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EF 507 QUANTITATIVE METHODS FOR ECONOMICS AND FINANCE FALL 2008 Chapter 3 Describing Data: Numerical Chap 3-1 Describing Data Numerically Describing Data Numerically Central Tendency Variation Arithmetic Mean Range Median Variance Mode Standard Deviation Coefficient of Variation Chap 3-2 Measures of Central Tendency Overview Central Tendency Mean Median Mode Midpoint of ranked values Most frequently observed value n x x i1 i n Arithmetic average Chap 3-3 Arithmetic Mean The arithmetic mean (mean) is the most common measure of central tendency For a population of N values: N x x1 x 2 x N μ N N i1 i Population values Population size For a sample of size n: n x x i1 n i x1 x 2 x n n Observed values Sample size Chap 3-4 Arithmetic Mean (continued) The most common measure of central tendency Mean = sum of values divided by the number of values Affected by extreme values (outliers) 0 1 2 3 4 5 6 7 8 9 10 Mean = 3 1 2 3 4 5 15 3 5 5 0 1 2 3 4 5 6 7 8 9 10 Mean = 4 1 2 3 4 10 20 4 5 5 Chap 3-5 Median In an ordered list, the median is the “middle” number (50% above, 50% below) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Median = 3 Median = 3 Not affected by extreme values Chap 3-6 Finding the Median The location of the median: n 1 Median position position in the ordered data 2 If the number of values is odd, the median is the middle number If the number of values is even, the median is the average of the two middle numbers n 1 is not the value of the median, only the 2 position of the median in the ranked data Note that Chap 3-7 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 be no mode There may be several modes 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 9 0 1 2 3 4 5 6 No Mode Chap 3-8 Review Example Five houses on a hill by the beach $2,000 K House Prices: $2,000,000 500,000 300,000 100,000 100,000 $500 K $300 K $100 K $100 K Chap 3-9 Review Example: Summary Statistics House Prices: $2,000,000 500,000 300,000 100,000 100,000 Mean: Median: middle value of ranked data = $300,000 Mode: most frequent value = $100,000 Sum 3,000,000 ($3,000,000/5) = $600,000 Chap 3-10 Which measure of location is the “best”? Mean is generally used, unless extreme values (outliers) exist If outliers exist, then median is often used, since the median is not sensitive to extreme values. Example: Median home prices may be reported for a region – less sensitive to outliers Chap 3-11 Shape of a Distribution Describes how data are distributed Measures of shape Symmetric or skewed Left-Skewed Symmetric Right-Skewed Mean < Median Mean = Median Median < Mean Chap 3-12 Measures of Variability Variation Range Variance Standard Deviation Coefficient of Variation Measures of variation give information on the spread or variability of the data values. Same center, different variation Chap 3-13 Range Simplest measure of variation Difference between the largest and the smallest observations: Range = Xlargest – Xsmallest Example: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Range = 14 - 1 = 13 Chap 3-14 Disadvantages of the Range Ignores the way in which data are distributed 7 8 9 10 11 12 Range = 12 - 7 = 5 7 8 9 10 11 12 Range = 12 - 7 = 5 Sensitive to outliers 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5 Range = 5 - 1 = 4 1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120 Range = 120 - 1 = 119 Chap 3-15 Quartiles Quartiles split the ranked data into 4 segments with an equal number of values per segment 25% Q1 25% 25% Q2 25% Q3 The first quartile, Q1, is the value for which 25% of the observations are smaller and 75% are larger Q2 is the same as the median (50% are smaller, 50% are larger) Only 25% of the observations are greater than the third quartile Chap 3-16 Population Variance Average of squared deviations of values from the mean (Why “N-1”?) N Population variance: σ 2 Where (x μ) i1 2 i N -1 μ = population mean N = population size xi = ith value of the variable x Chap 3-17 Sample Variance Average (approximately) of squared deviations of values from the mean n Sample variance: s 2 Where (x x) i1 2 i n -1 X = arithmetic mean n = sample size Xi = ith value of the variable X Chap 3-18 Population Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data Population standard deviation: N σ 2 (x μ) i i1 N -1 Chap 3-19 Sample Standard Deviation Most commonly used measure of variation Shows variation about the mean Has the same units as the original data n Sample standard deviation: S 2 (x x ) i i1 n -1 Chap 3-20 Calculation Example: Sample Standard Deviation Sample Data (xi) : 10 12 14 n=8 s 15 17 18 18 24 Mean = x = 16 (10 X)2 (12 x)2 (14 x)2 (24 x)2 n 1 (10 16)2 (12 16)2 (14 16)2 (24 16)2 8 1 126 7 4.2426 A measure of the “average” scatter around the mean Chap 3-21 Measuring variation Small standard deviation Large standard deviation Chap 3-22 Comparing Standard Deviations Data A 11 12 13 14 15 16 17 18 19 20 21 Mean = 15.5 s = 3.338 20 21 Mean = 15.5 s = 0.926 20 21 Mean = 15.5 s = 4.570 Data B 11 12 13 14 15 16 17 18 19 Data C 11 12 13 14 15 16 17 18 19 Chap 3-23 Advantages of Variance and Standard Deviation Each value in the data set is used in the calculation Values far from the mean are given extra weight (because deviations from the mean are squared) Chap 3-24 Chebyshev’s Theorem For any population with mean μ and standard deviation σ , and k > 1 , the percentage of observations that fall within the interval [μ + kσ] Is at least 100[1 (1/k )]% 2 Chap 3-25 Chebyshev’s Theorem (continued) Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k standard deviations of the mean (for k > 1) Examples: At least within (1 - 1/12) = 0% ……..... k=1 (μ ± 1σ) (1 - 1/22) = 75% …........ k=2 (μ ± 2σ) (1 - 1/32) = 89% ………. k=3 (μ ± 3σ) Chap 3-26 The Empirical Rule If the data distribution is bell-shaped, then the interval: μ 1σ contains about 68% of the values in the population or the sample 68% μ μ 1σ Chap 3-27 The Empirical Rule μ 2σ contains about 95% of the values in the population or the sample μ 3σ contains about 99.7% of the values in the population or the sample 95% 99.7% μ 2σ μ 3σ Chap 3-28 Coefficient of Variation Measures relative variation Always in percentage (%) Shows variation relative to mean Can be used to compare two or more sets of data measured in different units s CV 100% x Chap 3-29 Comparing Coefficient of Variation Stock A: Average price last year = $50 Standard deviation = $5 s $5 CVA 100% 100% 10% $50 x Stock B: Average price last year = $100 Standard deviation = $5 s $5 CVB 100% 100% 5% $100 x Both stocks have the same standard deviation, but stock B is less variable relative to its price Chap 3-30 Using Microsoft Excel Descriptive Statistics can be obtained from Microsoft® Excel Use menu choice: tools / data analysis / descriptive statistics Enter details in dialog box Chap 3-31 Using Excel Use menu choice: tools / data analysis / descriptive statistics Chap 3-32 Using Excel (continued) Enter dialog box details Check box for summary statistics Click OK Chap 3-33 Excel output Microsoft Excel descriptive statistics output, using the house price data: House Prices: $2,000,000 500,000 300,000 100,000 100,000 Chap 3-34 Weighted Mean The weighted mean of a set of data is n w x x w i1 i i w 1x1 w 2 x 2 w n x n wi Where wi is the weight of the ith observation Use when data is already grouped into n classes, with wi values in the ith class Chap 3-35 The Sample Covariance The covariance measures the strength of the linear relationship between two variables The population covariance: N Cov (x , y) xy (x i i1 x )(y i y ) N The sample covariance: n Cov (x , y) s xy (x x)(y y) i1 i i n 1 Only concerned with the strength of the relationship No causal effect is implied Chap 3-36 Interpreting Covariance Covariance between two variables: Cov(x,y) > 0 x and y tend to move in the same direction Cov(x,y) < 0 x and y tend to move in opposite directions Cov(x,y) = 0 x and y are independent Chap 3-37 Coefficient of Correlation Measures the relative strength of the linear relationship between two variables Population correlation coefficient (rho): Cov (x , y) ρ σXσY Sample correlation coefficient (r): Cov (x , y) r sX sY Chap 3-38 Features of Correlation Coefficient, r Unit free (difference with covariance) Ranges between –1 and 1 The closer to –1, the stronger the negative linear relationship The closer to 1, the stronger the positive linear relationship The closer to 0, the weaker any positive linear relationship Chap 3-39 Scatter Plots of Data with Various Correlation Coefficients Y Y Y X X r = -1 r = -0.6 Y r=0 Y Y r = +1 X X X r = +0.3 X r=0 Chap 3-40 Using Excel to Find the Correlation Coefficient Select Tools/Data Analysis Choose Correlation from the selection menu Click OK . . . Chap 3-41 Using Excel to Find the Correlation Coefficient (continued) Input data range and select appropriate options Click OK to get output Chap 3-42 Interpreting the Result Scatter Plot of Test Scores r = 0.733 100 There is a relatively strong positive linear relationship between test score #1 and test score #2 Test #2 Score 95 90 85 80 75 70 70 75 80 85 90 95 100 Test #1 Score Students who scored high on the first test tended to score high on second test Chap 3-43 Obtaining Linear Relationships An equation can be fit to show the best linear relationship between two variables: Y = β0 + β1 X Where Y is the dependent variable and X is the independent variable (Ch 12) Chap 3-44