Chapter 3 Describing Data Using Numerical Measures

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1 Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The Scentfc Method 1. Formulate a theory 2. Collect data to test the theory 3. Analyze the results 4. Interpret the results, and make decsons Fall 2006 Fundamentals of Busness Statstcs 2

2 Chapter 3 Student Lecture Notes 3-2 Summary Measures Descrbng Data Numercally Center and Locaton Mean Medan Mode Weghted Mean Other Measures of Locaton Percentles Quartles Varaton Range Interquartle Range Varance Standard Devaton Coeffcent of Varaton Fall 2006 Fundamentals of Busness Statstcs 3 Measures of Center and Locaton Overvew Center and Locaton x = μ = Mean Medan Mode Weghted Mean n = 1 N n = 1 N x x Fall 2006 Fundamentals of Busness Statstcs 4 X μ W W = = w x w w w x

3 Chapter 3 Student Lecture Notes 3-3 Mean (Arthmetc Average) The mean of a set of quanttatve data, X 1,X 2,,X n, s equal to the sum of the measurements dvded by the number of n = Sample Sze measurements. n Sample mean x = 1 x1 + x2 + L + xn x = = n n Populaton mean μ = N N x N = Populaton Sze + x Fall 2006 Fundamentals of Busness Statstcs 5 x + N = L = + x N Example Fnd the mean of the followng 5 numbers 5, 3, 8, 5, 6 Fall 2006 Fundamentals of Busness Statstcs 6

4 Chapter 3 Student Lecture Notes 3-4 Mean (Arthmetc Average) Affected by extreme values (outlers) For non-symmetrcal dstrbutons, the mean s located away from the concentraton of tems. (contnued) Mean = = = Mean = = = Fall 2006 Fundamentals of Busness Statstcs 7 YDI 5.1 and 5.2 Km s test scores are 7, 98, 25, 19, and 26. Calculate Km s mean test score. Does the mean do a good job of capturng Km s test scores? The mean score for 3 students s 54, and the mean score for 4 dfferent students s 76. What s the mean score for all 7 students? Fall 2006 Fundamentals of Busness Statstcs 8

5 Chapter 3 Student Lecture Notes 3-5 Medan The medan Md of a data set s the mddle number when the measurements are arranged n ascendng (or descendng) order. Calculatng the Medan: 1. Arrange the n measurements from the smallest to the largest. 2. If n s odd, the medan s the mddle number. 3. If n s even, the medan s the mean (average) of the mddle two numbers. Example: Calculate the medan of 5, 3, 8, 5, 6 Fall 2006 Fundamentals of Busness Statstcs 9 Medan Not affected by extreme values Medan = 3 Medan = 3 In an ordered array, the medan s the mddle number. What f the values n the data set are repeated? Fall 2006 Fundamentals of Busness Statstcs 10

6 Chapter 3 Student Lecture Notes 3-6 Mode Mode s the measurement that occurs wth the greatest frequency Example: 5, 3, 8, 6, 6 The modal class n a frequency dstrbuton wth equal class ntervals s the class wth the largest frequency. If the frequency polygon has only a sngle peak, t s sad to be unmodal. If the frequency polygon has two peaks, t s sad to be bmodal. Fall 2006 Fundamentals of Busness Statstcs 11 Revew Example Fve houses on a hll by the beach House Prces: $2,000 K $2,000, , , , ,000 $300 K $500 K $100 K $100 K Fall 2006 Fundamentals of Busness Statstcs 12

7 Chapter 3 Student Lecture Notes 3-7 Summary Statstcs House Prces: $2,000, , , , ,000 Sum 3,000,000 Mean: Medan: Mode: Fall 2006 Fundamentals of Busness Statstcs 13 Whch measure of locaton s the best? Mean s generally used, unless extreme values (outlers) exst Then medan s often used, snce the medan s not senstve to extreme values. Example: Medan home prces may be reported for a regon less senstve to outlers Fall 2006 Fundamentals of Busness Statstcs 14

