chap2 & 3-a

chap2 & 3-a - Business Statistics (BUSA 3101) Dr....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Business Statistics (BUSA 3101) Dr. Lari H. Arjomand lariarjomand@clayton.edu Slide 1 Chapters 2 & 3 (Part A) Descriptive Statistics: Tabular and Graphical Presentations s s Summarizing Qualitative Data Summarizing Quantitative Data Typ es o f Dat a Ty p D at Data Data Numerical Numerical Categorical Categorical (Quantitative) (Quantitative) Discrete Discrete (Qualitative) (Qualitative) Continuous Continuous Slide 2 Summarizing Qualitative Data s s s s s Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Bar Graph Pie Chart Slide 3 Construction of a Frequency Distribution Graph Raw data Question to be addressed Collect Collect data data Organize Organize data data Present Present data data Draw Draw conclusion conclusion Frequency distribution Slide 4 Frequency Distribution A frequency distribution is a tabular summary of A frequency distribution is a tabular summary of data showing the frequency (or number) of items data showing the frequency (or number) of items in each of several non­overlapping classes. in each of several non­overlapping classes. The objective is to provide insights about the data The objective is to provide insights about the data that cannot be quickly obtained by looking only at that cannot be quickly obtained by looking only at the original data. the original data. Slide 5 Example: Marada Inn Guests staying at Marada Inn were asked to rate the quality of their accommodations as being excellent, above average, average, below average, or poor. The ratings provided by a sample of 20 guests are: Below Average Above Average Above Average Average Above Average Average Above Average Average Above Average Below Average Poor Excellent Above Average Average Above Average Above Average Below Average Poor Above Average Average Average Slide 6 Frequency Distribution Rating Frequency 2 Poor 3 Below Average 5 Average 9 Above Average 1 Excellent Total 20 Slide 7 Relative Frequency Distribution The relative frequency of a class is the fraction or The relative frequency of a class is the fraction or proportion of the total number of data items proportion of the total number of data items belonging to the class. belonging to the class. A relative frequency distribution is a tabular A relative frequency distribution is a tabular summary of a set of data showing the relative summary of a set of data showing the relative frequency for each class. frequency for each class. Slide 8 Percent Frequency Distribution The percent frequency of a class is the relative The percent frequency of a class is the relative frequency multiplied by 100. frequency multiplied by 100. A percent frequency distribution is a tabular A percent frequency distribution is a tabular summary of a set of data showing the percent summary of a set of data showing the percent frequency for each class. frequency for each class. Slide 9 Relative Frequency and Percent Frequency Distributions Relative Frequency Rating .10 Poor .15 Below Average .25 Average .45 Above Average .05 Excellent Total 1.00 Percent Frequency 10 15 25 .10(100) = 10 45 5 100 1/20 = .05 Slide 10 Bar Graph A bar graph is a graphical device for presenting qualitative data. On one axis (usually the horizontal axis), we specify the labels that are used for each of the classes. A frequency, relative frequency, or percent frequency scale can be used for the other axis (usually the vertical axis). Using a bar of fixed width drawn above each class label, we extend the height appropriately. The bars are separated to emphasize the fact that each class is a separate category. Slide 11 Bar Graph Marada Inn Quality Ratings 10 9 Frequency 8 7 6 5 4 3 2 1 Poor Below Average Above Excellent Average Average Rating Slide 12 Pie Chart The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. s First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. s Since there are 360 degrees in a circle, a class with a relative frequency of .25 would consume .25(360) = 90 degrees of the circle. Slide 13 Pie Chart Marada Inn Quality Ratings Excellent 5% Poor 10% Below Above Average 45% Average 15% Average 25% Slide 14 Example: Marada Inn s Insights Gained from the Preceding Pie Chart • One­half of the customers surveyed gave Marada a quality rating of “above average” or “excellent” (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” rating, there were two customers who gave a “poor” rating (looking at the top of the pie). This should displease the manager. Slide 15 Summarizing Quantitative Data s s s s s s Frequency Distribution Relative Frequency and Percent Frequency Distributions