Dr Kafu Wong ECON 1003 Analysis of Economic

Dr. Ka-fu Wong ECON 1003 Analysis of Economic Data Ka-fu Wong © 2003 Chap 2 -1

Chapter Two Descriptive Statistics GOALS 1. Organize data into a frequency distribution. 2. Plots: n Histogram, n stem-and-leaf display. 3. Measures of central tendency 4. Measures of dispersion 5. Some properties of mean and standard deviations Ka-fu Wong © 2003 Chap 2 -2

Frequency Distribution A Frequency distribution is a grouping of data into mutually exclusive categories showing the number of observations in each class. Ka-fu Wong © 2003 Chap 2 -3

Construction of a Frequency Distribution 1. Question to be addressed. 2. Collect data (raw data). 3. Organize data. n Frequency distribution 4. Present data. 5. Draw conclusion. Ka-fu Wong © 2003 Chap 2 -4

Terms Related to Frequency Distribution n n In constructing a frequency distribution, data are divided into exhaustive and mutually exclusive classes. Mid-point: A point that divides a class into two equal parts. This is the average of the upper and lower class limits. Class frequency: The number of observations in each class. Class interval: The class interval is obtained by subtracting the lower limit of a class from the lower limit of the next class. Ka-fu Wong © 2003 Chap 2 -5

Four steps to construct frequency distribution 1. 2. 3. 4. Decide on the number of classes. Determine the class interval or width. Set the individual class limits. Tally the data into the classes and count the number of items in each. For illustration, it is convenient to carry an example with us – example 1. Ka-fu Wong © 2003 Chap 2 -6

EXAMPLE 1 n Dr. Tillman is Dean of the School of Business Socastee University. He wishes to prepare a report showing the number of hours per week students spend studying. He selects a random sample of 30 students and determines the number of hours each student studied last week. 15. 0, 23. 7, 19. 7, 15. 4, 18. 3, 23. 0, 14. 2, 20. 8, 13. 5, 20. 7, 17. 4, 18. 6, 12. 9, 20. 3, 13. 7, 21. 4, 18. 3, 29. 8, 17. 1, 18. 9, 10. 3, 26. 1, 15. 7, 14. 0, 17. 8, 33. 8, 23. 2, 12. 9, 27. 1, 16. 6. Organize the data into a frequency distribution. Ka-fu Wong © 2003 Chap 2 -7

Step 1: Decide on the number of classes. The goal is to use just enough groupings or classes to reveal the shape of the distribution. n “Just enough” Recipe –“ 2 to the k rule” n Select the smallest number (k) for the number of classes such that 2 k is greater than the number of observations (n). n Ka-fu Wong © 2003 Chap 2 -8

2 to the k rule n n Select the smallest number (k) for the number of classes such that 2 k is greater than the number of observations (n). Sample size (n) = 80 21=2; 22=4; 23=8; 24=16; 25=32; 26=64; 27=128; … The rule suggest 7 classes. Ka-fu Wong © 2003 n n n Sample size (n) = 1000 21=2; 22=4; 23=8; 24=16; 25=32; 26=64; 27=128; 28=256; 29=512; 210=1024 … The rule suggest 10 classes. Chap 2 -9

2 to the k rule n n Select the smallest number (k) for the number of classes such that 2 k is greater than the number of observations (n). Sample size (n)=10000 21=2; 22=4; 23=8; 24=16; 25=32; 26=64; 27=128; …, 213=8192; 214 = 16384 The rule suggest 14. Ka-fu Wong © 2003 n n Sample size (n)=100000 22=4; 23=8; 24=16; 25=32; 26=64; 27=128; … Chap 2 -10

2 to the k rule Select the smallest number (k) for the number of classes such that 2 k is greater than the number of observations (n). n We want to find smallest k such that 2 k > n. n Smallest k such that k log 2 > log n n Smallest k such that k > (log n) / (log 2) n Example: If n=10000, (log n) / (log 2) = 13. 28. Hence the recipe suggest 14 classes. n Note: Same result for base 10 log and natural log. Ka-fu Wong © 2003 Chap 2 -11