8 Chapter 3 Student Lecture Notes 3-8 Shape of a Dstrbuton Descrbes how data s dstrbuted Symmetrc or skewed Left-Skewed Symmetrc Rght-Skewed Mean < Medan < Mode (Longer tal extends to left) Mean = Medan = Mode Mode < Medan < Mean (Longer tal extends to rght) Fall 2006 Fundamentals of Busness Statstcs 15 Other Locaton Measures Other Measures of Locaton Percentles Quartles Let x 1, x 2,, x n be a set of n measurements arranged n ncreasng (or decreasng) order. The pth percentle s a number x such that p% of the measurements fall below the pth percentle. 1 st quartle = 25 th percentle 2 nd quartle = 50 th percentle = medan 3 rd quartle = 75 th percentle Fall 2006 Fundamentals of Busness Statstcs 16

9 Chapter 3 Student Lecture Notes 3-9 Quartles Quartles splt the ranked data nto 4 equal groups 25% 25% 25% 25% Q1 Q2 Q3 Example: Fnd the frst quartle Sample Data n Ordered Array: Fall 2006 Fundamentals of Busness Statstcs 17 Box and Whsker Plot A Graphcal dsplay of data usng 5-number summary: Mnmum -- Q1 -- Medan -- Q3 -- Maxmum Example: 25% 25% 25% 25% Mnmum 1st Medan 3rd Maxmum Mnmum Quartle 1st Medan 3rd Quartle Maxmum Quartle Quartle Fall 2006 Fundamentals of Busness Statstcs 18

10 Chapter 3 Student Lecture Notes 3-10 Shape of Box and Whsker Plots The Box and central lne are centered between the endponts f data s symmetrc around the medan A Box and Whsker plot can be shown n ether vertcal or horzontal format Fall 2006 Fundamentals of Busness Statstcs 19 Dstrbuton Shape and Box and Whsker Plot Left-Skewed Symmetrc Rght-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3 Fall 2006 Fundamentals of Busness Statstcs 20

11 Chapter 3 Student Lecture Notes 3-11 Box-and-Whsker Plot Example Below s a Box-and-Whsker plot for the followng data: Mn Q1 Q2 Q3 Max Ths data s very rght skewed, as the plot depcts Fall 2006 Fundamentals of Busness Statstcs 21 Measures of Varaton Varaton Range Interquartle Range Varance Standard Devaton Coeffcent of Varaton Populaton Varance Populaton Standard Devaton Sample Varance Sample Standard Devaton Fall 2006 Fundamentals of Busness Statstcs 22

12 Chapter 3 Student Lecture Notes 3-12 Varaton Measures of varaton gve nformaton on the spread or varablty of the data values. Same center, dfferent varaton Fall 2006 Fundamentals of Busness Statstcs 23 Range Smplest measure of varaton Dfference between the largest and the smallest observatons: Range = x maxmum x mnmum Example: Fall 2006 Fundamentals of Busness Statstcs 24

13 Chapter 3 Student Lecture Notes 3-13 Dsadvantages of the Range Consders only extreme values Wth a frequency dstrbuton, the range of orgnal data cannot be determned exactly. Fall 2006 Fundamentals of Busness Statstcs 25 Interquartle Range Can elmnate some outler problems by usng the nterquartle range Elmnate some hgh-and low-valued observatons and calculate the range from the remanng values. Interquartle range = 3 rd quartle 1 st quartle Fall 2006 Fundamentals of Busness Statstcs 26

14 Chapter 3 Student Lecture Notes 3-14 Interquartle Range Example: X mnmum Q1 Medan (Q2) Q3 X maxmum 25% 25% 25% 25% Fall 2006 Fundamentals of Busness Statstcs 27 YDI 5.8 Consder three samplng desgns to estmate the true populaton mean (the total sample sze s the same for all three desgns): 1. smple random samplng 2. stratfed random samplng takng equal sample szes from the two strata 3. stratfed random samplng takng most unts from one strata, but samplng a few unts from the other strata For whch populaton wll desgn (1) and (2) be comparably effectve? For whch populaton wll desgn (2) be the best? For whch populaton wll desgn (3) be the best? Whch stratum n ths populaton should have the hgher sample sze? Fall 2006 Fundamentals of Busness Statstcs 28

