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Chapter 12: Leaving Room for a Margin of Error

When you are working with categorical variables (those that record certain
characteristics that don’t involve measurements or counts; see Chapter 6), a
quick-and-dirty way to get a rough idea of the margin of error for proportions,
for any given sample size (n), is simply to find 1 divided by the square root of
n. For the Gallup poll example, n = 1,000, and its square root is roughly 31.62,
so the margin of error is roughly 1 divided by 31.62, or about 0.03, which is
equivalent to 3%. In the remainder of this chapter, you see how to get a more
accurate measure of the margin of error.
Finding the Margin of Error:
A General Formula
The margin of error is the amount of “plus or minus” that is attached to your
sample result when you move from discussing the sample itself to discussing
the whole population that it represents. Therefore, you know that the general 183
formula for the margin of error contains a “±” in front of it. So, how do you
come up with that plus or minus amount (other than taking a rough estimate,
as shown above)? This section shows you how.
Measuring sample variability
Sample results vary, but by how much? According to the Central Limit
Theorem (see Chapter 11), when sample sizes are large enough, the so-called
sampling distribution of the sample proportions (or the sample means) follows
a bell-shaped curve (or approximate normal distribution — see Chapter 9).
Some of the sample proportions (or sample means) overestimate the popula-
tion value and some underestimate it, but most are close to the middle.
And what’s in the middle of this sampling distribution? If you average out the
results from all the possible samples you could take, the average is the actual
population proportion, in the case of categorical data, or the actual population
average, in the case of numerical data. Normally, you don’t know all the values
of the population, so you can’t look at all of the possible sample results and
average them out — but knowing something about all the other sample pos-
sibilities does help you to measure the amount by which you expect your own
sample proportion (or average) to vary. (See Chapter 11 for more on sample
means and proportions.)
Standard errors are the basic building blocks of the margin of error. The stan-
dard error of a statistic is basically equal to the standard deviation of the pop-
ulation divided by the square root of n (the sample size). This reflects the fact
that the sample size greatly affects how much that sample statistic is going to
vary from sample to sample. (See Chapter 11 for more about standard errors.)

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