Step 1: Decide on the number of classes. n 2 to the k rule n Select the smallest number (k) for the number of classes such that 2 k is greater than the number of observations (n). n Example 1 (continued): n Sample size (n) = 30 n 21=2; 22=4; 23=8; 24=16; 25=32; 26=64; 27=128; … n The rule suggest 5 classes. n Alternative, by computing (Log 30/log 2) = 4. 91, we get the same suggestion of 5 classes. Ka-fu Wong © 2003 Chap 2 -12

Step 2: Determine the class interval or width. n n n Generally the class interval or width should be the same for all classes. The classes all taken together must cover at least the distance from the lowest value in the raw data to the highest value. The classes must be mutually exclusive and exhaustive. Class interval ≥ (Highest value – lowest value) / number of classes. Usually we will chose some convenient number as class interval that satisfy the inequality. Ka-fu Wong © 2003 Chap 2 -13

Step 2: Determine the class interval or width. Class interval ≥ (Highest value – lowest value) / number of classes. n Example 1 (continued): n Highest value = 33. 8 hours n Lowest value = 10. 3 hours n k=5. n Hence, class interval ≥ (33. 8 -10. 3)/5 = 4. 7 n We choose class interval to be 5, some convenient number. Ka-fu Wong © 2003 Chap 2 -14

Step 3: Set the individual class limits n n The class limits must be set so that the classes are mutually exclusive and exhaustive. Round up to some convenient numbers. Ka-fu Wong © 2003 Chap 2 -15

Step 3: Set the individual class limits n Example 1 (continue): n Highest value = 33. 8 hours. n Lowest value = 10. 3 hours. n Range = highest – lowest = 23. 5. n K=5; Interval = 5. n With k=5 and interval = 5, the classes will cover a n n Ka-fu Wong © 2003 range of 25. Let’s split the surplus in the lower and upper tail equally. (25 -23. 5)/2 = 0. 75. Hence, the lower limit of the first class should be around (10. 3 – 0. 75)=9. 55 and upper limit of the last class should be (33. 8 + 0. 75)=34. 55. 9. 55 and 34. 55 look odd. Some convenient and close numbers would be 10 and 35. Chap 2 -16

Step 3: Set the individual class limits n Example 1 (continue): n We have determined n K=5; Interval = 5. n The lower limit of the n first class = 10 and The upper limit of the last class = 35. “ 10 up to 15” means the interval from 10 to 15 that includes 10 but not 15. Ka-fu Wong © 2003 Chap 2 -17

Step 4: Tally the data into the classes and count the number of items in each 15. 0, 23. 7, 19. 7, 15. 4, 18. 3, 23. 0, 14. 2, 20. 8, 13. 5, 20. 7, 17. 4, 18. 6, 12. 9, 20. 3, 13. 7, 21. 4, 18. 3, 29. 8, 17. 1, 18. 9, 10. 3, 26. 1, 15. 7, 14. 0, 17. 8, 33. 8, 23. 2, 12. 9, 27. 1, 16. 6. Hours studying 7 10 up to 15 15 up to 20 12 20 up to 25 7 25 up to 30 30 up to 35 3 Ka-fu Wong © 2003 1 Chap 2 -18

EXAMPLE 1 (continued) Ka-fu Wong © 2003 Chap 2 -19

EXAMPLE 1 (continued) n A relative frequency distribution shows the percent of observations in each class. Ka-fu Wong © 2003 Chap 2 -20

Stem-and-leaf Displays n Stem-and-leaf display: A statistical technique for displaying a set of data. Each numerical value is divided into two parts: the leading digits become the stem and the trailing digits the leaf. n Note: n An advantage of the stem-and-leaf display over a frequency distribution is we do not lose the identity of each observation. n An disadvantage is that it is not good for large data sets. Ka-fu Wong © 2003 Chap 2 -21

EXAMPLE 2 n Colin achieved the following scores on his twelve accounting quizzes this semester: 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. n Construct a stem-and-leaf chart. Ka-fu Wong © 2003 Chap 2 -22

EXAMPLE 2 (continued) 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Stem Ka-fu Wong © 2003 Leaf 6 9 7 9 8 8 6 4 8 9 2 1 6 3 2 5 Chap 2 -23

EXAMPLE 2 (continued) 86, 79, 92, 84, 69, 88, 91, 83, 96, 78, 82, 85. Freq Stem Leaf 1 6 9 2 7 8 9 (6) 8 2 3 4 3 9 1 2 6 Ka-fu Wong © 2003 5 6 8 Chap 2 -24