15 Chapter 3 Student Lecture Notes 3-15 Varance Average of squared devatons of values from the mean n 2 Sample varance: (x x) 2 = 1 Example: 5, 3, 8, 5, 6 s = n -1 Fall 2006 Fundamentals of Busness Statstcs 29 Varance The greater the varablty of the values n a data set, the greater the varance s. If there s no varablty of the values that s, f all are equal and hence all are equal to the mean then s 2 = 0. The varance s 2 s expressed n unts that are the square of the unts of measure of the characterstc under study. Often, t s desrable to return to the orgnal unts of measure whch s provded by the standard devaton. The postve square root of the varance s called the sample standard devaton and s denoted by s s = Fall 2006 Fundamentals of Busness Statstcs 30 s 2

16 Chapter 3 Student Lecture Notes 3-16 Populaton Varance Populaton varance: σ 2 = N = 1 (x N μ) Populaton Standard Devaton 2 σ = N = 1 (x N μ) 2 Fall 2006 Fundamentals of Busness Statstcs 31 Comparng Standard Devatons Data A Mean = 15.5 s = Data B Data C Mean = 15.5 s =.9258 Mean = 15.5 s = 4.57 Fall 2006 Fundamentals of Busness Statstcs 32

17 Chapter 3 Student Lecture Notes 3-17 Coeffcent of Varaton Measures relatve varaton Always n percentage (%) Shows varaton relatve to mean Is used to compare two or more sets of data measured n dfferent unts Populaton Sample CV = σ μ 100% CV = s x 100% Fall 2006 Fundamentals of Busness Statstcs 33 YDI Stock A: Average prce last year = $50 Standard devaton = $5 Stock B: Average prce last year = $100 Standard devaton = $5 Fall 2006 Fundamentals of Busness Statstcs 34

18 Chapter 3 Student Lecture Notes 3-18 Lnear Transformatons The data on the number of chldren n a neghborhood of 10 households s as follows: 2, 3, 0, 2, 1, 0, 3, 0, 1, If there are two adults n each of the above households, what s the mean and standard devaton of the number of people (chldren + adults) lvng n each household? 2. If each chld gets an allowance of $3, what s the mean and standard devaton of the amount of allowance n each household n ths neghborhood? X = 1.6 s = 1.43 Fall 2006 Fundamentals of Busness Statstcs 35 Defntons Let X be the varable representng a set of values, and s x and X be the standard devaton and mean of X, respectvely. Let Y = ax + b, where a and b are constants. Then, the mean and standard devaton of Y are gven by Y = ax + b s = a Y s X Fall 2006 Fundamentals of Busness Statstcs 36

19 Chapter 3 Student Lecture Notes 3-19 Standardzed Data Values A standardzed data value refers to the number of standard devatons a value s from the mean Standardzed data values are sometmes referred to as z-scores Fall 2006 Fundamentals of Busness Statstcs 37 Standardzed Values A standardzed varable Z has a mean of 0 and a standard devaton of 1. where: z = x = orgnal data value x= sample mean x x s s = sample standard devaton z = standard score (number of standard devatons x s from the mean) Fall 2006 Fundamentals of Busness Statstcs 38

20 Chapter 3 Student Lecture Notes 3-20 YDI Durng a recent week n Europe, the temperature X n Celsus was as follows: Based on ths X = 40 s X = 1.14 Day 40 Calculate the mean and standard devaton n Fahrenhet. Calculate the standardzed score. X M T 41 W 39 H 41 F 41 S 40 S 38 Fall 2006 Fundamentals of Busness Statstcs 39

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