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Chapter 4 Numerical Descriptive Techniques 1 4.2 Measures of Central Location Usually, we focus our attention on two types of measures when describing population characteristics: Central location (e.g. average) Variability or spread The measure of central location reflects the locations of all the actual data points. 2 4.2 Measures of Central Location The measure of central location reflects the locations of all the actual data points. How? With two data points, the central location But if the third data With one data point should fall inpoint the middle on the leftthem hand-side clearly the centralappears between (in order of the midrange, it should “pull”of location is at the point to reflect the location the central location to the left. itself. both of them). 3 The Arithmetic Mean This is the most popular and useful measure of central location Sum of the observations Mean = Number of observations 4 The Arithmetic Mean Sample mean x n n ii11xxii nn Sample size Population mean N i1 x i N Population size 5 The arithmetic mean The Arithmetic Mean • Example 4.1 The reported time on the Internet of 10 adults are 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 hours. Find the mean time on the Internet. x 10 i 1 xi 10 0x1 7x2 ... 22 x10 11.0 10 • Example 4.2 Suppose the telephone bills of Example 2.1 represent the population of measurements. The population mean is 200 i1 x i x42.19 x38.45 ... x45.77 1 2 200 200 200 43.59 6 The Median The Median of a set of observations is the value that falls in the middle when the observations are arranged in order of magnitude. Example 4.3 Comment Find the median of the time on the internet Suppose only 9 adults were sampled (exclude, say, the longest time (33)) for the 10 adults of example 4.1 Even number of observations 0, 0, 5, 0, 7, 5, 8, 7, 8, 9, 12, 14,14, 22,22, 33 33 8.59,, 12, Odd number of observations 0, 0, 5, 7, 8 9, 12, 14, 22 7 The Mode The Mode of a set of observations is the value that occurs most frequently. Set of data may have one mode (or modal class), or two or more modes. The modal class For large data sets the modal class is much more relevant than a single-value mode. 8 The Mode The Mean, Median, Mode The Mode Example 4.5 Find the mode for the data in Example 4.1. Here are the data again: 0, 7, 12, 5, 33, 14, 8, 0, 9, 22 Solution • All observation except “0” occur once. There are two “0”. Thus, the mode is zero. • Is this a good measure of central location? • The value “0” does not reside at the center of this set (compare with the mean = 11.0 and the mode = 8.5). 9 Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincide If a distribution is asymmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) Mode Mean Median 10 Relationship among Mean, Median, and Mode If a distribution is symmetrical, the mean, median and mode coincide If a distribution is non symmetrical, and skewed to the left or to the right, the three measures differ. A positively skewed distribution (“skewed to the right”) A negatively skewed distribution (“skewed to the left”) Mode Mean Median Mean Mode Median 11 The Geometric Mean This is a measure of the average growth rate. Let Ri denote the the rate of return in period i (i=1,2…,n). The geometric mean of the returns R1, R2, …,Rn is the constant Rg that produces the same terminal wealth at the end of period n as do the actual returns for the n periods. 12 The Geometric Mean The Geometric Mean For the given series of rate of returns the nth period return is calculated by: If the rate of return was Rg in every period, the nth period return would be calculated by: n (1 R1 )(1 R 2 )...(1 R n ) (1 R g ) Rg is selected such that… R g n (1 R1 )(1 R2 )...(1 Rn ) 1 13 4.3 Measures of variability Measures of central location fail to tell the whole story about the distribution. A question of interest still remains unanswered: How much are the observations spread out around the mean value? 