Graphic Presentation of a Frequency Distribution n The three commonly used graphic forms are n histograms, n frequency polygons, and n a cumulative frequency distribution. n A Histogram is a graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. n The class frequencies are represented by the heights of the bars and the bars are drawn adjacent to each other. Ka-fu Wong © 2003 Chap 2 -25

Graphic Presentation of a Frequency Distribution n A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency. n A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value. Ka-fu Wong © 2003 Chap 2 -26

Histogram for Hours Spent Studying Example 1 (continued): A Histogram is a graph in which the classes are marked on the horizontal axis and the class frequencies on the vertical axis. Ka-fu Wong © 2003 Chap 2 -27

Frequency Polygon for Hours Spent Studying Example 1 (continued): A frequency polygon consists of line segments connecting the points formed by the class midpoint and the class frequency. Ka-fu Wong © 2003 Chap 2 -28

Cumulative Frequency Distribution For Hours Studying Example 1 (continued): A cumulative frequency distribution is used to determine how many or what proportion of the data values are below or above a certain value. Ka-fu Wong © 2003 Chap 2 -29

Bar Chart n A bar chart can be used to depict any of the levels of measurement (nominal, ordinal, interval, or ratio). Ka-fu Wong © 2003 Chap 2 -30

Example 3 Construct a bar chart for the number of unemployed per 100, 000 population for selected cities during 2001 Ka-fu Wong © 2003 Chap 2 -31

Bar Chart for the Unemployment Data Example 3 (continued): Ka-fu Wong © 2003 Chap 2 -32

Pie Chart n A pie chart is useful for displaying a relative frequency distribution. A circle is divided proportionally to the relative frequency and portions of the circle are allocated for the different groups. Ka-fu Wong © 2003 Chap 2 -33

EXAMPLE 4 A sample of 200 runners were asked to indicate their favorite type of running shoe. Draw a pie chart based on the following information. Ka-fu Wong © 2003 Chap 2 -34

EXAMPLE 4 (continued) Compute the percentage and degree each type occupy out of 360 o. Degree occupied in a circle = percentage x 360 Ka-fu Wong © 2003 Chap 2 -35

Pie Chart for Running Shoes Example 4 (continued): Ka-fu Wong © 2003 Chap 2 -36

Measures of Central Tendency 1. Mean 2. Median 3. Mode Ka-fu Wong © 2003 Chap 2 -37

Population Parameters and Sample Statistics A population parameter is number calculated from all the population measurements that describes some aspect of the population. n The population mean, denoted , is a population parameter and is the average of the population measurements. n A point estimate is a one-number estimate of the value of a population parameter. n A sample statistic is number calculated using sample measurements that describes some aspect of the sample. n Ka-fu Wong © 2003 Chap 2 -38

The Mean Population X 1, X 2, …, XN Sample x 1, x 2, …, xn Population Mean Ka-fu Wong © 2003 Sample Mean Chap 2 -39

Population Mean n For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values: where µ is the population mean. n. N is the total number of observations. n. X is a particular value. n indicates the operation of adding. Ka-fu Wong © 2003 Chap 2 -40

Sample Mean n For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values: Where n is the total number of values in the sample. This sample mean is also referred as arithmetic mean or simply sample average. Ka-fu Wong © 2003 Chap 2 -41

EXAMPLE 5 n A sample of five executives received the following bonus last year ($000): 14. 0, 15. 0, 17. 0, 16. 0, 15. 0 Ka-fu Wong © 2003 Chap 2 -42

Population and Sample Proportions Population X 1, X 2, …, XN Sample x 1, x 2, …, xn p Population Proportion Sample Proportion xi = 1 if characteristic present, 0 if not Ka-fu Wong © 2003 Chap 2 -43

Properties of the Arithmetic Mean 1. Every set of interval-level and ratio-level data has a mean. 2. All the values are included in computing the mean. 3. A set of data has a unique mean. 4. The mean is affected by unusually large or small data values. 5. The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean is zero. Ka-fu Wong © 2003 Chap 2 -44

