Margin of Error

In survey sampling, the margin of error is a measure of the uncertainty in a survey outcome, given a particular sampling plan. In combination with a confidence level, the margin of error defines a range within which the true population value may be found.

For example, if a survey reports a margin of error of ±3% at a 95% confidence level for a sample statistic, we can expect the sampling plan (sample size, sampling method, etc.) used by the researcher to yield a sample statistic within 3 percentage points of the true population parameter, 95% of the time.

Prerequisite

This lesson assumes that you are familiar with the standard deviation of a sampling distribution and the standard error, topics that we covered in the previous lesson.

If you are not familiar with these topics, click the link below for a quick review:

• Lesson 50: What is the Standard Error?

How to Compute the Margin of Error

The margin of error is calculated from either of the following equations:

ME = CV x SD

ME = CV x SE

where ME is the margin of error, CV is a critical value (described below) corresponding to the desired confidence level, SD is the standard deviation for the sampling distribution of a statistic, and SE is the standard error.

Note: We seldom know the values of population parameters required to compute the standard deviation for the sampling distribution of a statistic (SD); whereas the standard error (SE) can be easily calculated from sample data, so we use the second equation ( ME = CV x SE ) to compute margin of error most of the time.

Standard Deviation and Standard Error

Formulas for the standard deviation (SD) of a sampling distribution and for the standard error (SE), vary depending on the statistic in question. Let's begin with the formulas the sampling distribution of a mean.

SD = σ / sqrt(n)

SE = s / sqrt(n)

where σ is the standard deviation of the population, s is the standard deviation of the sample, and n is sample size. (We'll introduce other formulas for other statistics as they come up in this tutorial.)

The Critical Value

The critical value is a factor used in the calculation of the margin of error (ME) to reflect the desired level of confidence in the estimate. The critical value depends on:

• The confidence level. Researchers often choose confidence levels of 90%, 95%, or 99%.
• The shape of the sampling distribution. If the sampling distribution is shaped like a t distribution, the critical value is expressed as a t-score. If it is shaped like a normal distribution, the critical value is expressed as a z-score.

How to Express Critical Value as t-Score

To express the critical value as a t-score, follow these steps.

• Compute alpha (α): α = 1 - (confidence level / 100)
  • When the confidence level is 99%, α is 1 - 99/100 or 0.01.
  • When the confidence level is 95%, α is 1 - 95/100 or 0.05.
  • When the confidence level is 90%, α is 1 - 90/100 or 0.1.
• Find the critical probability (p*): p* = 1 - α/2
• Find the degrees of freedom (df): df = n - 1 (for a mean score from a single sample)

  When estimating a mean score from a single sample, degrees of freedom are equal to the sample size (n) minus one. For other applications, the degrees of freedom may be calculated differently. (Elsewhere on this site, we explain how to calculate degrees of freedom for different applications.)

• Find the t-score having degrees of freedom equal to df and a cumulative probability equal to the critical probability (p*).

To find the critical t-score, use an online calculator (e.g.,Stat Trek's t Distribution Calculator), a graphing calculator, or a t-distribution statistical table (found in the appendix of most introductory statistics texts).

How to Express Critical Value as z-Score

When the critical value is expressed as a z-score, its value depends on the confidence level. Common z-score critical values are 1.645 for a 90% confidence level, 1.96 for a 95% confidence level, and 2.576 for a 99% confidence level.

To express the critical value as a z-score when the confidence level is not 90%, 95%, or 99%, follow these steps.

• Compute alpha (α): α = 1 - (confidence level / 100)
• Find the critical probability (p*): p* = 1 - α/2
• Find the z-score having a cumulative probability equal to the critical probability (p*).

To find the critical z-score, use an online calculator (e.g, Stat Trek's Normal Distribution Calculator), a graphing calculator, or a normal distribution statistical table (found in the appendix of most introductory statistics texts).

t-Score vs. z-Score

Should you express the critical value as a t-score or as a z-score? The answer depends several factors, including the type of statistic (e.g., proportion vs. mean), sample size, and knowledge of population variance. Here are some rules of thumb to help you choose:

• For a sample mean, use z-scores when the population standard deviation is known and the sample size is large. Otherwise, we use t-scores.
• For a sample proportion, use z-scores when n * p ≥ 10 and n * (1-p) ≥ 10. (This condition will be satisfied when the sample includes at least 10 "successes" and at least 10 "failures".)

We use a z-score when the sampling distribution of the statistic more closely approximates a normal distribution; and we use a t-score when it more closely approximates a t distribution. Elsewhere on this site, we explain how to choose between the normal and t distributions.

Test Your Understanding

This problem demonstrates each step required to compute the margin of error.

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 sample 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)

  α = 1 - 0.95 = 0.05

• Find the critical probability (p*):

  p* = 1 - α/2

  p* = 1 - 0.05/2 = 0.975

• Find the degrees of freedom (df):

  df = n - 1 = 900 -1 = 899

• Find the critical value. Since we don't know the population standard deviation, we'll express the critical value as a t-score. For this problem, it will be the t-score having 899 degrees of freedom and a cumulative probability equal to 0.975. Using the t Distribution Calculator, we find that the critical value is about 1.96.

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

SE = s / sqrt( n )

SE = 0.4 / sqrt( 900 ) = 0.4 / 30 = 0.013

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

ME = Critical value x Standard error

ME = 1.96 * 0.013 = 0.025

So, for this study, we report that the average GPA is 2.7 ± 0.025.

Note: The larger the sample size, the more closely the t-distribution looks like the normal distribution. For this problem, since the sample size is very large, we would have found the same result with a z-score as we found with a t-score. That is, the critical value would still have been about 1.96. The choice of t-score versus z-score does not make much practical difference when the sample size is very large.