14 4.3 Measures of variability Observe two hypothetical data sets: Small variability The average value provides a good representation of the observations in the data set. This data set is now changing to... 15 4.3 Measures of variability Observe two hypothetical data sets: Small variability The average value provides a good representation of the observations in the data set. Larger variability The same average value does not provide as good representation of the observations in the data set as before. 16 The range The range of a set of observations is the difference between the largest and smallest observations. Its major advantage is the ease with which it can be But, how do all the observations spread out? computed. ? to provide Its major shortcoming?is its?failure Largest information onSmallest the dispersion of the observations observation observation between the two end points. The range cannot assistRange in answering this question 17 The Variance This measure reflects the dispersion of all the observations The variance of a population of size N x1, x2,…,xN whose mean is is defined as 2 2 N ( x ) i1 i N The variance of a sample of n observations x1, x2, …,xn whose mean is x is defined as s2 ni1( xi x)2 n 1 18 Why not use the sum of deviations? Consider two small populations: 9-10= -1 11-10= +1 8-10= -2 12-10= +2 A measure of dispersion A Can the sum of deviations agreesofwith this Be aShould good measure dispersion? The sum of deviations is observation. zero for both populations, 8 9 10 11 12 therefore, is not a good …but Themeasurements mean of both in B measure of arepopulations moredispersion. dispersed is 10... 4-10 = - 6 16-10 = +6 7-10 = -3 then those in A. B 4 Sum = 0 7 10 13 16 13-10 = +3 19 Sum = 0 The Variance Let us calculate the variance of the two populations 2 2 2 2 2 2 (8 10) (9 10) (10 10) (11 10) (12 10) A 2 5 2 2 2 2 2 2 ( 4 10) (7 10) (10 10) (13 10) (16 10) B 18 5 Why is the variance defined as the average squared deviation? Why not use the sum of squared deviations as a measure of variation instead? After all, the sum of squared deviations increases in magnitude when the variation of a data set increases!! 20 The Variance Let us calculate the sum of squared deviations for both data sets Which data set has a larger dispersion? Data set B is more dispersed around the mean A B 1 2 3 1 3 5 21 The Variance SumA = (1-2)2 +…+(1-2)2 +(3-2)2 +… +(3-2)2= 10 SumB = (1-3)2 + (5-3)2 = 8 SumA > SumB. This is inconsistent with the observation that set B is more dispersed. A B 1 2 3 1 3 5 22 The Variance However, when calculated on “per observation” basis (variance), the data set dispersions are properly ranked. A2 = SumA/N = 10/5 = 2 B2 = SumB/N = 8/2 = 4 A B 1 2 3 1 3 5 23 The Variance Example 4.7 The following sample consists of the number of jobs six students applied for: 17, 15, 23, 7, 9, 13. Finds its mean and variance Solution x i61 xi ni1( x i 6 17 15 23 7 9 13 84 14 jobs 6 6 2 x) 1 s (17 14)2 (15 14)2 ...(13 14)2 n 1 6 1 33.2 jobs 2 2 24 The Variance – Shortcut method n 2 n 1 ( x ) 2 2 i1 i s x i n 1 i1 n 2 1 2 17 15 ... 13 2 2 17 15 ... 13 6 1 6 33.2 jobs 2 25 Standard Deviation The standard deviation of a set of observations is the square root of the variance . Sample standard deviation : s s 2 Population standard deviation : 2 26 Standard Deviation Example 4.8 To examine the consistency of shots for a new innovative golf club, a golfer was asked to hit 150 shots, 75 with a currently used (7-iron) club, and 75 with the new club. The distances were recorded. Which 7-iron is more consistent? 