Guide to Computing
Margins of Error for Percentages and Means
In this guide, the phrase "margin
of error" is equivalent to "two standard errors" or "a 95% confidence level."
The textbook explains what this means.
First, determine if you need
the margin of error for a mean score or for a percentage. If
it is a mean score (an average of a continuous variable, e.g., income in
dollars, test score points, pounds, inches, etc.), skip to the end of the
page and use
Formula Four.
If you are dealing with percentages,
you must choose among three formulas depending on the information given
and requested in the question (if you are not told that it is a mean or
average, assume that percentages will be computed):

y ou are told only the size of
the sample and are asked to provide the margin of error for percentages
which are not (yet) known. This is typically the case when you are
computing the margin of error for a survey which is going to be conducted
in the future. It is also useful for getting a general "ballpark"
figure for a sample as a whole. In this case, you use
Formula
One.

You are given a percentage result,
e.g., 65% voted for Candidate Blowhard. In this case you use Formula
Two.

You are told the margin of error
which is acceptable, and asked to compute the sample size. In this
case you use Formula Three. If
not told otherwise, assume that any question which asks for a sample size
wants a margin of error for percentages.
Formula
One: This
is the easy one, you should try to learn to use it in your head: M =
1/SQRT(N). Caution: N refers to the sample which answered
the question at hand, e.g., if you are asked for the margin of error for
the Hispanic respondents, N refers to the number of Hispanics in the sample.
The
answers will be in proportions, to get percents move the decimal point
two digits to the right. The confidence interval is + or  M.
Thus if M = .04, the confidence interval is +/ 4%.
Formula
Two: In
this formula, "p" refers to the proportion (not the percentage) giving
a certain answer to a question. For example, if 65% voted for Blowhard,
p = .65. N, as always, refers to the sample which answered the question
at hand. M = 2 * SQRT((p * (1p))/N). You
must do this calculation in the proper order. First determine p and
1p. If p = .65, for example, 1p means 1.65 or .35.
Then multiply p times 1p, divide the result by N, take the square root
and multiply the result by 2 (or 1.96 if you are a perfectionist).
This is best done as a chain calculation in your calculator, without writing
any of the intermediate steps down. To get confidence intervals,
take p and add M to get the upper bound, subtract M to get the lower bound.
It is conventional to use percentages in reporting the confidence interval.
Formula
Three: This
formula is used whenever you are asked to compute how large a sample will
be needed. The unexpected thing here is that the size of the population
does not matter! All that matters is how much error you
can tolerate. Take the margin of error you can tolerate, e.g., 4%,
and convert it to a proportion, e.g., .04. This is your "M".
Then use the formula N = 1/(M*M).
Just multiply M by itself
(square it) and divide the result into 1 (also called taking the reciprocal).
Caution:
In this formula, "N" refers to the number of respondents who will answer
the question you will ask or who will be used to compute the statistics
you need. Thus, if you need a result for the residents of Camden
County, N refers to the number of Camden County residents you must sample.
If you need a 5% margin of error for each of five counties, the total sample
size must be 5 * N.
Formula
Four for Mean Scores: If
you need a margin of error for a mean score (an average such as income
in dollars or scores on a test), you need to know the standard deviation
(sd) and the sample size (N). Ignore any other information
you are given, including the size of the population.
Use the following formula:
M
= 2 * sd / SQRT(N)
In this formula
"*" means multiply. "/" means divide.
"SQRT" means take the square root.
To get the confidence interval,
add M to the mean to get the upper bound and subtract M from the mean to
get the lower bound.
Caution: Your results
in this case are not in proportions. They are in whatever unit the
mean score was measured in,.e.g, dollars, inches, pounds, test score points,
etc. Your answer will be marked wrong if you convert the results
for a mean score to percents.