Statistics For Managers 5th Edition Chapter 3 Numerical Descriptive Measures

Chapter Topics Measures of central tendency Measure of variation Shape Mean, median, mode, weighted mean, geometric mean, quartiles Measure of variation Range, interquartile range, average deviation, variance and standard deviation, coefficient of variation, standard units, Sharpe ratio, Sortino ratio Shape

Coefficient of Variation Summary Measures Summary Measures Central Tendency Variation Quartile Mean Mode Coefficient of Variation Median Range Variance Standard Deviation Geometric Mean

Measures of Central Tendency Average Median Mode Geometric Mean

Mean (Arithmetic Mean) Mean (arithmetic mean) of data values Sample mean Population mean Sample Size Population Size

Mean (Arithmetic Mean) (continued) The most common measure of central tendency Most commonly used average 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Mean = 5 Mean = 6

Mean of Grouped Data = 20 660 33 X S (M • F) n (F) (M) Class Frequency Mid-Point 10 but under 20 3 15 20 but under 30 6 25 30 but under 40 5 35 40 but under 50 4 45 50 but under 60 2 55 20 M • F 45 150 175 180 110 660 = 20 660 33 X S (M • F) n

Advantages and Disadvantages of the Arithmetic Mean Familiar and Easy to Understand Easy to Calculate Always Exists Is Unique Lends Itself to Further Calculation Affected by Extreme Values

Median Robust measure of central tendency In an ordered array, the median is the “middle” number If n or N is odd, the median is the middle number If n or N is even, the median is the average of the two middle numbers 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 Median = 5 Median = 5

Median of Group Data Median Class Step 1: Locate Median Term MT = n 2 Frequency 10 but under 20 20 but under 30 30 but under 40 40 but under 50 50 but under 60 3 6 Median Class 5 4 2 20 Step 1: Locate Median Term MT = n 2 = 20 = 10 Step 2: Assign a Value to the Median Term MD = L+ (MT - SFP) FMD •(i) = 30+ 5 (10 - 9) •10 = 32

Advantages and Disadvantages of the Median Easy to Understand Easy to Calculate Always Exists Is Unique Not Affected by Extreme Values Only Indicates Middle Value

Mode A measure of central tendency Value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may may be no mode There may be several modes 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 No Mode Mode = 9

Weighted Mean Used when observations differ in relative importance Xw = X1W1 + X2W2 + …….. + XnWn W1 + W2 + …….. + Wn

Weighted Mean If you bought 100 shares of a stock at $20 per share, 400 shares at $30 per share, and 500 shares at $40 per share, what would your average cost per share be? Xw = 100(20) + 400(30)+500(40) =$34 100 + 400 +500

Geometric Mean Useful in the measure of rate of change of a variable over time Geometric mean rate of return Measures the status of an investment over time

Example An investment of $100,000 declined to $50,000 at the end of year one and rebounded to $100,000 at end of year two:

Quartiles Split Ordered Data into 4 Quarters Position of i-th Quartile and Are Measures of Noncentral Location = Median, A Measure of Central Tendency 25% 25% 25% 25% Data in Ordered Array: 11 12 13 16 16 17 18 21 22

Measures of Variation Variation Variance Standard Deviation Coefficient of Variation Range Population Variance Population Standard Deviation Sample Variance Sample Standard Deviation Interquartile Range

Range Measure of variation Difference between the largest and the smallest observations: Ignores the way in which data are distributed Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 12 7 8 9 10 11 12

Interquartile Range Measure of variation Also known as midspread Spread in the middle 50% Difference between the first and third quartiles Not affected by extreme values Data in Ordered Array: 11 12 13 16 16 17 17 18 21

Average Deviation X = S X n 25 5 = 5 = AD 12 5 = 2.4 n S X 1 3 6 9 25 - 4 - 2 + 1 + 4 (X-X) 4 2 1 12 X-X X = S X n 25 5 = 5 = AD 12 5 = 2.4 n X-X S

Standard Deviation Most important measure of variation Shows variation about the mean Has the same units as the original data Sample standard deviation: Population standard deviation:

Calculation Example: Sample Standard Deviation Sample Data (Xi) : 10 12 14 15 17 18 18 24 n = 8 Mean = X = 16 A measure of the “average” scatter around the mean

Standard Deviation of Grouped Data 2 324 64 4 144 484 (M - X) 972 384 20 576 968 2920 2•F (M - X) 10 but under 20 3 15 20 but under 30 6 25 30 but under 40 5 35 40 but under 50 4 45 50 but under 60 2 55 20 Classes F M -18 -8 +2 +12 +22 (M - X) s = 1 - n å [ ( M ) X 2 • F ] = 2920 19 12.39

Comparing Standard Deviations Data A Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21

Variance Important measure of variation Shows variation about the mean Sample variance: Population variance:

Coefficient of Variation Used to Compare Relative Variation in Two or More Data Sets Measures relative variation Always in percentage (%) Shows variation relative to mean Is used to compare two or more sets of data measured in different units

Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 Stock B: Average price last year = $100 Coefficient of variation:

Using z scores to evaluate performance (Example) The industry in which sales rep Bill works has average annual sales of $2,500,000 with a standard deviation of $500,000. The industry in which sales rep Paula works has average annual sales of $4,800,000 with a standard deviation of $600,000. Last year Rep Bill’s sales were $4,000,000 and Rep Paula’s sales were $6,000,000. Which of the representatives would you hire if you had one sales position to fill?

Standard Units Used to Compare Relative Positions of Individual Observations in Two or More Data Sets Sales person Bill mB= $2,500,000 sB= $500,000 XB= $4,000,000 Sales person Paula mP=$4,800,000 sP= $600,000 XP= $6,000,000 ZB XB - mB sB = 4,000,000 – 2,500,000 500,000 +3 ZP = XP - mP sP 6,000,000 – 4,800,000 600,000 +2

SHARPE RATIO Sharpe ratio = (Prr – RFrr)/Srr Where: Prr = Annualized average return on the portfolio RFrr = Annualized average return on risk free proxy Srr = Annualized standard deviation of average returns Sharpe R = (10.5 – 2.5)/ 3.5 = 2.29 Generally, the higher the better.

SORTINO RATIO Sortino Ratio = (Prr – RFrr)/Srr(downside) Where: Prr = Annualized rate of return on portfolio RFrr= Annualized risk free annualized rate of return on portfolio Srr(downside) = downside semi-standard deviation Sortino = (10.5-2.5)/ 2.5 = 3.20 Doesn’t penalize for positive upside returns which the Sharpe ratio does

Shape of a Distribution Describes how data is distributed Measures of shape Symmetric or skewed Left-Skewed Symmetric Right-Skewed Mean < Median < Mode Mean = Median =Mode Mode < Median < Mean

Ethical Considerations Numerical descriptive measures: Should document both good and bad results Should be presented in a fair, objective and neutral manner Should not use inappropriate summary measures to distort facts

Chapter Summary Described measures of central tendency Mean, median, mode, geometric mean Discussed quartile Described measure of variation Range, interquartile range, average deviation, variance, and standard deviation, coefficient of variation, standard units, Sharp ratio, Sortino ratio Illustrated shape of distribution Symmetric, skewed