EXAMPLE 5 (continued) n A sample of five executives received the following bonus last year ($000): 7. 0, 15. 0, 17. 0, 16. 0, 15. 0 n Changing the first observation from 14. 0 to 7. 0 will change the sample mean. Ka-fu Wong © 2003 Chap 2 -45

EXAMPLE 6 n Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property: Ka-fu Wong © 2003 Chap 2 -46

Weighted Mean n The weighted mean of a set of numbers X 1, X 2, . . . , Xn, with corresponding weights w 1, w 2, . . . , wn, is computed from the following formula: Ka-fu Wong © 2003 Chap 2 -47

EXAMPLE 7 n During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0. 50, fifteen for $0. 75, fifteen for $0. 90, and fifteen for $1. 10. Compute the weighted mean of the price of the drinks. Ka-fu Wong © 2003 Chap 2 -48

The Median n The Median is the midpoint of the values after they have been ordered from the smallest to the largest. n There as many values above the median as below it in the data array. n For an even set of values, the median will be the arithmetic average of the two middle numbers. Ka-fu Wong © 2003 Chap 2 -49

EXAMPLE 8 n The ages for a sample of five college students are: 21, 25, 19, 20, 22 n Arranging the data in ascending order gives: 19, 20, 21, 22, 25. Thus the median is 21. n. The heights of four basketball players, in inches, are: 76, 73, 80, 75 n. Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75. 5 Ka-fu Wong © 2003 Chap 2 -50

Properties of the Median 1. There is a unique median for each data set. 2. It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. 3. It can be computed for ratio-level, intervallevel, and ordinal-level data. 4. It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class. Ka-fu Wong © 2003 Chap 2 -51

The Mode n The mode is the value of the observation that appears most frequently. n EXAMPLE 6: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode. Ka-fu Wong © 2003 Chap 2 -52

Geometric Mean n The geometric mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The formula is: The geometric mean is used to average percents, indexes, and relatives. n The geometric mean is not applicable when some numbers are negative. Ka-fu Wong © 2003 Chap 2 -53

EXAMPLE 9 n The interest rate on three bonds were 5, 21, and 4 percent. n The geometric mean is n The arithmetic mean is (5+21+4)/3 =10. 0 n The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 21 percent. Ka-fu Wong © 2003 Chap 2 -54

Geometric Mean continued n Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another. Ka-fu Wong © 2003 Chap 2 -55

EXAMPLE additional n Suppose I deposited all my lai-see money of 120 to the bank two years ago. I got 132. 3 in my bank account today. What was the average annual interest rate in these two years? Ka-fu Wong © 2003 Chap 2 -56

EXAMPLE additional n Suppose I deposited all my lai-see money of “value at beginning of period” to the bank k years ago. I got “value at end of period” in my bank account today. What was the average annual interest rate in these k years? Ka-fu Wong © 2003 Chap 2 -57

EXAMPLE 10 n The total number of females enrolled in American colleges increased from 755, 000 in 1992 to 835, 000 in 2000. That is, the geometric mean rate of increase is 1. 27%. Ka-fu Wong © 2003 Chap 2 -58

EXAMPLE 11 n A banker wants to get an annual return of 100% on its loan in credit card business. What monthly interest rate should he charge? A monthly interest rate of 5. 9%. Ka-fu Wong © 2003 Chap 2 -59

EXAMPLE 12 n The Chinese government claimed in 1990 that their GDP will double in 20 years. What must the annual GDP growth rate be for this dream to come true? A annual GDP growth of 3. 5%. Ka-fu Wong © 2003 Chap 2 -60

EXAMPLE 13 n If we want to double our wealth in k years. What should the annual return rate be? Ka-fu Wong © 2003 k Annual return rate 1 100% 2 41. 4% 3 26. 0% 4 18. 9% 5 14. 8% 10 7. 17% 15 4. 7% 20 3. 5% Chap 2 -61

The Mean of Grouped Data n The mean of a sample of data organized in a frequency distribution is computed by the following formula: Ka-fu Wong © 2003 Chap 2 -62

EXAMPLE 14 n A sample of ten movie theaters in a large metropolitan area tallied the total number of movies showing last week. Compute the mean number of movies showing. Ka-fu Wong © 2003 Chap 2 -63

