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2 Contents 2.1 Organizing Categorical Variables. 2.2 Organizing Numerical Variables Classes and Excel Bins Stacked and Unstacked Data 2.3 Visualizing Categorical Variables 2.4 Visualizing Numerical Variables 2.5 Visualizing Two Numerical Variables!(4#)3%*,0*=>&83*,

3 Objectives!To construct tables and charts for categorical data!to construct tables and charts for numerical data!the principles of properly presenting graphs!to organize and analyze many variables!(4#)3%*,0*=>&83*c

4 How to proceed with This Chapter E For Categorical Variables:! Summary table, contingency table(in Section 2.1)! Bar chart, pie chart, Pareto chart, side-by-side bar chart(in Section 2.3) E For Numerical Variables:! Ordered array, frequency distribution, relative frequency distribution, percentage distribution, cumulative percentage distribution (in Section 2.2)! Stem-and-leaf display, histogram, polygon, cumulative percentage polygon(in Section 2.4) E For Two Numerical Variables:! Scatter plot, time-series plot (in Section 2.5)!(4#)3%*,0*=>&83*D

5 2.1 Organizing Categorical Variables You organize categorical variables by tallying the values of a variable by categories and placing the results in tables. Typically, you construct a contingency table to organize the data for a single categorical variable and you construct a contingency table to organize the data from two or more categorical variables.!(4#)3%*,0*=>&83*/

6 Categorical Data Are Organized By Utilizing Tables!#%&*+',-#.( /#%# 0#..1,2*(/#%# G!?AH 72&(!#%&*+',-#.( 8#',#6.& 3455#'1( 0#6.& 09+(!#%&*+',-#.( 8#',#6.&:!+2%,2*&2-1( 0#6.&!(4#)3%*,0*=>&83*F

7 Summary Table G!?AH A summary table tallies the values as frequencies or percentages for each category. A summary table helps you see the differences among the categories by displaying the frequency, amount, or percentage of items in a set of categories in a separate column. ;#,2(<&#:+2(=+42*(>?4.%:(3"+$(72.,2& Reason For Shopping Online? Percent Better Prices 37% Avoiding holiday crowds or hassles 29% Convenience 18% Better selection 13% Ships directly 3% ="9%:3J*G4)4*3K)%4:)38*468*484#)38*L%"M*NO4&6*P345"6*Q"96'*H89>)5*=("#*?6>&63RS!"#$%&'()*$+,-,./,0$1*$2342*$56$4#6!(4#)3%*,0*=>&83*I

8 Summary Table G!?AH A summary table tallies the values as frequencies or percentages for each category. A summary table helps you see the differences among the categories by displaying the frequency, amount, or percentage of items in a set of categories in a separate column. The next table presents a summary table that tallies responses to a recent survey that asked young adults about the main reason that they shop online. ="9%:3J*G4)4*3K)%4:)38*468*484#)38*L%"M*NO4&6*P345"6*Q"96'*H89>)5*=("#*?6>&63RS!"#$%&'()*$+,-,./,0$1*$2342*$56$4#6!(4#)3%*,0*=>&83*T

9 ;#,2(<&#:+2(=+42*(>?4.%:(3"+$(72.,2& G!?AH Reason For Shopping Online? Percent Better Prices 37% Avoiding holiday crowds or hassles 29% Convenience 18% Better selection 13% Ships directly 3% From the table, stored in online shopping, you can conclude that 37% shop online mainly for better prices and convenience and that 29% shop online mainly to avoid holiday crowds and hassles ="9%:3J*G4)4*3K)%4:)38*468*484#)38*L%"M*NO4&6*P345"6*Q"96'*H89>)5*=("#*?6>&63RS!"#$%&'()*$+,-,./,0$1*$2342*$56$4#6!(4#)3%*,0*=>&83*1

10 The sample of 316 retirement funds for the Choice Is yours scenario includes the variable risk that has the defined categories low, average, and high. Construct a summary table of the retirement funds, categorized by risk. Fund Risk Level Number of Funds Percentage of Funds Low % Average % High % Total % You can see that about two-thirds of the funds have low risk. About 30% of the funds have average risk. Very few funds have high risk!(4#)3%*,0*=>&83*.-

11 A Contingency Table Helps Organize Two or More Categorical Variables G!?AH!Used to study patterns that may exist between the responses of two or more categorical variables!cross tabulates or tallies jointly the responses of the categorical variables!for two variables the tallies for one variable are located in the rows and the tallies for the second variable are located in the columns!(4#)3%*,0*=>&83*..

