Clinical Trials

❘❙❚■ Chapter 17 | Types of Data and Normal DistributionFigure 6. Histograms for random samples from normal distributions.N(0,0.5 2 ) N(0,1 2 ) N(0,5 2 )n = 20n = 20n = 20–2 –1 0 1 2–4 –2 0 2 4–20 –10 0 10 20n = 40n = 40n = 40–2 –1 0 1 2–4 –2 0 2 4–20 –10 0 10 20n = 100n = 100n = 100–2 –1 0 1 2–4 –2 0 2 4–20 –10 0 10 20n = 400n = 400n = 400–2 –1 0 1 2–4 –2 0 2 4–20 –10 0 10 20The sampling distribution is a distribution of a sample statistic (eg, the mean).The theorem says that if we draw N samples (each of size n) from a populationand create a new variable, X (sampling distribution of mean), taking N values, themeans for N samples will be X 1, X 2, …, X Nand the distribution of X will be normalif the sample size n is large enough, regardless of whether the populationdistribution is normal. This theorem is fundamental to statistical inference [3,4].What is a standard normal distribution?The standard normal distribution is a special normal distribution with a mean of0 and a standard deviation of 1 or N(0,1). This is a unique distribution whosedistribution table is given in almost all statistical textbooks.The standard normal distribution is important because any other normaldistribution, N(μ,σ 2 ), can be converted to a standard normal distribution, N(0,1).178