1. Univariate Statistics Demonstration Day

2. Topics Questions Calculating measures of central tendency & dispersion by hand and in Excel Summation Notation & Rules Skewness

3. 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)?

4. 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) : the sample mean : the population mean s 2 : the sample variance σ 2 : the population variance s: the sample standard deviation σ : the population standard deviation

5. Equations Review Sample mean Sample standard deviation

6. Summation Notation The order of operations for statistical equations Similar to Please Excuse My Dear Aunt Sally from algebra

7. Summation Notation: Examples 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Example I: All observations are included in the sum: Example II: Only observations 3 through 5 are included in the sum:

8. Summation Notation: Rules Rule I: Summing a constant n times yields a result of n*b: Here we are simply using the summation notation to carry out a multiplication, e.g.:

9. Summation Notation: Rules Rule II: Constants may be taken outside of the summation sign

10. Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x 1 = 4, x 2 = 5, x 3 = 6 y 1 = 7, y 2 = 8, y 3 = 9 Rule II: Constants may be taken outside of the summation sign

11. Summation Notation: Rules Rule III: The order in which addition operations are carried out is unimportant +

12. Example: Now let a = 3, and let the values of a set (n = 3) of x and y values be: x 1 = 4, x 2 = 5, x 3 = 6 y 1 = 7, y 2 = 8, y 3 = 9 Rule III: The order in which addition operations are carried out is unimportant

13. Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum Summation Notation: Rules

14. = Example: Now let the values of a set (n = 3) of x values be: x 1 = 4, x 2 = 5, x 3 = 6 Rule IV: Exponents are handled differently depending on whether they are applied to the observation term or the whole sum

15. Rule V: Products are handled much like exponents Summation Notation: Rules

16. Example: Now let the values of a set (n = 3) of x and y values be: x 1 = 4, x 2 = 5, x 3 = 6 y 1 = 7, y 2 = 8, y 3 = 9 Rule V: Products are handled much like exponents

17. 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: Summation Notation: Compound Sums Rows Columns

18. Excel’s Skew Equation Pearson’s Skew Equation