27 The Standard Deviation Standard Deviation Example 4.8 – solution Excel printout, from the “Descriptive Statistics” submenu. The innovation club is more consistent, and because the means are close, is considered a better club Current Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Innovation 150.5467 0.668815 151 150 5.792104 33.54847 0.12674 -0.42989 28 134 162 11291 75 Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 150.1467 0.357011 150 149 3.091808 9.559279 -0.88542 0.177338 12 144 156 11261 75 28 Interpreting Standard Deviation The standard deviation can be used to compare the variability of several distributions make a statement about the general shape of a distribution. The empirical rule: If a sample of observations has a mound-shaped distribution, the interval ( x s, x s) contains approximat ely 68% of the measuremen ts ( x 2s, x 2s) contains approximat ely 95% of the measuremen ts ( x 3s, x 3s) contains approximat ely 99.7% of the measuremen ts 29 Interpreting Standard Deviation Example 4.9 A statistics practitioner wants to describe the way returns on investment are distributed. The mean return = 10% The standard deviation of the return = 8% The histogram is bell shaped. 30 Interpreting Standard Deviation Example 4.9 – solution The empirical rule can be applied (bell shaped histogram) Describing the return distribution Approximately 68% of the returns lie between 2% and 18% [10 – 1(8), 10 + 1(8)] Approximately 95% of the returns lie between -6% and 26% [10 – 2(8), 10 + 2(8)] Approximately 99.7% of the returns lie between -14% and 34% [10 – 3(8), 10 + 3(8)] 31 The Chebysheff’s Theorem The proportion of observations in any sample that lie within k standard deviations of the mean is at least 1-1/k2 for k > 1. This theorem is valid for any set of measurements (sample, population) of any shape!! K Interval Chebysheff Empirical Rule 1 2 3 x s, x s x 2 s, x 2 s x 3 s, x 3 s at least 0% at least 75% at least 89% (1-1/12) (1-1/22) (1-1/32) approximately 68% approximately 95% approximately 99.7% 32 The Chebysheff’s Theorem Example 4.10 The annual salaries of the employees of a chain of computer stores produced a positively skewed histogram. The mean and standard deviation are $28,000 and $3,000,respectively. What can you say about the salaries at this chain? Solution At least 75% of the salaries lie between $22,000 and $34,000 28000 – 2(3000) 28000 + 2(3000) At least 88.9% of the salaries lie between $$19,000 and $37,000 28000 – 3(3000) 28000 + 3(3000) 33 The Coefficient of Variation The coefficient of variation of a set of measurements is the standard deviation divided by the mean value. s Sample coefficient of variation: cv x P opulationcoefficient of variation: CV This coefficient provides a proportionate measure of variation. A standard deviation of 10 may be perceived large when the mean value is 100, but only moderately large when the mean value is 500 34 4.4 Measures of Relative Standing and Box Plots Percentile The pth percentile of a set of measurements is the value for which • p percent of the observations are less than that value • 100(1-p) percent of all the observations are greater than that value. Example • Suppose your score is the 60% percentile of a SAT test. Then 40% 60% of all the scores lie here Your score 35 Quartiles Commonly used percentiles First (lower)decile First (lower) quartile, Q1, Second (middle)quartile,Q2, Third quartile, Q3, Ninth (upper)decile = 10th percentile = 25th percentile = 50th percentile = 75th percentile = 90th percentile 36 Quartiles Example Find the quartiles of the following set of measurements 7, 8, 12, 17, 29, 18, 4, 27, 30, 2, 4, 10, 21, 5, 8 37 Quartiles Solution Sort the observations 2, 4, 4, 5, 7, 8, 10, 12, 17, 18, 18, 21, 27, 29, 30 The first quartile 15 observations At most (.25)(15) = 3.75 observations should appear below the first quartile. Check the first 3 observations on the left hand side. At most (.75)(15)=11.25 observations should appear above the first quartile. Check 11 observations on the right hand side. Comment:If the number of observations is even, two observations remain unchecked. In this case choose the midpoint between these two observations. 