# Chi-square Goodness-of-fit

## Chi-square goodness-of-fit compares expected proportions to observed proportions

The Chi-square goodness-of-fit test is used to answer research questions about the dispersal of a categorical outcome across a given population. To conduct a chi-square goodness-of-fit test, researchers first hypothesize the proportion of the population that will "fit" into each level of the categorical outcome. Then, the researcher tests the observed proportions in each level against the hypothesized proportions in each level to see if they are significantly different.

For instance, let's say that researchers hypothesize that 90% of a population will possess a certain characteristic. After collecting data from a representative sample from the population, however, they find that only 67% of the population has the characteristic. Chi-square goodness-of-fit generates evidence that the observed proportion (67%) was statistically different from the hypothesized proportion (90%) with an effect size of 23% (90% - 67% = 23%).

For instance, let's say that researchers hypothesize that 90% of a population will possess a certain characteristic. After collecting data from a representative sample from the population, however, they find that only 67% of the population has the characteristic. Chi-square goodness-of-fit generates evidence that the observed proportion (67%) was statistically different from the hypothesized proportion (90%) with an effect size of 23% (90% - 67% = 23%).

**The most important part of chi-square goodness-of-fit test is to state the hypothesis for the expected proportion in an***a priori*fashion.### The steps for conducting a Chi-square goodness-of-fit test in SPSS

1. The data is entered in a between subjects fashion and the outcome is codified as

2. Click

3. Drag the cursor over the

4. Click on

5. Click on the

6. Under the

7. Click on the

8. Here is where the

However, many hypotheses will specify a proportion of an outcome to be some other value besides 50%. Researchers can specify any proportion across any number of levels or groups of a categorical outcome. Let's use the example from above to show how this works.

First, researchers have a

9. Click on the

10. Type the numerical designation of having the outcome or characteristic into the

11. Type

12. Type the numerical designation of

13. Type

14. Click

15. Click

**nominal**in**Variable View**.2. Click

**.**__A__nalyze3. Drag the cursor over the

**drop-down menu.**__N__onparametric Tests4. Click on

**.**__O__ne Sample5. Click on the

**Settings**tab.6. Under the

**Customize tests**marker, click on the**box to select it.**__C__ompare observed probabilities to hypothesized (Chi-Square test)7. Click on the

**Options**button.8. Here is where the

*a priori*hypothesized probability comes into play. If researchers believe that there is a 50/50 chance of a person being in either of the categories, then just click Run.However, many hypotheses will specify a proportion of an outcome to be some other value besides 50%. Researchers can specify any proportion across any number of levels or groups of a categorical outcome. Let's use the example from above to show how this works.

First, researchers have a

**null hypothesis that 90%**will possess the characteristic. The null hypothesis also states to the**10% that will not possess**the characteristic so as to generalize back to the population.9. Click on the

**marker to select it.**__C__ustomize expected probability10. Type the numerical designation of having the outcome or characteristic into the

**Category**column in the**Expected probabilities:**table.11. Type

**".90"**into the**Relative Frequency**column next to the outcome designation.12. Type the numerical designation of

**NOT**having the outcome or characteristic into the**Category**column in the**Expected probabilities:**table.13. Type

**".10"**into the**Relative Frequency**column next to the**NOT**having the outcome designation.14. Click

**OK**.15. Click

**Run**.### The steps for interpreting the SPSS output for chi-square goodness-of-fit

1. In the

If the

If the

**Hypothesis Test Summary**table, look under the**Sig.**column. This is the*p*-value that is interpreted.If the

*p*-value is**LESS THAN .05**, then the observed proportion was significantly different than the hypothesized proportion.If the

*p*-value is**MORE THAN .05**, then the observed proportion was**NOT**significantly different than the hypothesized proportion.Click on the

**Download Database**and**Download Data Dictionary**buttons for a configured database and data dictionary for Chi-square goodness-of-fit.## Hire A Statistician - Statistical Consulting for Students

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