quantitative (numerical) or categorical (labels)? Are there any outliers? Often a graphical dis- play reveals many patterns. There might be clusters or outliers. The data could be symmetric or skewed; unimodal or multi-modal. If the data are paired, then a scatterplot may reveal a relationship: linear or otherwise. Also, it is straightforward to compute the summary statistics (as appropriate): mean, median, mode, standard deviation, interquartile range, maximum, min- imum, range, correlation and/or covariance. Most of this is easily accomplished with a computer and statistical software such as SAS. (2.4) A statistic is a number computed from data, while a parameter is a number associated with a population. Most often, a parameter is unknown, and a statistic is used to estimate a particular parameter. (2.5) By definition, s 2 = 1 n - 1 n X i =1 ( y i - y ) 2 . Squaring, we see that ( y i - y ) 2 = y 2 i - 2 y i y + y 2 , and distributing the sum gives s 2 = 1 n - 1 n X i =1 ( y i - y ) 2 = 1 n - 1 " n X i =1 y 2 i - 2 y n X i =1 y i + n X i =1 y 2 . # Since y is constant, and since ∑ y i = n y , we see that 1 n - 1 " n X i =1 y 2 i - 2 y n X i =1 y i + n X i =1 y 2 # = 1 n - 1 " n X i =1 y 2 i - 2 y · n y + n y 2 # = 1 n - 1 " n X i =1 y 2 i - n y 2 # as required.
(2.13) Very briefly: (a) The average is M/R , and each average is exact (having factored the billions and thousands into the computation). Ordered by state we have: