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Session 2 Handout Clarification of Some Common Misconceptions • Mean, median, variance and other descriptive statistics are NOT exclusive to the normal distribution. • The normal distribution is a case of symmetrical distribution in which theoretically, mean=median • “(Empirical) distribution” aka sampling distribution, means where the data points are; theoretical distribution means where the data points “should be”. • In empirical distributions, mean & median rarely exactly equal, so whereas it cannot be PROVED that a set of data points is normally distributed, it can be shown that the set of data points is not too deviant from the (theoretical) normal distribution. • Descriptive statistics of a sample are conceptually and computationally different from descriptive statistics of a population, ie the sample mean is defined and calculated differently from population mean, the sample variance is defined and calculated differently from population variance, and so on. Demo – Mean & Median http://opl.apa.org/contributions/Rice/rvls_sim/stat_sim/descriptive/index.html Descriptive Statistics Exercise • Suppose a waitress recorded the amount of tips she got from a few tables at a birthday party. Calculate & interpret the following statistics: Sample Median, Sample Mean, Sample Variance, Sample Standard Deviation Data Set Index A Data points (ie Sample) 17, 15, 23, 7, 9, 13, 39, 5 Sample median 14 Sample mean ( ) 16 Sample 2 variance (s ) 120 Sample standard deviation (s) ≈10.95445 B 18, 16, 24, 8, 10, 14, 40, 6 15 17 120 ≈10.95445 C 19, 17, 25, 9, 11, 15, 41, 7 16 18 120 ≈10.95445 D 20, 18, 26, 10, 12, 16, 42, 8 17 19 120 ≈10.95445 In this exercise, all data points were systematically increased by 1 unit across data sets, hence only sample medians and sample means changed. In general, if the sample distribution only shifts location across samples and the relative distances between data points within the sample do not change (ie the change of data points between samples is systematic), then only the sample median and sample mean change; Measures of within-sample variability (sample variance and sample standard deviation) do not change, as the within-sample relative distances between individual data points with respect to the sample mean do not change.