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
nearly normal, if the sample size is large enough. As a rough guide, many
statisticians say that a sample size of 30 is large enough when the population
distribution is bell-shaped.
But if the original population is badly skewed, has
multiple peaks, and/or has outliers, researchers like the sample size to be even larger.
When the sampling distribution is nearly normal, the critical value
can be expressed as a
or as a
When the sample size is smaller,
the critical value should only be expressed as a t statistic.
To find the critical value, follow these steps.
- The critical t statistic (t*) is
the t statistic having degrees of freedom equal to DF and a
equal to the critical probability (p*).
T-Score vs. Z-Score
Should you express the critical value as a t statistic or as a z-score? One way to answer
this question focuses on the population standard deviation.
- If the population standard deviation is unknown, use the t statistic.
Another approach focuses on sample size.
- If the sample size is small, use the t statistic.
In practice, researchers employ a mix of the above guidelines. On this site, we use z-scores
when the population standard deviation is known and the sample size is large.
Otherwise, we use the t statistics, unless the sample size is small and the underlying
distribution is not normal.
Warning: If the sample size is small and the population distribution is not normal,
we cannot be confident that the sampling distribution of the statistic will be normal. In this
situation, neither the t statistic nor the z-score should be used to compute critical values.
You can use the
Normal Distribution Calculator
to find the critical z score, and the
t Distribution Calculator to find
the critical t statistic. 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 value.
- Find the critical value.
Since we don't know the population standard deviation, we'll express the
critical value as a t statistic. For this problem, it will
be the t statistic 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 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
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 statistic. That is,
the critical value would still have been 1.96. The choice of t statistic versus z-score does not
make much practical difference when the sample size is very large.