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12.3 Mathematical Analysis

To further investigate the nature of the relationships in your data you may continue with more quantitative, mathematical techniques, such as the ones presented in Chapter 8. When working with categorical data, the classical significance tests are often useful for demonstrating effects in a more unambiguous way than diagrams do. In this book we have only utilized the most fundamental tests. If these are not sufficient for your needs, the statistical literature contains a wealth of significance tests for various types of data.

For continuous numerical data, correlation and regression analysis are useful techniques for quantifying relationships between variables. Let us look at an example of how regression analysis was applied to strengthen the conclusions from Experiment 2.

Example 12.4: As explained in Chapter 10, the measured lift-off lengths in Experiment 2 were compared with an empirical formula, based on an extensive database from a spray chamber. According to this formula, the temperature, T, of the air surrounding the spray had a dominant effect on the lift-off length. More specifically, the lift-off length was proportional to T−3.74. In Experiment 2, the lift-off length shortened with increasing air temperature, but less than the empirical formula predicted. Regression analysis yielded a temperature exponent −1.55, substantially lower than the value of −3.74. The temperature dependence of the lift-off length was clearly weaker in the optical engine ...

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