Chapter 3 Student Lecture Notes 3-1

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1 Chapter 3 Student Lecture otes 3-1 Busness Statstcs: A Decson-Makng Approach 6 th Edton Chapter 3 Descrbng Data Usng umercal Measures 005 Prentce-Hall, Inc. Chap 3-1 Chapter Goals After completng ths chapter, you should be able to: Compute and nterpret the mean, medan, and mode for a set of data Compute the range, varance, and standard devaton and know what these values mean Construct and nterpret a box and whskers plot Compute and explan the coeffcent of varaton and z scores Use numercal measures along wth graphs, charts, and tables to descrbe data 005 Prentce-Hall, Inc. Chap 3- Chapter Topcs Measures of Center and Locaton Mean, medan, mode, geometrc mean, mdrange Other measures of Locaton Weghted mean, percentles, quartles Measures of Varaton Range, nterquartle range, varance and standard devaton, coeffcent of varaton 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

2 Chapter 3 Student Lecture otes 3- Summary Measures Descrbng Data umercally Center and Locaton Mean Medan Mode Weghted Mean Other Measures of Locaton Percentles Quartles Varaton Range Interquartle Range Varance Standard Devaton Coeffcent of Varaton 005 Prentce-Hall, Inc. Chap 3-4 Measures of Center and Locaton Overvew Center and Locaton x = µ = Mean Medan Mode Weghted Mean n n x = 1 x = Prentce-Hall, Inc. Chap 3-5 X µ W W = = w x w w w x Mean (Arthmetc Average) The Mean s the arthmetc average of data values Sample mean n = Sample Sze n x = 1 x1 + x + L + xn x = = n n Populaton mean µ = = Populaton Sze x x1 + x + = = 1 L + x 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

3 Chapter 3 Student Lecture otes 3-3 Mean (Arthmetc Average) The most common measure of central tendency (contnued) Mean = sum of values dvded by the number of values Affected by extreme values (outlers) Mean = = = Mean = = = Prentce-Hall, Inc. Chap 3-7 Medan ot affected by extreme values Medan = 3 Medan = 3 In an ordered array, the medan s the mddle number If n or s odd, the medan s the mddle number If n or s even, the medan s the average of the two mddle numbers 005 Prentce-Hall, Inc. Chap 3-8 Mode A measure of central tendency Value that occurs most often ot affected by extreme values Used for ether numercal or categorcal data There may may be no mode There may be several modes Mode = o Mode 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

4 Chapter 3 Student Lecture otes 3-4 Weghted Mean Used when values are grouped by frequency or relatve mportance Example: Sample of 6 Repar Projects Days to Frequency Complete Weghted Mean Days to Complete: wx (4 5) + (1 6) + (8 7) + ( 8) XW = = w = = days 005 Prentce-Hall, Inc. Chap 3-10 Revew Example Fve houses on a hll by the beach House Prces: $,000 K $,000, , , , ,000 $300 K $500 K $100 K $100 K 005 Prentce-Hall, Inc. Chap 3-11 Summary Statstcs House Prces: $,000, , , , ,000 Sum 3,000,000 Mean: ($3,000,000/5) = $600,000 Medan: mddle value of ranked data = $300,000 Mode: most frequent value = $100, Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

5 Chapter 3 Student Lecture otes 3-5 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 005 Prentce-Hall, Inc. Chap 3-13 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) 005 Prentce-Hall, Inc. Chap 3-14 Other Locaton Measures Other Measures of Locaton Percentles Quartles The p th percentle n a data array: p% are less than or equal to ths value (100 p)% are greater than or equal to ths value (where 0 p 100) 1 st quartle = 5 th percentle nd quartle = 50 th percentle = medan 3 rd quartle = 75 th percentle 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

6 Chapter 3 Student Lecture otes 3-6 Percentles The p th percentle n an ordered array of n values s the value n th poston, where p = (n + 1) 100 Example: The 60 th percentle n an ordered array of 19 values s the value n 1 th poston: p 60 = (n + 1) = (19 + 1) = Prentce-Hall, Inc. Chap 3-16 Quartles Quartles splt the ranked data nto 4 equal groups 5% 5% 5% 5% Q1 Q Q3 Example: Fnd the frst quartle Sample Data n Ordered Array: (n = 9) Q1 = 5 th percentle, so fnd the 5 (9+1) =.5 poston 100 so use the value half way between the nd and 3 rd values, so Q1 = Prentce-Hall, Inc. Chap 3-17 Box and Whsker Plot A Graphcal dsplay of data usng 5-number summary: Mnmum -- Q1 -- Medan -- Q3 -- Maxmum Example: 5% 5% 5% 5% Mnmum 1st Medan 3rd Maxmum Mnmum Quartle 1st Medan 3rd Quartle Maxmum Quartle Quartle 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

7 Chapter 3 Student Lecture otes 3-7 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 005 Prentce-Hall, Inc. Chap 3-19 Dstrbuton Shape and Box and Whsker Plot Left-Skewed Symmetrc Rght-Skewed Q1 Q Q3 Q1 Q Q3 Q1 Q Q3 005 Prentce-Hall, Inc. Chap 3-0 Box-and-Whsker Plot Example Below s a Box-and-Whsker plot for the followng data: Mn Q1 Q Q3 Max Ths data s very rght skewed, as the plot depcts 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