12 Contingency Table - Example G!?AH! A random sample of 400 invoices is drawn.! Each invoice is categorized as a small, medium, or large amount.! Each invoice is also examined to identify if there are any errors.! This data are then organized in the contingency table to the right.!+2%,2*&2-1(0#6.&(3"+9,2* B'&C4&2-1(+D(E2F+,-&:(!#%&*+',G&? H1(3,G&(#2?(0"&(I'&:&2-&(7D(A''+': =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) A''+': A''+': 0+%#..I-, D-.D- F/ / I- CC/ F/ D--!(4#)3%*,0*=>&83*.,

13 =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) Contingency Table Based On Percentage Of Overall A''+': A''+': 0+%#..I-, D-.D- F/ / I- V")4> CC/ F/ D-- TC<I/W*"L*54M#>38*&6Z"&:35* (4Z3*6"*3%%"%5*468*DI</-W* "L*54M#>38*&6Z"&:35*4%3*L"%* 5M4>>*4M"96)5< =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) D,</-W*X*.I-*Y*D--,/<--W*X*.--*Y*D--.F<,/W*X***F/*Y*D-- A''+': A''+': 0+%#. D,</-W /<--W DI</-W,/<--W.-<--W C/<--W.F<,/W.<,/W.I</-W V")4> TC<I/W.F<,/W.--<-W!(4#)3%*,0*=>&83*.C

14 =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) Contingency Table Based On Percentage of Row A''+': A''+': 0+%#..I-,-.1- F/ / I- V")4> CC/ F/ D-- O38&9M*&6Z"&:35*(4Z3*4*>4%'3%* :(46:3*[,T</IW\*"L*(4Z&6'* 3%%"%5*)(46*5M4>>*[.-</CW\*"%* >4%'3*[I<.DW\*&6Z"&:35< =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96).-- D-.D- T1<DIW*X*.I-*Y*.1- I.<DCW*X*.--*Y*.D- 1,<TFW*X***F/*Y*I- A''+': A''+': 0+%#. T1<DIW.-</CW.--<-W I.<DCW,T</IW.--<-W 1,<TFW I<.DW.--<-W V")4> TC<I/W.F<,/W.--<-W!(4#)3%*,0*=>&83*.D

15 =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) Contingency Table Based On Percentage Of Column A''+': A''+': 0+%#..I-, D-.D- F/ / I- V")4> CC/ F/ D-- V(3%3*&5*4*F.</DW*:(46:3* )(4)*&6Z"&:35*]&)(*3%%"%5*4%3* "L*M38&9M*5&@3< =M4>> HM"96) O38&9M HM"96) U4%'3 HM"96) /-<I/W*X*.I-*Y*CC/ C-<IIW*X***,-*Y*F/ A''+': A''+': 0+%#. /-<I/W C-<IIW DI</-W,1<T/W F.</DW C/<--W.1<D-W I<F1W.I</-W V")4>.--<-W.--<-W.--<-W!(4#)3%*,0*=>&83*./

16 Problem 2.2 page 68 The following data represent the responses to two questions asked in a survey of 40 college students majoring in business: what is your gender? (M=male; F=female) and what is your major? (A= Accounting; C=Computer Information System; M=Marketing): Gender: M M M F M F F M F M Major: A C C M A C A A C C Gender: F M M M M F F M F F Major: A A A M C M A A A C Gender: M M M M F M F F M M Major: C C A A M M C A A A Gender: F M M M M F M F M M Major: C C A A A A C C A C!(4#)3%*,0*=>&83*.F

17 a.v4>>$*)(3*84)4*&6)"*4*:"6)&6'36:$*)4b>3*](3%3*)(3*)]"* %"]5*%3#%3536)*)(3*'3683%*:4)3'"%&35*468*)(3*)(%33* :">9M65*%3#%3536)*)(3*4:483M&:*M4^"%*:4)3'"%&35< b. :"65)%9:)*:"6)&6'36:$*)4B>35*B4538*"6*#3%:36)4'35* "L*4>>*D-*5)9836)*%35#"65350*B4538*"6*%"]*#3%:36)4'35* 468*B4538*"6*:">9M6*#3%:36)4'35<!(4#)3%*,0*=>&83*.I

18 2.2 Organizing Numerical Variables Tables Used For Organizing Numerical Data could G!?AH 7'?&'&?(>''#1 B'&C4&2-1 /,:%',64%,+2:!454.#%,F& /,:%',64%,+2:!(4#)3%*,0*=>&83*.T