EXAMPLE 14 continued Ka-fu Wong © 2003 Chap 2 -64

The Median of Grouped Data n The median of a sample of data organized in a frequency distribution is computed by: where L is the lower limit of the median class, CF is the cumulative frequency preceding the median class, f is the frequency of the median class, and i is the median class interval. Ka-fu Wong © 2003 Chap 2 -65

Finding the Median Class n To determine the median class for grouped data: n Construct a cumulative frequency distribution. n Divide the total number of data values by 2. n Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25 th value. Ka-fu Wong © 2003 Chap 2 -66

EXAMPLE 15 Ka-fu Wong © 2003 Chap 2 -67

Cumulative frequency The idea behind the formula 7 6 5 3 1 1 -3 3 -5 5 -7 7 -9 9 -11 Ka-fu Wong © 2003 x Chap 2 -68

The idea behind the formula Cumulative frequency (5 – 3)/(6 -3) = k/(7 -5) (n/2 – CF)/f = k/i k =(n/2 – CF)/f (i) Median = 5+k Median = L+k 6 5 3 5 7 k Ka-fu Wong © 2003 x Chap 2 -69

EXAMPLE 15 continued n From the table, L=5, n=10, f=3, i=2, CF=3 Ka-fu Wong © 2003 Chap 2 -70

The Mode of Grouped Data n The mode for grouped data is approximated by the midpoint of the class with the largest class frequency. n Example 13 (continued): The modes in EXAMPLE 13 are 6 and 10. When two values occur a large number of times, the distribution is called bimodal, as in Example 13. Ka-fu Wong © 2003 Chap 2 -71

Symmetric Distribution zero skewness: mode = median = mean Density Distribution (the height may be interpreted as relative frequency) The area under the density distribution is 1. The sum of relative frequency is 1. Thus median always splits the density distribution into two equal areas. Ka-fu Wong © 2003 Chap 2 -72

Right Skewed Distribution Positively skewed: (Skew to the right) Mean and Median are to the right of the Mode<Median<Mean Ka-fu Wong © 2003 Chap 2 -73

Left Skewed Distribution Negatively Skewed: Mean and Median are to the (skew to the left) left of the Mode. Mean<Median<Mode Ka-fu Wong © 2003 Chap 2 -74

Measures of dispersion 1. 2. 3. 4. 5. l Range Mean Deviation Variance and standard deviation Coefficient of variation Box plots. Ka-fu Wong © 2003 Chap 2 -75

Range n The range is the difference between the largest and the smallest value. n Only two values are used in its calculation. n It is influenced by an extreme value. n It is easy to compute and understand. Ka-fu Wong © 2003 Chap 2 -76

Mean Deviation n The Mean Deviation is the arithmetic mean of the absolute values of the deviations from the arithmetic mean. n All values are used in the calculation. n It is not influenced too much by large or small values. n The absolute values are difficult to manipulate. Mean deviation is also known as Mean Absolute Deviation (MAD). Ka-fu Wong © 2003 Chap 2 -77

EXAMPLE 16 n The weights of a sample of crates containing books for the bookstore (in pounds ) are: 103, 97, 101, 106, 103 Find the range and the mean deviation. Range = 106 – 97 = 9 Ka-fu Wong © 2003 Chap 2 -78

Example 16 The first step is to find the mean weight. The mean deviation is: Ka-fu Wong © 2003 Chap 2 -79

Population Variance n The population variance is the arithmetic mean of the squared deviations from the population mean. n All values are used in the calculation. n More likely to be influenced by extreme values than mean deviation. n The units are awkward, the square of the original units. Ka-fu Wong © 2003 Chap 2 -80

The Variance Population X 1, X 2, …, XN s 2 Population Variance Sample x 1, x 2, …, xn s 2 Sample Variance Note in the sample variance formula the sum of deviation is divided by (n-1) instead of n in order to yield an unbiased estimator of the population variance. Ka-fu Wong © 2003 Chap 2 -81

EXAMPLE 17 n The ages of the Dunn family are: 2, 18, 34, 42 What is the population variance? Ka-fu Wong © 2003 Chap 2 -82

The Population Standard Deviation n The population standard deviation (σ) is the square root of the population variance. n For EXAMPLE 2, the population standard deviation is 15. 36, found by Ka-fu Wong © 2003 Chap 2 -83

