Transcript 01/18/2008
Univariate Statistics Demonstration Day Topics Questions Calculating measures of central tendency & dispersion by hand and in Excel Summation Notation & Rules Skewness Questions Which measure of central tendency is most appropriate for the following distributions and why? – – – – Bimodal distribution? Skewed distribution? Dataset with an outlier (an extreme value)? Normal distribution (unimodal)? Symbols Review n : the number of observations in a sample N : the number of elements in the population Σ : this (capital sigma) is the symbol for sum i : the starting point of a series of numbers X : one element in our dataset, usually has a subscript (e.g., i, min, max) x : the sample mean : the population mean s2 : the sample variance σ2 : the population variance s : the sample standard deviation σ : the population standard deviation Equations Review Sample mean Sample standard deviation n x x i 1 n n i s ( x x) i 1 i n 1 Summation Notation The order of operations for statistical equations Similar to Please Excuse My Dear Aunt Sally from algebra Summation Notation: Examples Example I: All observations are included in the sum: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 10 x i 1 i x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 1 2 3 4 5 6 7 8 9 10 Example II: Only observations 3 through 5 are included in the sum: 5 x i 3 i x3 x4 x5 3 4 5 12 Summation Notation: Rules Rule I: Summing a constant n times yields a result of n*b: n b b b b nb i 1 Here we are simply using the summation notation to carry out a multiplication, e.g.: 5 4 i 1 4 4 4 4 4 4 5 20 Summation Notation: Rules Rule II: Constants may be taken outside of the summation sign n ax i 1 n ax i 1 i i n a xi i 1 ax1 ax2 axn n a( x1 x2 xn ) a xi i 1 • Rule II: Constants may be taken outside of the summation sign Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9 3 3 ax i 2 a xi 3(5 6) 33 i i 2 ax i a xi 3(4 5 6) 45 Summation Notation: Rules Rule III: The order in which addition operations are carried out is unimportant n (x i 1 i n n i 1 i 1 yi ) xi yi ( x1 x2 x3 xn 1 xn ) + ( y1 y2 y3 yn1 yn ) • Rule III: The order in which addition operations are carried out is unimportant Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9 2 2 2 (x y ) x y i 1 i i i i 1 2 i 1 i (4 5) (7 8) 24 ( x y ) (4 7) (5 8) 24 i 1 i i Summation Notation: Rules Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum n k k k k x x x x i 1 2 n i 1 k k x ( x x x ) i 1 2 n i 1 n • Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum Example: Now let the values of a set (n = 3) of x values be: x1 = 4, x2 = 5, x3 = 6 2 2 2 2 x 4 5 6 77 i = ( xi ) 2 (4 5 6) 2 225 Summation Notation: Rules Rule V: Products are handled much like exponents n x y i i 1 i ( x1 y1 x2 y2 xn yn ) n n n x y x y i 1 n n x y i 1 i i 1 i i i i 1 i i 1 i ( x1 x2 xn ) ( y1 y2 yn ) • Rule V: Products are handled much like exponents Example: Now let the values of a set (n = 3) of x and y values be: x1 = 4, x2 = 5, x3 = 6 y1 = 7, y2 = 8, y3 = 9 x y i 4 7 5 8 6 9 122 i x y i i (4 5 6) (7 8 9) 360 Summation Notation: Compound Sums We frequently use tabular data (or data drawn from matrices), with which we can construct sums of both the rows and the columns (compound sums), using subscript i to denote the row index and the subscript j to denote the column index: Columns Rows 2 3 x i 1 j 1 ij x11 x12 x13 x21 x22 x23 ( x11 x12 x13 x21 x22 x23 ) Pearson’s Skew Equation 3( x median) s Excel’s Skew Equation