Testing Claims about the Goodness-of-Fit

**Purpose:** To test whether an unknown distribution (or
population) is the same as a "known" distribution (or population).

**Requirements:** Let
be
a random variable having the unknown distribution. Set up a two-way table. One
of the ways will be the factor “distribution” and the other will be the factor
“values of ”.
Enter into the table, in the row (or column) for the unknown distribution, the
observed counts of the various values of
.
(Note: You may need to bin the values of
.)
Find the total number of observations of
,
and call it .
For the “known” distribution, enter in each cell of the row (or column) the
number of observations you could *expect* to find IF
had
the “known” distribution.

(Example: Say you have 120 observations of
,
and suppose you need to compute the expected count for the bin
.
Use the “known” distribution to find the percentage of times
*should*
be between 4 (inclusive) and 8 (exclusive). Say it’s 20%. Multiply that
percentage by 120. The result (24) is the expected count for the bin
.
Now do the same sort of calculation for all the cells in the row (or column) for
the “known” distribution.)

(1) The values of are from a (simple) random sample.

(2) All expected counts are higher than 1.

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

**Hypotheses:**

*There
is no difference between the two distributions*. That is, the two
distributions are really the same.

*The
two distributions are not the same*.

**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, and get the 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.** If
you’re using a hand calculator, we recommend using the above formula instead of
the one usually found in textbooks, as it suffers less from this problem. 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
goodness-of-fit:** Click here.

**Examples of testing claims about several proportions: **Click here.

**Related topics:**

Chi-squared tests master reference page

Tests of homogeneity (several proportions)

**Statistics Reference Pages page:**
Click here

______________________________________________________________

BYU-Idaho mailto:brownd@byui.edu

232 Ricks Building 208-496-1839 voice

Rexburg, ID 83460 208-496-2005 fax

**Please
do not call me at home.**