Experimental Design and Statistical Analysis for Pharmacology and the Biomedical Sciences. Paul J. Mitchell

4.8 The Chi‐square distribution. The probability density function for the Chi‐squared distribution with 1 (bold solid line), 2 (thin solid line), 5 (dashed line), and 10 (dotted line) degrees of freedom. X‐axis values indicate Chi‐squared (χ 2) and Y‐axis indicates probability (see also Appendix A.4).

      9 Student‐t distribution (Figure 4.9)The Student t‐distribution is derived from the Chi‐square and normal distributions. The distribution is symmetrical and bell‐shaped, very much like the Normal Distribution (see Figure 4.7) but with greater area under the curve in the tails of the distribution. The t‐distribution arises when the mean of a set of data that follows a normal distribution is estimated where the sample size is small and the population standard deviation (σ) is unknown. As the sample size increases so the t‐distribution approximates more closely to the standard normal distribution. The t‐distribution plays an important role in assessing the probability that two sample means arise from the same population, in determining the confidence intervals for the difference between two population means (see Chapters 11 and 12) and in linear regression analysis (see Chapter 20).

      10 F distribution (Figure 4.10)The F distribution (named after Sir Ronald Fisher, who developed the F distribution for use in determining the critical values for the Analysis of Variance (ANOVA) models; see Chapters 15, 16 and 17) is a function of the ratio of two independent random variables (each of which has a Chi‐square distribution) divided by its respective number of Degrees of Freedom. It is used in several applications including assessing the equality of two or more population variances and the validity of equations following multiple regression analysis. The F‐distribution has two very important properties; first, it is defined for positive values only (this makes sense since all variance values are positive!), and second, unlike the t‐distribution, it is not symmetrical about its mean but instead is positively skewed.

So why do we need to understand data distribution?

      Of the data distributions briefly described above, the majority are only of value to understand the theoretical basis of statistical analysis (the Chi square, t‐ and F‐distributions are important once we get into inferential statistics, but only if we wish to understand the process of how such tests work). In contrast, the most important distribution for the experimental pharmacologist is the Normal Distribution which shall be discussed in detail later (see Chapters 6 and 7).

Figure 4.9 The t‐distribution. The probability density function of the t‐distribution with 1 (dotted line), 2 (dashed line), and 4 (thin solid line) degrees of freedom compared with the Standard Normal Distribution (bold solid line; N(0,1)). For the t‐distribution data the X‐axis values indicate the value of t. For the Standard Normal Distribution, the X values indicate the mean of zero with standard deviations either side of the mean. Y‐axis indicates probability in all cases. Note as the degrees of freedom increase so the probability density function of the t‐distribution approximates towards the Standard Normal Distribution (see also Appendix A.5).

Figure 4.10 The F distribution. The probability density function of the F distribution with 1, 4 (bold solid line), 2, 8 (thin solid line), 4, 20 (dashed line), and 8, 32 (dotted line) degrees of freedom. X‐axis values indicate the F ratio, while Y‐axis values indicate probability density function (see also Appendix A.6)

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