EXAMPLE 18 The hourly wages earned by a sample of five students are: $7, $5, $11, $8, $6. Find the variance. Ka-fu Wong © 2003 Chap 2 -84

Sample Standard Deviation n The sample standard deviation is the square root of the sample variance. n In EXAMPLE 3, the sample standard deviation is 2. 30 Ka-fu Wong © 2003 Chap 2 -85

Sample Variance For Grouped Data n The formula for the sample variance for grouped data is: Ka-fu Wong © 2003 Chap 2 -86

Interpretation and Uses of the Standard Deviation n Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least: where k is any constant greater than 1. Ka-fu Wong © 2003 Chap 2 -87

Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least 1 - 1/k 2 Ka-fu Wong © 2003 K Coverage 1 0% 2 75. 00% 3 88. 89% 4 93. 75% 5 96. 00% 6 97. 22% Chap 2 -88

Interpretation and Uses of the Standard Deviation n Empirical Rule: For any symmetrical, bell-shaped distribution: n About 68% of the observations will lie within 1 s the mean, n About 95% of the observations will lie within 2 s of the mean n Virtually all the observations will be within 3 s of the mean Empirical rule is also known as normal rule. Ka-fu Wong © 2003 Chap 2 -89

Bell-shaped Curve showing the relationship between σ and μ Ka-fu Wong © 2003 3 s -2 s -1 s +2 s + 3 s Chap 2 -90

Why are we concern about dispersion? n Dispersion is used as a measure of risk. n Consider two assets of the same expected (mean) returns. n -2%, 0%, +2% n -4%, 0%, +4% n The dispersion of returns of the second asset is larger then the first. Thus, the second asset is more risky. n Thus, the knowledge of dispersion is essential for investment decision. And so is the knowledge of expected (mean) returns. Ka-fu Wong © 2003 Chap 2 -91

Relative Dispersion n The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage: Ka-fu Wong © 2003 Chap 2 -92

Sharpe Ratio and Relative Dispersion n Sharpe Ratio is often used to measure the performance of investment strategies, with an adjustment for risk. n If X is the return of an investment strategy in excess of the market portfolio, the inverse of the CV is the Sharpe Ratio. n An investment strategy of a higher Sharpe Ratio is preferred. http: //www. stanford. edu/~wfsharpe/art/sr/sr. htm Ka-fu Wong © 2003 Chap 2 -93

Skewness n Skewness is the measurement of the lack of symmetry of the distribution. n The coefficient of skewness can range from 3. 00 up to 3. 00. n A value of 0 indicates a symmetric distribution. n It is computed as follows: Or Ka-fu Wong © 2003 Chap 2 -94

Why are we concerned about skewness? n Skewness measures the degree of asymmetry in risk. n Upside risk n Downside risk n Consider the distribution of asset returns: n Right skewed implies higher upside risk than downside risk. n Left skewed implies higher downside risk than upside risk. Ka-fu Wong © 2003 Chap 2 -95

Percentiles and Quartiles n For a set of measurements arranged in increasing order, the pth percentile is a value such that p percent of the measurements fall at or below the value and (100 -p) percent of the measurements fall at or above the value. n The first quartile Q 1 is the 25 th percentile n The second quartile (or median) Md is the 50 th percentile n The third quartile Q 3 is the 75 th percentile. n The interquartile range IQR is Q 3 - Q 1 Ka-fu Wong © 2003 Chap 2 -96

Example: Quartiles 20 customer satisfaction ratings: 1 3 5 5 7 8 8 8 9 9 9 10 10 Md = (8+8)/2 = 8 Q 1 = (7+8)/2 = 7. 5 Q 3 = (9+9)/2 = 9 IRQ = Q 3 - Q 1 = 9 - 7. 5 = 1. 5 Ka-fu Wong © 2003 Chap 2 -97

Box Plots n A box plot is a graphical display, based on quartiles, that helps to picture a set of data. n Five pieces of data are needed to construct a box plot: the Minimum Value, the First Quartile, the Median, the Third Quartile, and the Maximum Value. Ka-fu Wong © 2003 Chap 2 -98

