Social Research Methods: Qualitative and Quantitative Approaches

ANALYSIS OF QUANTITATIVE DATA

For example, a sample shows that college men

and women differ in how many hours they study. Is

the result due to having an unusual sample, and in

reality there is no difference in the population, or

does it reflect a true difference between the men and

women? (See Example Box 5, Chi-Square.)

Type I and Type II Errors

The logic of statistical significance rests on whether

chance factors might have produced the results. You

may ask, why use the .05 level? We use it to mean

a 5 percent chance that randomness could cause the

results. Why not use a more certain standard—for

example, a 1 in 1,000 probability of random chance?

This gives a smaller chance that randomness ver-

sus a true relationship caused the results.

There are two answers to this way of thinking.

The simple answer is that the scientific community

has informally agreed to use .05 as a rule of thumb

for most purposes. Being 95 percent confident of

results is the accepted standard for explaining the

social world. A second, more complex answer

involves a trade-off between making Type I and

Type II errors. We can make two kinds of logical

mistakes. A Type I erroroccurs when we say that

a relationship exists when in fact none exists. It

means falsely rejecting a null hypothesis. A Type

II erroroccurs when we say that a relationship does

not exist, when in fact it does. It means falsely

accepting a null hypothesis (see Table 6). Of course,

we want to avoid both errors and say a relationship

is in the data only when it does indeed exist and

there is no relationship only when there really is

none. However, we face a dilemma: As the odds of

making one type of error decline, the odds of mak-

ing the opposite error increase.

You may find the ideas of Type I and Type II

errors difficult at first, but the same logical dilemma

appears outside research settings. For example, a

jury can err by deciding that an accused person is

guilty when in fact he or she is innocent, or the jury

Type I Error The mistake made in saying that a rela-

tionship exists when in fact none exists; a false rejec-

tion of a null hypothesis.

Type II Error The mistake made in saying that a rela-

tionship does not exist when in fact it does; false accept-

ance of a null hypothesis.

EXAMPLE BOX 5

Chi-Square

The chi-square (X^2 ) is used in two ways. This creates confusion. As a descriptive statistic,
it tells us the strength of the association between two variables; as an inferential statistic,
it tells us the probability that any association we find is likely to be due to chance factors.
The chi-square is a widely used and powerful way to look at variables measured at the
nominal or ordinal level. It is a more precise way to tell whether there is an association in
a bivariate percentaged table than by just “eyeballing” it.
Logically, we first determine “expected values” in a table. We do this based on infor-
mation from the marginals alone. Recall that marginals are frequency distributions of each
variable alone. An expected value can be thought of as our “best guess” without exam-
ining the body of the table. Next we consider the data to see how much differs from the
“expected value.” If they differ a lot, then there may be an association between the vari-
ables. If the data in a table are identical or very close to the expected values, then the vari-
ables are not associated; they are independent. In other words, independencemeans
“what is going on” in a table is what we would expect based on the marginals alone. Chi-
square is zero if there is independence increases as the association gets stronger. If the
data in the table greatly differ from the expected values, then we know something is
“going on” beyond what we would expect from the marginals alone (i.e., an association
between the variables). See the example of an association between height and grade.