8 Chapter 3 Student Lecture otes 3-8 Measures of Varaton Varaton Range Interquartle Range Varance Standard Devaton Coeffcent of Varaton Populaton Varance Populaton Standard Devaton Sample Varance Sample Standard Devaton 005 Prentce-Hall, Inc. Chap 3- Varaton Measures of varaton gve nformaton on the spread or varablty of the data values. Same center, dfferent varaton 005 Prentce-Hall, Inc. Chap 3-3 Range Smplest measure of varaton Dfference between the largest and the smallest observatons: Range = x maxmum x mnmum Example: Range = 14-1 = Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

9 Chapter 3 Student Lecture otes 3-9 Dsadvantages of the Range Ignores the way n whch data are dstrbuted Range = 1-7 = Range = 1-7 = 5 Senstve to outlers 1,1,1,1,1,1,1,1,1,1,1,,,,,,,,,3,3,3,3,4,5 Range = 5-1 = 4 1,1,1,1,1,1,1,1,1,1,1,,,,,,,,,3,3,3,3,4,10 Range = 10-1 = Prentce-Hall, Inc. Chap 3-5 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 005 Prentce-Hall, Inc. Chap 3-6 Interquartle Range Example: X mnmum Q1 Medan (Q) Q3 X maxmum 5% 5% 5% 5% Interquartle range = = Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

10 Chapter 3 Student Lecture otes 3-10 Varance Average of squared devatons of values from the mean Sample varance: n (x x) = 1 s = n -1 Populaton varance: (x µ) 005 Prentce-Hall, Inc. Chap 3-8 σ = = 1 Standard Devaton Most commonly used measure of varaton Shows varaton about the mean Has the same unts as the orgnal data Sample standard devaton: s = n = 1 (x x) n -1 Populaton standard devaton: σ = (x µ) 005 Prentce-Hall, Inc. Chap 3-9 = 1 Calculaton Example: Sample Standard Devaton Sample Data (X ) : n = 8 Mean = x = 16 s = = (10 x ) + (1 x ) + (14 x ) + L + (4 x ) n 1 (10 16) + (1 16) + (14 16) L + (4 16) = 16 7 = Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

11 Chapter 3 Student Lecture otes 3-11 Comparng Standard Devatons Data A Data B Data C Mean = 15.5 s = Mean = 15.5 s =.958 Mean = 15.5 s = Prentce-Hall, Inc. Chap 3-31 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 CV Populaton = σ µ 100% CV Sample s = 100% x 005 Prentce-Hall, Inc. Chap 3-3 Stock A: Comparng Coeffcent of Varaton Average prce last year = $50 Standard devaton = $5 s $5 CV A = 100% = 100% = 10% x $50 Stock B: Average prce last year = $100 Standard devaton = $5 s $5 CV B = 100% = 100% = 5% x $100 Both stocks have the same standard devaton, but stock B s less varable relatve to ts prce 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

12 Chapter 3 Student Lecture otes 3-1 The Emprcal Rule If the data dstrbuton s bell-shaped, then the nterval: µ ± 1σ contans about 68% of the values n the populaton or the sample X 68% µ µ ±1σ 005 Prentce-Hall, Inc. Chap 3-34 The Emprcal Rule µ ± σ contans about 95% of the values n the populaton or the sample µ ± 3σ contans about 99.7% of the values n the populaton or the sample 95% µ ± σ 99.7% µ ± 3σ 005 Prentce-Hall, Inc. Chap 3-35 Tchebysheff s Theorem Regardless of how the data are dstrbuted, at least (1-1/k ) of the values wll fall wthn k standard devatons of the mean Examples: At least wthn (1-1/1 ) = 0%... k=1 (µ ± 1σ) (1-1/ ) = 75%... k= (µ ± σ) (1-1/3 ) = 89%. k=3 (µ ± 3σ) 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

13 Chapter 3 Student Lecture otes 3-13 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 005 Prentce-Hall, Inc. Chap 3-37 Standardzed Populaton Values where: x µ z = σ x = orgnal data value µ = populaton mean σ = populaton standard devaton z = standard score (number of standard devatons x s from µ) 005 Prentce-Hall, Inc. Chap 3-38 Standardzed Sample Values where: x x z = s x = orgnal data value x = sample mean s = sample standard devaton z = standard score (number of standard devatons x s from µ) 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

14 Chapter 3 Student Lecture otes 3-14 Usng Mcrosoft Excel Descrptve Statstcs are easy to obtan from Mcrosoft Excel Use menu choce: tools / data analyss / descrptve statstcs Enter detals n dalog box 005 Prentce-Hall, Inc. Chap 3-40 Usng Excel Use menu choce: tools / data analyss / descrptve statstcs 005 Prentce-Hall, Inc. Chap 3-41 Usng Excel (contnued) Enter dalog box detals Check box for summary statstcs Clck OK 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.

15 Chapter 3 Student Lecture otes 3-15 Excel output Mcrosoft Excel descrptve statstcs output, usng the house prce data: House Prces: $,000, , , , , Prentce-Hall, Inc. Chap 3-43 Chapter Summary Descrbed measures of center and locaton Mean, medan, mode, geometrc mean, mdrange Dscussed percentles and quartles Descrbed measure of varaton Range, nterquartle range, varance, standard devaton, coeffcent of varaton Created Box and Whsker Plots 005 Prentce-Hall, Inc. Chap 3-44 Chapter Summary (contnued) Illustrated dstrbuton shapes Symmetrc, skewed Dscussed Tchebysheff s Theorem Calculated standardzed data values 005 Prentce-Hall, Inc. Chap Prentce-Hall, Inc.