38 Location of Percentiles Find the location of any percentile using the formula P LP (n 1) 100 where LP is the location of the P th percentile Example 4.11 Calculate the 25th, 50th, and 75th percentile of the data in Example 4.1 39 Location of Percentiles Example 4.11 – solution After sorting the data we have 0, 0, 5, 7, 8, 9, 22, 33. L 25 (10 1) 25 2.75 100 Values 0 0 Location 2 Location 1 3.75 5 2.75 3 Location 3 The 2.75th location Translates to the value (.75)(5 – 0) = 3.75 40 Location of Percentiles Example 4.11 – solution continued 50 L 50 (10 1) 5.5 100 The 50th percentile is halfway between the fifth and sixth observations (in the middle between 8 and 9), that is 8.5. 41 Location of Percentiles Example 4.11 – solution continued 75 L 75 (10 1) 8.25 100 The 75th percentile is one quarter of the distance between the eighth and ninth observation that is 14+.25(22 – 14) = 16. Eighth observation Ninth observation 42 Quartiles and Variability Quartiles can provide an idea about the shape of a histogram Q1 Q2 Positively skewed histogram Q3 Q1 Q2 Q3 Negatively skewed histogram 43 Interquartile Range This is a measure of the spread of the middle 50% of the observations Large value indicates a large spread of the observations Interquartile range = Q3 – Q1 44 Box Plot This is a pictorial display that provides the main descriptive measures of the data set: • • • • • L - the largest observation Q3 - The upper quartile Q2 - The median Q1 - The lower quartile S - The smallest observation 1.5(Q3 – Q1) S Whisker 1.5(Q3 – Q1) Q1 Q2 Q 3 Whisker L 45 Box Plot Example 4.14 (Xm02-01) Bills 42.19 38.45 29.23 89.35 118.04 110.46 . Smallest =. 0 . Q1 = 9.275 Median = 26.905 Q3 = 84.9425 Largest = 119.63 IQR = 75.6675 Outliers = () Left hand boundary = 9.275–1.5(IQR)= -104.226 Right hand boundary=84.9425+ 1.5(IQR)=198.4438 -104.226 0 9.275 84.9425 119.63 26.905 198.4438 No outliers are found 46 Box Plot Additional Example - GMAT scores Create a box plot for the data regarding the GMAT scores of 200 applicants (see GMAT.XLS) GMAT 512 531 461 515 . . . Smallest = 449 Q1 = 512 Median = 537 Q3 = 575 Largest = 788 IQR = 63 Outliers = (788, 788, 766, 763, 756, 719, 712, 707, 703, 694, 690, 675, ) 417.5 449 512-1.5(IQR) 512 537 575 669.5 575+1.5(IQR) 47 788 Box Plot GMAT - continued Q1 512 449 25% Q2 537 50% Q3 575 669.5 25% Interpreting the box plot results • The scores range from 449 to 788. • About half the scores are smaller than 537, and about half are larger than 537. • About half the scores lie between 512 and 575. • About a quarter lies below 512 and a quarter above 575. 48 Box Plot GMAT - continued The histogram is positively skewed Q1 512 449 25% Q2 537 50% Q3 575 669.5 25% 50% 25% 25% 49 Box Plot Example 4.15 (Xm04-15) A study was organized to compare the quality of service in 5 drive through restaurants. Interpret the results Example 4.15 – solution Minitab box plot 50 Box Plot Jack in the Box5 Jack in the box is the slowest in service Hardee’s Hardee’s service time variability is the largest C7 McDonalds 4 3 Wendy’s 2 Popeyes 1 Wendy’s service time appears to be the shortest and most consistent. 100 300 200 C6 51 Box Plot Times are symmetric Jack in the Box5 Jack in the box is the slowest in service Hardee’s Hardee’s service time variability is the largest C7 McDonalds 4 3 Wendy’s 2 Popeyes 1 Wendy’s service time appears to be the shortest and most consistent. 100 300 200 C6 Times are positively skewed 52 4.5 Measures of Linear Relationship The covariance and the coefficient of correlation are used to measure the direction and strength of the linear relationship between two variables. Covariance - is there any pattern to the way two variables move together? Coefficient of correlation - how strong is the linear relationship between two variables 53 Covariance Populationcovariance COV(X,Y) (x i x )( y i y ) N x (y) is the population mean of the variable X (Y). N is the population size. (xi x)(y i y ) Sample covariance cov(x, y) n-1 x (y) is the sample mean of the variable X (Y). n is the sample size. 54 Covariance Compare the following three sets xi yi (x – x) (y – y) (x – x)(y – y) 2 6 7 13 20 27 -3 1 2 x=5 y =20 xi yi (x – x) (y – y) (x – x)(y – y) 2 6 7 27 20 13 -3 1 2 x=5 y =20 -7 0 7 21 0 14 Cov(x,y)=17.5 7 0 -7 -21 0 -14 Cov(x,y)=-17.5 xi yi 2 6 7 20 27 13 Cov(x,y) = -3.5 x=5 y =20 55 Covariance If the two variables move in the same direction, (both increase or both decrease), the covariance is a large positive number. If the two variables move in opposite directions, (one increases when the other one decreases), the covariance is a large negative number. If the two variables are unrelated, the covariance will be close to zero. 56 The coefficient of correlation Population coefficient of correlation COV( X, Y) x y Sample coefficient of correlation cov(X, Y) r sx sy This coefficient answers the question: How strong is the association between X and Y. 57 The coefficient of correlation +1 Strong positive linear relationship COV(X,Y)>0 or r = or 0 No linear relationship -1 Strong negative linear relationship COV(X,Y)=0 COV(X,Y)<0 58 The coefficient of correlation If the two variables are very strongly positively related, the coefficient value is close to +1 (strong positive linear relationship). If the two variables are very strongly negatively related, the coefficient value is close to -1 (strong negative linear relationship). No straight line relationship is indicated by a coefficient close to zero. 59 The coefficient of correlation and the covariance – Example 4.16 Compute the covariance and the coefficient of correlation to measure how GMAT scores and GPA in an MBA program are related to one another. Solution We believe GMAT affects GPA. Thus • GMAT is labeled X • GPA is labeled Y 60 The coefficient of correlation and the covariance – Example 4.16 Student 1 x 599 y 9.6 x2 y2 xy 358801 92.16 5750.4 2 689 8.8 474721 77.44 6063.2 cov(x,y)=(1/12-1)[67,559.2-(7587)(106.4)/12]=26.16 3 584 7.4 341056 54.76 4321.6 Sx = {(1/12-1)[4,817,755-(7587)2/12)]}.5=43.56 4 100 6310 Sy =…………………………………………………. similar631 to Sx =10 1.12 398161 593 xSy = 26.16/(43.56)(1.12) 8.8 351649 77.44 r = 11 cov(x,y)/S = .5362 5218.4 12 683 8 466489 64 5464 Total 7,587 106.4 4,817,755 957.2 67,559.2 Shortcut Formulas cov(x, y ) xi y i 1 xi yi n 1 n 2 1 x 2 s2 x i n 1 n 61 The coefficient of correlation and the covariance – Example 4.16 – Excel Use the Covariance option in Data Analysis If your version of Excel returns the population covariance and variances, multiply each one by n/n-1 to obtain the corresponding sample values. Use the Correlation option to produce the correlation matrix. Variance-Covariance Matrix Population values GPA GPA 1.15 GMAT 23.98 GMAT 1739.52 Sample values 12 GPA 12-1 GMAT GPA GMAT 1.25 26.16 1897.66 62 The coefficient of correlation and the covariance – Example 4.16 – Excel Interpretation The covariance (26.16) indicates that GMAT score and performance in the MBA program are positively related. The coefficient of correlation (.5365) indicates that there is a moderately strong positive linear relationship between GMAT and MBA GPA. 63 The Least Squares Method We are seeking a line that best fits the data when two variables are (presumably) related to one another. We define “best fit line” as a line for which the sum of squared differences between it and the data points is n minimized. 2 Minimize ( y i yˆ i ) i1 The actual y value of point i The y value of point i calculated from the equation yˆ i b 0 b1x i 64 The least Squares Method Y Errors Errors X Different lines generate different errors, thus different sum of squares of errors. There is a line that minimizes the sum of squared errors 65 The least Squares Method The coefficients b0 and b1 of the line that minimizes the sum of squares of errors are calculated from the data. n b1 cov(x, y ) s x2 ( x x )( y y) i i i 1 , n ( xi x ) 2 i 1 b0 y b1 x n where y y i 1 n n i and x x i 1 n i 66 The Least Squares Method Example 4.17 b1 x Find the least squares line for Example 4.16 (Xm04-16.xls) cov(x, y ) s x2 xi n y y 26.16 .0138 1897.2 Scatter Diagram 12 7,587 632.25 12 y = 0.1496 + 0.0138x 10 8 6 106.4 500 8.87 n 12 b0 y b1 x 8.87 (.0138)(632.25) .145 i 600 700 800 67