Some Words of Comfort
Do not worry if you find the theory underlying probability distributions complex. Our experience demonstrates that you want to know only when and how to use these distributions. We have therefore outlined the essentials and omitted the equations that define the probability distributions. You will find that you only need to be familiar with the basic ideas, the terminology and, perhaps (although infrequently in this computer age), know how to refer to the tables.
More Continuous Probability Distributions
These distributions are based on continuous random variables. Often it is not a measurable variable that follows such a distribution but a statistic derived from the variable. The total area under the probability density function represents the probability of all possible outcomes, and is equal to one (Chapter 7). We discussed the Normal distribution in Chapter 7; other common distributions are described in this chapter.
The t-Distribution (Appendix A2, Fig. 8.1)
- Derived by W.S. Gossett, who published under the pseudonym ‘Student’; it is often called Student’s t-distribution.
- The parameter that characterizes the t-distribution is the degrees of freedom (df), so we can draw the probability density function if we know the equation of the t-distribution and its degrees of freedom. We discuss degrees of freedom in Chapter 11; note that they are often closely affiliated to sample size.
- Its shape is similar to that of the Standard Normal distribution, but it is more spread out, with longer tails. Its shape approaches Normality as the degrees of freedom increase.
- It is particularly useful for calculating confidence intervals for and testing hypotheses about one or two means (Chapters 19–21).
The Chi-Squared (χ2) distribution (Appendix A3, Fig. 8.2)
- It is a right-skewed distribution taking positive values.
- It is characterized by its degrees of freedom (Chapter 11).
- Its shape depends on the degrees of freedom; it becomes more symmetrical and approaches Normality as the degrees of freedom increases.
- It is particularly useful for analysing categorical data (Chapters 23–25).
Figure 8.2 Chi-squared distributions with degrees of freedom (df) = 1, 2, 5 and 10.
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