Margin of Error
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
How to Compute the Margin of Error
The margin of error can be defined by either of the following
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
of the statistic is
or nearly normal.
central limit theorem
states that the sampling distribution of a statistic will be
normal or nearly normal, if any of the following
- The sample size is greater than 40, without outliers.
When one of these conditions is satisfied, the critical value
can be expressed as a
or as a
To find the critical value, follow these steps.
- The critical t score (t*) is
the t score having degrees of freedom equal to DF and a
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
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?
(E) None of the above.
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.
- 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
SEx = 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