Outline I. Summation Operation and Descriptive Statistics Basic Mathematical Tools II. Properties of Linear Functions III. Proportions and Percentages Read Wooldridge, Appendix A IV. Special Functions V. Differential Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 2 Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Summation Operation • Summation operator () involves the sum of many numbers. • Property s.1: For any constant c, n • Given a sequence of n numbers cnc {xi; i=1, …, n} i1 • The sum of these numbers • The sum of n constants (c) equals the product of n and c n xi = x1 + x2 + …. + xn i1 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 3 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 4 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation Summation Operation • Property s.2: • Property s.3: If {(xi,yi): i=1, …,n} is a set of n pairs of numbers and a and b are constants, then nn nnn cxii c x ii11 ()axii by a x i b y i iii111 The sum of c times xi equals c time the sum of xi. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 5 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 6 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation Summation Operation • Notes that the sum of ratios is not the ratio of the sums. • Note that the sum of the squares is not the square of the sum. n nn x x 22 ()x n x i ii ii1 ii11 n i1 yi yi i1 • Example: n = 2 2 2 2 • Example: n = 2 x1 + x2 (x1 + x2 ) I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 7 8 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation and Descriptive Statistics: Sample average Descriptive Statistics • When the xi are a sample of data on a particular variable, we • Given a sequence of n numbers {xi; i=1, …, n}, the average call this the sample average or sample mean. or mean can be written as 1 n • Sample average is an example of a descriptive statistic. x xi n i 1 • Sample average is a statistic that describes the central tendency of the set of n points. • Average is computed by adding them up and dividing by n I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 9 10 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Sample median Descriptive Statistics: Sample median • Other measure of central tendency is sample • Steps in finding sample median median. Step 1: order the values of the xi from smallest to largest. • Example: Given numbers, {‐4, 8, 2, 0, 21, ‐10, 18} Step 2: if n is odd, the sample median is the middle number of – Sample mean = 35/7 = 5 the ordered observations. – Sample median = 2 • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 21} Step 3: if n is even, the median is defined to be the average of the two middle values. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 11 12 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Sample median Summation Operation and Descriptive Statistics • If 21 is changed to 42 • Numbers {‐4, 8, 2, 0, 42, ‐10, 18} • Deviations – Sample mean = 56/7 = 8 – Sample median = 2 • Deviations can be found by taking each observation and subtracting off • Ordered sequence {‐10, ‐4, 0, 2, 10, 18, 42} the sample average • Sample median: dxxii Good point: it is less sensitive than sample average to changes in the extreme values in a list of numbers. Examples are median housing values or median income. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 13 14 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summation Operation and Descriptive Statistics Deviations and Demean Sample Example: n =5 • Properties d1: Given {x ; i=1, …, n}, i x = 6, x = 1, x = ‐2, x = 0, x = 5 The sum of the deviations equal zero. 1 2 3 4 5 nn x ? dxxii0 Demean sample is {4, ‐1, ‐4, ‐2, 3} ii11 n xxi 0 i1 I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 15 16 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Descriptive Statistics: Algebraic Fact Descriptive Statistics: Algebraic Fact • Properties d2: Given {xi; i=1, …, n}, • Properties d3: Given {(xi,yi): i=1, …,n}, the sum of squared deviations is the sum of squared xi minus It can be shown that n times the squared of sample mean. nn ()()xiixy y xy ii () y ii11 nn 222 nn ()xxii x nx () ()xxyii xynxy ii () ii11 ii11 • Show! I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 17 18 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Summary: Summation and Deviation Problem A.1 Summation Deviation A.1 The following table contains monthly housing s.1 d.1 expenditures for 10 families. 2 s.2 d.2 2 (i) Find the average monthly housing expenditure. [ans.] s.3 d.3 (ii) Find the median monthly housing expenditure. [ans.] I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 19 20 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Housing Problem A.1 continue Solution A.1 (i) Family Expenditures 1 300 • (iii) If monthly housing expenditures were measured in 2 440 (i) $566. 3 350 , rather than in dollars, what hundreds of dollars 4 1100 n would be the average and median expenditures? 1 5 640 x x [ans.] i 6 480 n i1 7 450 • (iv) Suppose that family number 8 increases its monthly 8 700 housing expenditure to $900 dollars, but the 9 670 expenditures of all other families remain the same. 10 530 Sum 5,660 Compute the average and median housing Mean 566 expenditures. [ans.] I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 21 22 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat Solution A.1 (ii) Solution A.1 (iii) (ii) 505 (iii) Steps in finding sample mean • $566 and $505 (in dollars), respectively • Step 1: order the values of the xi from smallest to largest. • 5.66 and 5.05 (in hundreds of dollars), {300, 350, 440, 450, 480, 530, 640, 670, 700, 1100,} respectively. • Step 3: if n is even, the median is defined to be the average of the two middle values. The two middle numbers are 480 and 530; when these are averaged, we obtain 505, or $505. I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus I. Summation II. Linear III. Prop&Perc IV. SpecFunc V. Calculus 23 24 I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat I. Summation Operation and Descriptive Statistics Basic Mathematical Tools . Intensive Course in Mathematics and Statistics . Chairat Aemkulwat II. Properties of Linear Functions Solution A.1 (iv) Housing Housing Family Expenditures Expenditures (iv) 1 300 300 A linear function can be written as • 2 440 440 The average increases to y = 0 + 1x $586 from $566. 3 350 350 4 1100 1100 • y and x are variables; • while the median is 5 640 640 • 0 and 1 are parameters; unchanged ($505). 6 480 480 – 0 is called the intercept; 7 450 450 – 1 is called the slope. {300, 350, 440, 450, 8 700 900 480, 530, 640, 900, 9 670 670 • We say that y is a linear function of x.