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Goodness of fit

The goodness of fit is a test performed on sample data to find out if it fits a distribution from a given population (population with a normal distribution or Weibull distribution). In other words, it determines if the sample collected from a population represents the same data as the population being studied.

The goodness of Fit Tests

The most commonly used goodness of fit tests in statistics include:

  • Kolmogorov – Smirnov test
  • Bayesian information criterion
  • Anderson – Darling test
  • Cramer – von Mises criterion
  • Chi-Square test
  • Shapiro – Wilk test
  • Hosmer – Lemeshow test
  • Akaike information criterion
  • Kernelized Stein discrepancy
  • Kuiper’s test
  • Moran test
  • Zhang’s tests
The goodness of Fit Tests

Chi-square goodness of fit test

Of all the goodness of fit tests listed above, chi-square is the most popular and is the one you will most likely be familiarized with within elementary and advanced placement statistics. Chi-square test is used for discrete distributions such as the Poisson distribution and binomial distribution. The Anderson – Darling, Kolmogorov – Smirnov, and the other goodness of fit tests are only used for continuous distributions such as beta distribution, uniform distribution, and normal distribution. There are two potential downsides of the chi-square test, however:
Chi-square goodness of fit test
  • Chi-square requires enough sample data for the test approximation to be valid
  • The test can only be carried out if the data being observed has been put into (bins) classes. If you have not binned (classified) your data, then you will need to make a histogram or frequency table before performing the test.
There is a different type of chi-square test known as the chi-square test for independence.  The difference between the two is that the chi-square test for independence checks two data sets to determine whether there is a relationship, while the chi-square goodness of fit tests for categorical variables in a distribution. For more information on chi-square tests, contact our goodness of fit online tutors.

Understanding goodness of fit

The goodness of fit tests is usually performed in business projects to aid in decision-making. To calculate the chi-square goodness of fit, which is the most common test in business decision-making, the null hypothesis, and the alternative hypothesis must be stated first. After that, one must choose a significance level and identify the critical value.

The Chi-square goodness of fit test is no just a statistical measure – it is also a confidence measure. Why? When you have put together a formula that checks for almost all variations in a given group of data, say product observations, you have also put together a formula that can reliably predict how the product will behave in the future. This would not be possible without well-defined goodness of fit, test like chi square.

Example of the goodness of fit test

Example of the goodness of fit test.
To better understand the goodness of fit, let’s consider a community gym that operates under the assumption that Mondays, Tuesdays, and Saturdays have the highest attendance, Wednesdays and Thursdays have an average attendance, and Fridays and Sundays have the lowest attendance. Using these assumptions, the gym hires a specific number of staff members every day to attend to the people visiting the gym, offer training and teaching, and clean the equipment.

Nevertheless, the gym is not doing well financially and the management wants to find out if the attendance assumptions and the levels of staffing are correct and whether something needs to be adjusted. The management decides to calculate the number of people who visit the gym every day for seven weeks. To obtain the most appropriate results, the management can apply the chi-square goodness of fit test to compare the assumed attendance with the attendance observed during the seven weeks. With the results obtained, the management can come up with strategies for operating the gym better to improve profitability.

The gym owner can also use the Kolmogorov – Smirnov goodness of fit test to manage the gym operations. While this test is performed to determine normality in data, it does not really show whether a given sample was obtained from a normal population. Instead, it shows when it is unlikely to have a normal distribution. The upside of the Kolmogorov – Smirnov goodness of fit test is that it does not make assumptions about data distributions, which means the results are based on actual observations.

If you would like to learn more about the goodness of fit test and how it is applied in real life, reach out to our goodness of fit homework help experts.