Clinical Trials

Clinical Trials: A Practical Guide ■❚❙❘Some key properties of the normal distribution are as follows [2,3]:• The curve has a single peak at the center; this peak occurs at the mean (μ).• The curve is symmetrical about the mean (μ).• The median is the value above and below which there is an equalnumber of values (or the mean of the two middle values if thereis no middle number); hence, the median is equal to the mean.• The total area under the curve is equal to 1.• The spread of the curve is described by the standard deviation (σ)(the square of σ is the variance [σ 2 ]).• 95% of the observations lie between μ – 1.96σ and μ + 1.96σ.Examples of random samples from normal distributionsFor a variable measured from the population to be distributed normally, the aboveproperties should be met. In clinical studies, we are usually interested in a set ofvalues of a variable (or a sample) from a population with a certain disease. In thiscase, the distribution of the sample values might not exactly meet the aboverequirements. In fact, samples from a normal distribution will not necessarily seemto display a normal distribution themselves, especially if the sample size is small.Figure 6 displays the histograms of samples of different sizes (n = 20, 40, 100,and 400) drawn randomly from three normal distributions: N(0,0.5 2 ), N(0,1 2 ),and N(0,5 2 ). The graphs show that few of the small samples display a normaldistribution, but that closeness to a normal distribution increases with sample size.Why is normal distribution important?The normal distribution is statistically important for three reasons:• Firstly, most biological, medical, and psychological variables such asheight, weight, and SBP have approximately normal distributions.• Secondly, many statistical tests assume that a quantitative outcomevariable will have a normal distribution. Fortunately, these tests work verywell even if the distribution is only approximately normally distributed.• Thirdly, the sampling distribution of a mean is approximately normaleven when the individual observations are not normally distributed,given a sufficiently large sample (such as >200) [3]. This particularnotion is known as the central limit theorem [4].177