### What is Goodness-Of-Fit?

The goodness of fit test is a statistical hypothesis test to see how well sample data fit a distribution from a population with a normal distribution. Put differently, this test shows if your sample data represents the data you would expect to find in the actual population or if it is somehow skewed. Goodness-of-fit establishes the discrepancy between the observed values and those that would be expected of the model in a normal distribution case.

There are multiple methods for determining goodness-of-fit. Some of the most popular methods used in statistics include the chi-square, the Kolmogorov-Smirnov test, the Anderson-Darling test and the Shipiro-Wilk test.

### Key Takeaways

- Goodness-of-fit tests are statistical tests aiming to determine whether a set of observed values match those expected under the applicable model.
- There are multiple types of goodness-of-fit tests, but the most common is the chi-square test.
- These tests can show you whether your sample data fit an expected set of data from a population with normal distribution.

### Understanding Goodness-Of-Fit

Goodness-of-fit tests are often used in business decision making. In order to calculate a chi-square goodness-of-fit, it is necessary to first state the null hypothesis and the alternative hypothesis, choose a significance level (such as α = 0.5) and determine the critical value.

The most common goodness-of-fit test is the chi-square test, typically used for discrete distributions. The chi-square test is used exclusively for data put into classes (bins), and it requires a sufficient sample size in order to produce accurate results.

Goodness-of-fit tests are commonly used to test for the normality of residuals or to determine whether two samples are gathered from identical distributions.

### Example of a Goodness-Of-Fit Test

For example, a small community gym might be operating under the assumption that it has its highest attendance on Mondays, Tuesdays and Saturdays, average attendance on Wednesdays and Thursdays, and lowest attendance on Fridays and Sundays. Based on these assumptions, the gym employs a certain number of staff members each day to check in members, clean facilities, offer training services and teach classes.

However, the gym is not performing well financially and the owner wants to know if these attendance assumptions and staffing levels are correct. The owner decides to count the number of gym attendees each day for six weeks. He can then compare the gym's assumed attendance with its observed attendance using a chi-square goodness-of-fit test for example. With the new data, he can determine how to best manage the gym and improve profitability.