers can check the margin of error’s accuracy against empirical data once it becomes available.
The margin of error calculator below crunches all the data for you. We’ll also break the formula down, so you know exactly how it works and can run it yourself.
[Embed: 45yUexVf7QP3ao5ufRcMEA]
The margin of error formula
MOE = z × ( σ / √ n )
MOE = margin of error
z = z-score | σ = population standard deviation | n = sample size
Z, or the z-score, is another way of representing the confidence level, which relates to the confidence interval—a range of expected answers that widens with higher confidence levels (and vice versa). In this context, confidence essentially means accuracy. It lets the researcher say, “Our survey results were calculated with 90% confidence.”
The confidence level is expressed as a percentage, but for this equation, it must be written as a numeral/decimal. Use the table below to select the z-score that represents your desired confidence level.
Next, find the population standard deviation, σ (the lowercase Greek letter “sigma”). Standard deviation is itself a formula that measures the data’s variance—how widely the data is distributed, which gives a sense of how varied the sample data is.
The σ calculation is as follows:
- Start by finding the mean of all values in your data set. Imagine we have four measurements of 3, 6, 9, and 12 inches. The mean is: (3 + 6 + 9 + 12) / 4 = 30 / 4 = 7.5
- Subtract the mean from each value: 7.5 - 3 = 4.5 7.5 - 6 = 1.5 7.5 - 9 = -1.5 7.5 - 12 = -4.5
- Square each of the above answers, resulting in 20.25, 2.25, 2.25, and 20.25.
- Sum these numbers together, which gets us 45.
- Next, divide by the number of values minus one—so 45 / (4-1) = 15. This is the “data variance.”
- The square root of this number is the standard deviation:
√15 = 3.873
How to calculate margin of error with your survey data
Let’s put it all together. A 95% confidence level is the most common standard, so we’ll use a z-score of 1.96. In our example above, σ was 3.873, and our sample size (n) was 4. The margin of error calculation would be:
MOE = 1.96 × (3.873 / √4)
= 1.96 × 1.937
= 3.8 (rounded)
What we have is a key indication of how accurate our sample measurements were. In our example, we can say, with 95% confidence, that our four survey measurements have a margin of error of about ±3.8. The population’s true value has a 95% chance of falling within 3.8 units of the sample’s observed mean value.
Our mean value was 7.5, so the confidence interval runs between:
7.5 ± 3.8 = 3.7 and 11.3
How sample size affects margin of error
A margin of error of ±3.8 on a mean of 7.5—roughly half the mean’s value—is very wide. That’s largely because our survey had only four data points with high variance, which isn’t very good. For better results, we need more sample data.
A larger sample size results in higher accuracy and a lower margin of error. The sample can then be considered more representative of the larger group—or, said mathematically, the observed score is more likely to match the true score (and within a narrower range).
[Embed: 3yjx5U6DUE5ihfEYcM4m81]
How to reduce the margin of error in your survey results
Armed with your knowledge of the margin of error, you can design surveys for more reliable results. To achieve the slimmest margins of error and maximum accuracy, adjust your surveys and margin of error calculation in the following ways:
- Reduce the variables
- Increase total sample size
- Use a lower confidence level
- Use a one-sided confidence interval
The standard confidence interval is two-sided, meaning it uses a range with lower and upper values. If either of those limits is irrelevant, you can use only the upper or lower value. The sample mean becomes the new opposite limit, essentially halving the margin of error.
Should you be using a customer intelligence platform?
Do you want to discover previous survey findings faster?
Do you share your survey findings with others?
Do you analyze survey data?