Scientists use margins of error to quantify how much the estimates from their research could differ from the “true” value. This uncertainty might seem like a weakness of science, but in reality, the ability to explicitly estimate a margin of error is one of its biggest strengths. Uncertainty can’t be avoided, but recognizing that it exists is essential. You can focus on the mean for many purposes, but if you want to draw any conclusions about the difference in means between different populations, the margins of error become absolutely essential. Learning how to calculate the margin of error is a crucial skill for scientists in any field.

#### TL;DR (Too Long; Didn't Read)

Find the margin of error by multiplying the critical value of (z), for large samples where the population standard deviation is known, or (t), for smaller samples with a sample standard deviation, for your chosen confidence level by the standard error or population standard deviation. Your result ± this result defines your estimate and its margin of error.

## Margins of Error Explained

When scientists calculate a mean (i.e., an average) for a population, they base this on a sample taken from the population. However, not all samples are perfectly representative of the population, and so the mean might not be accurate for the whole population. In general, a bigger sample and a set of results with a smaller spread about the mean make the estimate more reliable, but there will always be some possibility that the result isn’t quite accurate.

Scientists use confidence intervals to specify a range of values in which the true mean should fall. This is usually done at a 95 percent confidence level, but it may be done at 90 percent or 99 percent confidence in some cases. The range of values between the mean and the edges of the confidence interval are known as the margin of error.

## Calculating Margin of Error

Calculate the margin of error using the standard error or standard deviation, your sample size and an appropriate “critical value.” If you know the standard deviation of the population and you have a big sample (generally considered to be anything over 30), you can use a z-score for your chosen level of confidence and simply multiply this by the standard deviation to find the margin of error. So for 95 percent confidence, z = 1.96, and the margin of error is:

Margin of error = 1.96 × population standard deviation

This is the amount you add to your mean for the upper bound and subtract from the mean for the lower bound of your margin of error.

Most of the time, you won’t know the population standard deviation, so you should use the standard error of the mean instead. In this case (or with small sample sizes), you use a t-score instead of a *z*-score. Follow these steps to calculate your margin of error.

Subtract 1 from your sample size to find your degrees of freedom. For example, a sample size of 25 has df = 25 – 1 = 24 degrees of freedom. Use a t-score table to find your critical value. If you want a 95 percent confidence interval, use the column labeled 0.05 on a table for two-tailed values or the 0.025 column on a one-tailed table. Look for the value that intersects your confidence level and your degrees of freedom. With df = 24 and at 95 percent confidence, t = 2.064.

Find the standard error for your sample. Take the sample standard deviation, (s), and divide it by the square root of your sample size, (n). So in symbols:

Standard error = s ÷ √*n*

So for a standard deviation of s = 0.5 for a sample size of n = 25:

Standard error = 0.5 ÷ √25 = 0.5 ÷ 5 = 0.1

Find the margin of error by multiplying your standard error by your critical value:

Margin of error = standard error × t

In the example:

Margin of error = 0.1 × 2.064 = 0.2064

This is the value you add to the mean to find the upper limit for your margin of error and subtract from your mean to find the lower limit.

## Margin of Error for a Proportion

For questions involving a proportion (e.g., the percentage of respondents to a survey giving a specific answer), the formula for the margin of error is a little different.

First, find the proportion. If you surveyed 500 people to find out how many supported a political policy, and 300 did, you divide 300 by 500 to find the proportion, often called p-hat (because the symbol is a “p” with an accent over it, p̂).

p̂ = 300 ÷ 500 = 0.6

Choose your confidence level and look up the corresponding value of (z). For a 90 percent confidence level, this is z = 1.645.

Use the formula below to find the margin of error:

Margin of error = z × √(p̂( 1 – p̂) ÷ n)

Using our example, z = 1.645, p̂ = 0.6 and n = 500, so

Margin of error = 1.645 × √(0.6( 1 –* *0.6) ÷ 500)

= 1.645 × √(0.24÷ 500)

= 1.645 × √0.00048

= 0.036

Multiply by 100 to turn this into a percentage:

Margin of error (%) = 0.036 × 100 = 3.6%

So the survey found that 60 percent of people (300 out of 500) supported the policy with a 3.6 percent margin of error.