19 Organizing Numerical Data: Ordered Array " An ordered array is a sequence of data, in rank order, from the smallest value to the largest value. " Shows range (minimum value to maximum value) " May help identify outliers (unusual observations) G!?AH Age of Surveyed College Students Day Students Night Students !(4#)3%*,0*=>&83*.1

20 Organizing Numerical Data: Frequency Distribution G!?AH " The frequency distribution is a summary table in which the data are arranged into numerically ordered classes. " You must give attention to selecting the appropriate number of class groupings for the table, determining a suitable width of a class grouping, and establishing the boundaries of each class grouping to avoid overlapping. " The number of classes depends on the number of values in the data. With a larger number of values, typically there are more classes. In general, a frequency distribution should have at least 5 but no more than 15 classes. " To determine the width of a class interval, you divide the range (Highest value Lowest value) of the data by the number of class groupings desired.!(4#)3%*,0*=>&83*,-

21 Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27 Construct Frequency distribution Solution " Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 " Find range: = 46!(4#)3%*,0*=>&83*,.

22 " Select number of classes: 5 (usually between 5 and 15) " Compute class interval (width): 10 (46/5 then round up) Where Interval width =!"#!$%&'()*+$',*-.$%&'()*+$ /+01$2'-3'4*)%%$% " Determine class boundaries (limits): " Class 1: 10 to less than 20 " Class 2: 20 to less than 30 " Class 3: 30 to less than 40 " Class 4: 40 to less than 50 " Class 5: 50 to less than 60 " Compute class midpoints: 15, 25, 35, 45, 55 " Count observations & assign to classes!(4#)3%*,0*=>&83*,,

23 /#%#(,2(+'?&'&?(#''#1J K)L(KML(KNL()KL()OL()OL()PL()NL()NL(MQL(M)L(MRL(MNL(MSL(OKL(OML(OOL(OPL(RML(RS!.#::(((((((((((((((((((((((((((;,?$+,2%: B'&C4& but less than but less than but less than but less than but less than Total 20!(4#)3%*,0*=>&83*,C

24 /#%#(,2(+'?&'&?(#''#1J K)L(KML(KNL()KL()OL()OL()PL()NL()NL(MQL(M)L(MRL(MNL(MSL(OKL(OML(OOL(OPL(RML(RS!.#::(((((((((((((((((((((((((((B'&C4&2-1 <&.#%,F& B'&C4&2-1 I&'-&2%#*& 10 but less than % 20 but less than % 30 but less than % 40 but less than % 50 but less than % Total %!(4#)3%*,0*=>&83*,D

25 Organizing Numerical Data: Cumulative Frequency Distribution Example /#%#(,2(+'?&'&?(#''#1J G!?AH K)L(KML(KNL()KL()OL()OL()PL()NL()NL(MQL(M)L(MRL(MNL(MSL(OKL(OML(OOL(OPL(RML(RS!.#:: B'&C4&2-1 I&'-&2%#*&!454.#%,F&( B'&C4&2-1!454.#%,F&( I&'-&2%#*& 10 but less than % 3 15% 20 but less than % 9 45% 30 but less than % 14 70% 40 but less than % 18 90% 50 but less than % T 0+%#. )Q(((((((((((((((((KQQ )Q KQQT!(4#)3%*,0*=>&83*,/

26 Why Use a Frequency Distribution?!It condenses the raw data into a more useful form G!?AH!It allows for a quick visual interpretation of the data!it enables the determination of the major characteristics of the data set including where the data are concentrated / clustered!(4#)3%*,0*=>&83*,f

27 Frequency Distributions: Some Tips G!?AH! Different class boundaries may provide different pictures for the same data (especially for smaller data sets)! Shifts in data concentration may show up when different class boundaries are chosen! As the size of the data set increases, the impact of alterations in the selection of class boundaries is greatly reduced! When comparing two or more groups with different sample sizes, you must use either a relative frequency or a percentage distribution!(4#)3%*,0*=>&83*,i

28 Problem 2.16 page 78 The file utility contains the following data about the cost of electricity (in $) during July 2013 for a random sample of 50 one-bedroom apartments in a large city a. Construct a frequency distribution and a percentage distribution that have class intervals with the upper class boundaries $99, $119, and so on. b. Construct a cumulative percentage distribution. c. Around what amount does the monthly electricity cost seem to be concentrated?!(4#)3%*,0*=>&83*,t

29 2.3 Visualizing Categorical Variables The chart you choose to visualize the data for a single categorical variable depends on whether you seek to emphasize how categories directly compare to each other (bar chart) or how categories from parts of a whole (pie chart), or whether you have data that are concentrated in only a few of your categories (Pareto chart). To visualize the data for two categorical variables, you use a side-by-side bar chart!(4#)3%*,0*=>&83*,1

