# Margin of Error

In a
confidence interval, the range of values above and below the
sample statistic is called the **margin of error**.

For example, suppose we wanted to know the percentage of adults that exercise daily. We could devise a sample design to ensure that our sample estimate will not differ from the true population value by more than, say, 5 percent (the margin of error) 90 percent of the time (the confidence level).

## How to Compute the Margin of Error

The margin of error can be defined by either of the following equations.

Margin of error = Critical value x Standard deviation of the statistic

Margin of error = Critical value x Standard error of the statistic

If you know the standard deviation of the statistic, use the first equation to compute the margin of error. Otherwise, use the second equation. Previously, we described how to compute the standard deviation and standard error.

## How to Find the Critical Value

The **critical value** is a factor used to compute
the margin of error. This section describes how to find the
critical value, when the
sampling distribution
of the statistic is
normal
or nearly normal.

The central limit theorem states that the sampling distribution of a statistic will be normal or nearly normal, if any of the following conditions apply.

- The population distribution is normal.
- The sampling distribution is symmetric, unimodal, without outliers, and the sample size is 15 or less.
- The sampling distribution is moderately skewed, unimodal, without outliers, and the sample size is between 16 and 40.
- The sample size is greater than 40, without outliers.

When one of these conditions is satisfied, the critical value can be expressed as a t score or as a z score. To find the critical value, follow these steps.

- Compute alpha (α): α = 1 - (confidence level / 100)
- Find the critical probability (p*): p* = 1 - α/2
- To express the critical value as a z score, find the z score having a cumulative probability equal to the critical probability (p*).
- To express the critical value as a t score, follow these steps.
- Find the degrees of freedom (DF). When estimating a mean score or a proportion from a single sample, DF is equal to the sample size minus one. For other applications, the degrees of freedom may be calculated differently. We will describe those computations as they come up.
- The critical t score (t*) is the t score having degrees of freedom equal to DF and a cumulative probability equal to the critical probability (p*).

Should you express the critical value as a t score or as a z score? There are several ways to answer this question. As a practical matter, when the sample size is large (greater than 40), it doesn't make much difference. Both approaches yield similar results. Strictly speaking, when the population standard deviation is unknown or when the sample size is small, the t score is preferred. Nevertheless, many introductory statistics texts use the z score exclusively. On this web site, we provide sample problems that illustrate both approaches.

You can use the Normal Distribution Calculator to find the critical z score, and the t Distribution Calculator to find the critical t score. You can also use a graphing calculator or standard statistical tables (found in the appendix of most introductory statistics texts).

## Test Your Understanding

**Problem 1**

Nine hundred (900) high school freshmen were randomly selected for a national survey. Among survey participants, the mean grade-point average (GPA) was 2.7, and the standard deviation was 0.4. What is the margin of error, assuming a 95% confidence level?

(A) 0.013

(B) 0.025

(C) 0.500

(D) 1.960

(E) None of the above.

**Solution**

The correct answer is (B). To compute the margin of error, we need to find the critical value and the standard error of the mean. To find the critical value, we take the following steps.

- Compute alpha (α): α = 1 - (confidence level / 100) = 1 - 0.95 = 0.05
- Find the critical probability (p*): p* = 1 - α/2 = 1 - 0.05/2 = 0.975
- Find the critical z score. Since the sample size is large, the sampling distribution will be roughly normal in shape. Therefore, we can express the critical value as a z score. For this problem, it will be the z score having a cumulative probability equal to 0.975. Using the Normal Distribution Calculator, we find that the critical value is 1.96.

Next, we find the standard error of the mean, using the following equation:

SE_{x} = s / sqrt( n )
= 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

And finally, we compute the margin of error (ME).

ME = Critical value x Standard error = 1.96 * 0.013 = 0.025

This means we can be 95% confident that the mean grade point average in the population is 2.7 plus or minus 0.025, since the margin of error is 0.025.