EXAMPLE 19 n Based on a sample of 20 deliveries, Buddy’s Pizza determined the following information. The minimum delivery time was 13 minutes and the maximum 30 minutes. The first quartile was 15 minutes, the median 18 minutes, and the third quartile 22 minutes. Develop a box plot for the delivery times. min Q 1 median 12 14 16 18 Ka-fu Wong © 2003 20 Q 3 22 max 24 26 28 30 32 Chap 2 -99

Box and Whiskers Plots Ka-fu Wong © 2003 Chap 2 -100

Pareto Chart of Labeling Defects Ka-fu Wong © 2003 Chap 2 -101

Scatter Plots Restaurant Ratings: Mean Preference vs. Mean Taste Ka-fu Wong © 2003 Chap 2 -102

Misleading Graphs and Charts: Scale Break Mean Salaries at a Major University, 1999 - 2002 Ka-fu Wong © 2003 Chap 2 -103

Misleading Graphs and Charts: Horizontal Scale Effects Mean Salary Increases at a Major University, 1999 - 2002 Ka-fu Wong © 2003 Chap 2 -104

Working with mean and Standard Deviation Set Data Mean St Dev (1) 19 20 21 20. 00 0. 82 (2) -1 0 1 0. 00 0. 82 (3) 19 20 20 20. 00 0. 71 (4) 38 40 42 40. 00 1. 63 (5) 57 60 63 60. 00 2. 45 (6) 19 19 20 20. 00 0. 82 (7) 3 5 8 5. 33 2. 05 (8) 4 7 9 6. 67 2. 05 (9) 7 12 17 12. 00 4. 08 12 20 21 35. 56 18. 04 (10) Ka-fu Wong © 2003 21 20 27 21 32 21 35 45 56 72 Chap 2 -105

Working with mean and Standard Deviation Set Data Mean St Dev (1) 19 20 21 20. 00 0. 82 (2) -1 0 1 0. 00 0. 82 (3) 19 20 20 20. 00 0. 71 (4) 38 40 42 40. 00 1. 63 (5) 57 60 63 60. 00 2. 45 21 n (2) = (1) – mean(1): n Mean(2)=0; Stdev(2)=Stdev(1) n (3) = (1) + mean(1) n Mean(3)=Mean(1); Stdev(3)<Stdev(1). n (4) = (1)*2; (5) = (1)*3 n Mean(4)=mean(1)*2; mean(5)=mean(1)*3 n Stdev(4)=stdev(1)*2; stdev(5)=stdev(1)*3 Ka-fu Wong © 2003 Chap 2 -106

Working with mean and Standard Deviation Set Data Mean St Dev 20. 00 0. 82 (1) 19 20 21 (6) 19 19 20 (7) 3 5 8 5. 33 2. 05 (8) 4 7 9 6. 67 2. 05 (9) 7 12 17 12. 00 4. 08 12 20 21 35. 56 18. 04 (10) 20 27 21 32 21 35 45 56 72 n (6)=(1) multiplied by some frequency n Mean(6)=Mean(1); Stdev(6)=Stdev(1). n (9) = (7)+(8) n Mean(9)=mean(7)+mean(8) n (10) = (7) *(8) n Mean(10)=mean(7)*mean(8) Ka-fu Wong © 2003 Chap 2 -107

Further results about mean and variance of transformed variables n n n E(X) = mean or expected values V(X) = E[(X-E(X))2]=E(X 2) – E(X)2 E(a+b. X) = a+b. E(X) E(X+Y) = E(X) + E(Y) V(X+Y) = V(X) + V(Y) if X and Y are independent. Ka-fu Wong © 2003 Chap 2 -108

Further results about mean and variance of transformed variables n n E(a+b. X) = a+b. E(X) E(X+Y) = E(X) + E(Y) Suppose we invest $1 in two assets. $a in asset X and $(1 -a) in asset Y. Their expected returns are respectively E(X) and E(Y). We will expect a return of E(a. X+(1 -a)Y) = a. E(X) + (1 -a)E(Y) for this investment portfolio. If these two assets are independent or uncorrelated so that C(X, Y) =0, then the variance is V(a. X+(1 -a)Y) = a 2 V(X) + (1 -a)2 V(Y) Ka-fu Wong © 2003 Chap 2 -109

Chapter Two Descriptive Statistics - END - Ka-fu Wong © 2003 Chap 2 -110