30 !#%&*+',-#.( /#%# 8,:4#.,G,2*(/#%# 3455#'1( 0#6.&(B+'(72&( 8#',#6.&!+2%,2*&2-1( 0#6.&(B+'(09+( 8#',#6.&: H#'!"#'% I,&(!"#'% I#'&%+!"#'% 3,?&(H1(3,?&( H#'(!"#'%!(4#)3%*,0*=>&83*C-

31 The Bar Chart " A bar chart visualizes a categorical variable as a series of bars, with each bar representing the tallies for a single category. In a bar chart, the length of each bar represents either the frequency or percentage of values for a category and each bar is separated by space, called a gap. Reason For Shopping Online? Percent Better Prices 37% Avoiding holiday crowds or hassles 29% Convenience 18% Better selection 13% Ships directly 3%!(4#)3%*,0*=>&83*C.

32 pie chart The pie chart is a circle broken up into slices that represent categories. The size of each slice of the pie varies according to the percentage in each category. To represent each category as a pie slice, you multiply its percentage by the 360 degrees that makes up a circle to get a pie slice. Reason For Shopping Online? Percent Better Prices 37% Avoiding holiday crowds or hassles 29% Convenience 18% Better selection 13% Ships directly 3%!(4#)3%*,0*=>&83*C,

33 The Pareto Chart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

34 A Pareto chart presents the bars vertically, along with a cumulative percentage line. The cumulative line is plotted at the midpoint of each category, at a height equal to the cumulative percentage. In order for a Pareto chart to include all categories, even those with few defects, in some situations, you need to include a category labeled Other or Miscellaneous. If you include such a category, you place the bar that represents that category at the far end (to the right) of the X axis.! Used to portray categorical data (nominal scale)! A vertical bar chart, where categories are shown in descending order of frequency! A cumulative polygon is shown in the same graph! Used to separate the vital few from the trivial many!(4#)3%*,0*=>&83*cd

35 7'?&'&?(3455#'1(0#6.&(B+'(!#4:&: 7D(E2-+5$.&%&(>0;(0'#2:#-%,+2:!454.#%,F&!#4:&( B'&C4&2-1( I&'-&2% I&'-&2% `4%#38*:4%8*^4MM38 CF/**********/-<D.W /-<D.W!4%8*96%3484B>3*,CD**********C,<C,W T,<ICW HVO*M4>L96:)&"65 C,* D<D,W TI<./W HVO*"9)*"L*:45(,T C<TIW 1.<-,W ;6Z4>&8*4M"96)*%3_935)38,C* C<.TW 1D<,-W `%"6'*a3$5)%"a3,C* C<.TW 1I<CTW U4:a*"L*L9685*&6*4::"96).1*,<F,W.--<--W 0+%#.( N)O( KQQUQQT ="9%:3J*G4)4*3K)%4:)38*L%"M*H<*b(4>>40*NG"6c)*O&5953*)(3*24%3)"*2%&6:&#>30S*"78$"79.($:&0;.

36 V(3*NA&)4> d3]s!(4#)3%*,0*=>&83*cf

37 The Side-by-Side Bar Chart A side-by-side bar chart uses sets of bars to show the joint responses from two categorical variables. The side by side bar chart represents the data from a contingency table. =M4>> HM"96) O38&9M HM"96) U4%'3 A''+': A''+': 0+%#. /-<I/W C-<IIW DI</-W,1<T/W F.</DW C/<--W.1<D-W I<F1W.I</-W V")4>.--<-W.--<-W.--<-W 7%%"%5 e"*7%%"%5 E2F+,-&(3,G&(3$.,%(74%(H1(A''+':( V(@+(A''+': -<-W.-<-W,-<-W C-<-W D-<-W /-<-W F-<-W I-<-W U4%'3 O38&9M =M4>> E2F+,-&:(9,%"(&''+':(#'&(54-"(5+'&(.,W&.1(%+(6&(+D 5&?,45(:,G&(XPKUROT(F:(MQUNNT(#2?(NUPYTZ!(4#)3%*,0*=>&83*CI

38 2.4 Visualizing Numerical Variables You visualize the data for a numerical variable through a variety of techniques that show the distribution of values. These techniques include the stem-and-leaf display, the histogram, the percentage polygon, and the cumulative percentage polygon (ogive), and the boxplot!(4#)3%*,0*=>&83*ct

