## What is a Margin of Error?

A **margin of error** tells you **how many percentage points your results will differ **from the real population value. For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be within 4 percentage points of the real population value 95% of the time.

However, there’s a little more to the formal definition. The **margin of error **is defined a 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 (i.e. from a poll or survey).

For example, a poll might state that there is a 98% confidence interval of 4.88 and 5.26. So we can say that if the poll is repeated using the same techniques, 98% of the time the true population parameter (parameter vs. statistic) will fall within the interval estimates (i.e. between 4.88 and 5.26) 98% of the time.

## Statistics Aren’t Always Right!

The idea behind confidence levels and margins of error is that any survey or poll will differ from the true population by a certain amount. However, confidence intervals and margins of error reflect the fact that there

*is*room for error. So although 95% or 98% confidence with a 2 percent Margin of Error might sound like a very good statistic, room for error is built in, which means sometimes statistics are wrong.

For example, a Gallup poll in 2012 (incorrectly) stated that Romney would win the 2012 election with Romney at 49% and Obama at 48%. The stated confidence level was 95% with a margin of error of ± 2. We can conclude that the results were calculated to be accurate to within 2 percentages points 95% of the time.

In comparison, the real results from the election were: Obama 51%, Romney 47%. So this result was outside the range of the Gallup poll’s margin of error (2 percent). In addition to statistics being wrong, polls can be wrong as well.

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## How to Calculate Margin of Error

As an example, a poll might report that a certain candidate is going to win an election with 51 percent of the vote. Plus, the confidence level is 95 percent and the error is 4 percent. If we assume that the poll was repeated using the same techniques, then the pollsters would expect the results to be within 4 percent of the stated result (51 percent) 95 percent of the time. In other words, 95 percent of the time they would expect the results to be between:

- 51 – 4 = 47 percent and
- 51 + 4 = 55 percent.

**The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample**:

- Margin of error (parameter) = Critical value x Standard deviation for the population.
- Margin of error (statistic) = Critical value x Standard error of the sample.

## How to Calculate Margin of Error: Steps

Step 1: **Find the critical value**. The critical value is either a **t-score** or a **z-score**. If you aren’t sure which score you should be using, see: T-score vs z-score. However, in general, for small sample sizes (under 30) or when you don’t know the population standard deviation, use a t-score. Otherwise, use a z-score.

Step 2: **Find the Standard Deviation or the Standard Error.** So although these are essentially the same thing, only you *must* know your population parameters in order to calculate standard deviation. Otherwise, calculate the standard error.

Step 3: **Multiply the critical value **from Step 1** by the standard deviation** or standard error from Step 2. For example, if your CV is 1.95 and your SE is 0.019, then:

1.95 * 0.019 = 0.03705

**Example question:** 900 students were surveyed and had an average GPA of 2.7 with a standard deviation of 0.4. Calculate the margin of error for a 90% confidence level:

- The critical value is 1.645 (see this video for the calculation)
- The standard deviation is 0.4 (from the question), but as this is a sample, we need the standard error for the mean. The formula for the SE of the mean is
*standard deviation / √(sample size)*, so: 0.4 / √(900) = 0.013. - 1.645 * 0.013 = 0.021385

*That’s how to calculate margin of error! *

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**Tip**: You can use the t-distribution calculator on this site to find the t-score. In addition, the variance and standard deviation calculator will calculate the standard deviation from a sample.

## Margin of Error for a Proportion

The formula is a little different for proportions:

Where:

**Example question:** 1000 people were surveyed and 380 thought that climate change was not caused by human pollution. Find the MoE for a 90% confidence interval.

Step 1: **Find P-hat** by dividing the number of people who responded positively. “Positively” in this sense doesn’t mean that they gave a “Yes” answer; It means that they answered according to the statement in the question. In this case, 380/1000 people (38%) responded positively.

Step 2: **Find the z-score that goes with the given confidence interval.** But, you’ll need to reference this chart of common critical values. A 90% confidence interval has a z-score (a critical value) of 1.645.

Step 3: **Insert the values into the formula and solve:**

= 1.645 * 0.0153

= 0.0252

Step 4: **Turn Step 3 into a percentage**:

0.0252 = 2.52%

In conclusion, the margin of error is 2.52%.

Plus, check out our Youtube channel for video tips on statistics!

## References

Moore, D. S. and McCabe G. P. Introduction to the Practice of Statistics. New York: W. H. Freeman, p. 443, 1999.