Comparing Two or More Unknown Populations Reference Page

**Purpose:** To test whether two or more unknown
populations have the same distribution.

**Requirements:** Let
be
a random variable having the first unknown distribution,
a
random variable having the second unknown distribution, and so on, up to
.
Set up a two-way table of observed counts. One way is for the factor
“population,” with the populations as its levels. The other way is for values of
,
,
and so on, up to
.
Use the table to compute the expected counts for each cell in the table, thus:.
The conditions for the test are

(1) The data for each population are from a (simple) random sample taken from that population

(2) The samples are all independent of each other.

(3) All expected counts are greater than 1.

(4) No more than 20% of the expected counts are less than 5.

**Hypotheses:**

*The
populations all have the same distribution*.
The alternative hypothesis is
*At
least one of the populations is different from at least one of the others*.
(Hopefully, this reminds you of the ANOVA hypotheses.)

**Test Statistic:** We recommend using software such as
SPSS to calculate the
test
statistic, the degrees of freedom, and the p-value. However, you __can__ use
the formula ,
with degrees of freedom equal to ,
and get the (one-tailed) p-value from a
table
such as Table F in Moore & McCabe’s __The Practice of Business
Statistics__.

**WARNING: The test
statistic is very sensitive to round-off errors made during calculation. **
We
recommend using the above formula instead of the one usually found in textbooks,
as it suffers less from this problem than they do. Also, we recommend entering
the entire calculation in your calculator all at once, rather than computing
each term separately, to help minimize round-off error.

** **

**SPSS instructions for testing claims about several
proportions:** Click here.

**Examples of testing claims about several proportions:**

Click here for examples based on
Moore & McCabe’s __The Practice of Business Statistics__.

Click here for examples based on the Sullivan text.

**Related topics:**

Chi-squared tests master reference page

Tests of homogeneity (several proportions)

Goodness-of-fit (comparing populations when one population is “known”)

**Statistics Reference Pages page:**
Click here

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