39 Visualizing Numerical Data By Using Graphical G!?AH 7'?&'&?(>''#1 3%&5[#2?[\&#D /,:$.#1 B'&C4&2-1(/,:%',64%,+2:( #2?!454.#%,F&(/,:%',64%,+2: ],:%+*'#5 I+.1*+2 7*,F&!(4#)3%*,0*=>&83*C1

40 Stem-and-Leaf Display G!?AH! A simple way to see how the data are distributed and where concentrations of data exist METHOD: Separate the sorted data series into leading digits (the stems) and the trailing digits (the leaves)!(4#)3%*,0*=>&83*d-

41 Organizing Numerical Data: Stem and Leaf Display " A stem-and-leaf display organizes data into groups (called stems) so that the values within each group (the leaves) branch out to the right on each row. Age of College Students G!?AH Age of Surveyed College Students Day Students Night Students Day Students Stem Leaf Night Students Stem Leaf !(4#)3%*,0*=>&83*D.

42 Visualizing Numerical Data: The Histogram G!?AH " A vertical bar chart of the data in a frequency distribution is called a histogram. " In a histogram there are no gaps between adjacent bars. " The class boundaries (or class midpoints) are shown on the horizontal axis. " The vertical axis is either frequency, relative frequency, or percentage. " The height of the bars represent the frequency, relative frequency, or percentage.!(4#)3%*,0*=>&83*d,

43 Visualizing Numerical Data: The Histogram G!?AH!.#::((((((((((((((((((((((((B'&C4&2-1 <&.#%,F& B'&C4&2-1 I&'-&2%#*& 10 but less than but less than but less than but less than but less than Total XE2(#($&'-&2%#*&( ",:%+*'#5((%"&(F&'%,-#.( #^,:(9+4.?(6&(?&D,2&?(%+( :"+9(%"&($&'-&2%#*&(+D( +6:&'F#%,+2:($&'(-.#::Z!"#$%#&'( % $ # "! )*+,-."/0123.#24526,%7#&,+ & '& "& (& #& && )*+,!(4#)3%*,0*=>&83*DC

44 Visualizing Numerical Data: The Polygon G!?AH " A percentage polygon is formed by having the midpoint of each class represent the data in that class and then connecting the sequence of midpoints at their respective class percentages. " The cumulative percentage polygon, or ogive, displays the variable of interest along the X axis, and the cumulative percentages along the Y axis. " Useful when there are two or more groups to compare.!(4#)3%*,0*=>&83*dd

45 Visualizing Numerical Data: The Frequency Polygon _:&D4.(`"&2(!+5$#',2*(09+(+'(;+'&(a'+4$: G!?AH!(4#)3%*,0*=>&83*D/

46 Visualizing Numerical Data: The Percentage Polygon G!?AH!(4#)3%*,0*=>&83*DF

47 2.5 Visualizing Two Numerical Variables Visualizing two numerical variables together can reveal possible relationships between two variables. To visualize two numerical variables, you construct a scatter plot. For the special case in which one of the two variables represents the passage of time, you construct a time-series plot.!(4#)3%*,0*=>&83*di

48 8#',#6.&: 3-#%%&'( I.+% 0,5&[ 3&',&:( I.+%!(4#)3%*,0*=>&83*DT

49 The Scatter Plot " Scatter plots are used for numerical data consisting of paired observations taken from two numerical variables " One variable is measured on the vertical axis and the other variable is measured on the horizontal axis " Scatter plots are used to examine possible relationships between two numerical variables!(4#)3%*,0*=>&83*d1

50 Scatter Plot Example G!?AH 8+.45&( $&'(?#1!+:%($&'(?#1!"#$%&'(%)*+%,#-%.("/01$2"3%4"506'%,C.,/,F.D-,1.DF CC.F- CT.FI D,.I- /-.TT //.1/ F-,--!"#$%&'(%)*+% $"! $!! #"! #!! "!! $! %! &! "! '! (! 4"506'%&'(%)*+!(4#)3%*,0*=>&83*/-

51 The Time Series Plot!A Time-Series Plot is used to study patterns in the values of a numeric variable over time! The Time-Series Plot:!Numeric variable is measured on the vertical axis and the time period is measured on the horizontal axis!(4#)3%*,0*=>&83*/.

52 Time Series Plot Example G!?AH Q34% e9mb3%* "L* d%46:(&535.11f DC.11I /D.11T F-.111 IC,--- T,,--. 1/,--,.-I,--C 11,--D 1/!"#$%&'()' *&+,-./0%0'!"#$%&'()'*&+,-./0%01' ' &"! &!! %! $! #! "!! &''# &''$ &''% "!!! "!!" "!!# "!!$ 9%+&!(4#)3%*,0*=>&83*/,