Dot Plot Histogram Cumulative Distributions Ogive N u m er ic al (Quantitative) Nu (Quantitative) D at a Pr esen t at io n Numerical Numerical Data Data Ordered Ordered Array Array Stem-&-Leaf Stem-&-Leaf Display Display Frequency Frequency Distributions Distributions HistoHistogram gram Polygon Polygon Ogive Ogive Slide 16 Frequency Distribution Table Steps s s s s s s 1­ Determine range 2­ Select number of classes • Usually between 5 and 20 inclusive 3­ Compute class intervals (width) 4­ Determine class boundaries (limits) 5­ Compute class midpoints 6­ Count observations & assign to classes Slide 17 Example: Hudson Auto Repair The manager of Hudson Auto would like to have a better understanding of the cost of parts used in the engine tune­ups performed in the shop. She examines 50 customer invoices for tune­ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. Slide 18 Example: Hudson Auto Repair s Sample of Parts Cost for 50 Tune­ups 91 71 104 85 62 78 69 74 97 82 93 72 62 88 98 57 89 68 68 101 75 66 97 83 79 52 75 105 68 105 99 79 77 71 79 80 75 65 69 69 97 72 80 67 62 62 76 109 74 73 Slide 19 Frequency Distribution s Guidelines for Selecting Number of Classes • Use between 5 and 20 classes. • Data sets with a larger number of elements usually require a larger number of classes. • Smaller data sets usually require fewer classes Slide 20 Frequency Distribution (Continued) s Guidelines for Selecting Width of Classes •Use classes of equal width. •Approximate Class Width = Largest Data Value − Smallest Data Value Number of Classes Slide 21 Example: Frequency Distribution For Hudson Auto Repair, if we choose six classes: Approximate Class Width = (109 ­ 52)/6 = 9.5 ≅ 10 10 Frequency Parts Cost ($) 50­59 2 60­69 13 70­79 16 80­89 7 90­99 7 100­109 5 Total 50 Slide 22 Solution Using SWStat+ s After putting your data into Excel, then create Data Area: Slide 23 Solution Using SWStat (cont.) s s SwStat Statistics Tabulations and Histograms Statistics Tabulations Note that here we are using 10 classes—you can use 6 classes if you Note 10 would like to do so. would Slide 24 Solution Using SWStat (cont.) s Results: Slide 25 Relative Frequency and Percent Frequency Distributions Parts Relative Percent Cost ($) Frequency Frequency 50­59 .04 4 60­69 .26 2/50 26 .04(100) 70­79 .32 32 80­89 .14 14 90­99 .14 14 100­109 .10 10 Total 1.00 100 Slide 26 Relative Frequency and Percent Frequency Distributions s Insights Gained from the Percent Frequency Distribution • Only 4% of the parts costs are in the $50­59 class. • 30% of the parts costs are under $70. • The greatest percentage (32% or almost one­third) of the parts costs are in the $70­79 class. • 10% of the parts costs are $100 or more. Slide 27 Dot Plot s s s One of the simplest graphical summaries of data is a dot plot. A horizontal axis shows the range of data values. Then each data value is represented by a dot placed above the axis. Slide 28 Dot Plot Tune­up Parts Cost . . .. . . . . .. .. .. .. . . . . . ..... .......... .. . .. . . ... . .. . 50 60 70 80 90 100 110 Cost ($) Slide 29 Histogram Another common graphical presentation of quantitative data is a histogram. The variable of interest is placed on the horizontal axis. A rectangle is drawn above each class interval with its height corresponding to the interval’s frequency, relative frequency, or percent frequency. Unlike a bar graph, a histogram has no natural separation between rectangles of adjacent classes. Slide 30 Histogram 18 Tune­up Parts Cost 16 Frequency 14 12 10 8 6 4 2 50−59 60−69 70−79 80−89 90−99 100­110 Parts Cost ($) Slide 31 Histogram (Continued) Symmetric • Left tail is the mirror image of the right tail • Example: heights and weights of people .35 Relative Frequency s .30 .25 .20 .15 .10 .05 0 Slide 32 Histogram (Continued) Moderately Skewed Left • A longer tail to the left • Example: exam scores .35 Relative Frequency s .30 .25 .20 .15 .10 .05 0 Slide 33 Histogram (Continued) Moderately Right Skewed • A Longer tail to the right • Example: housing values .35 Relative Frequency s .30 .25 .20 .15 .10 .05 0 Slide 34 Histogram (Continued) Highly Skewed Right • A very long tail to the right • Example: executive salaries .35 Relative Frequency s .30 .25 .20 .15 .10 .05 0 Slide 35 Cumulative Distributions Cumulative frequency distribution − shows the Cumulative frequency distribution − shows the number of items with values less than or equal to number of items with values less than or equal to the upper limit of each class.. the upper limit of each class.. Cumulative relative frequency distribution – shows Cumulative relative frequency distribution – shows the proportion of items with values less than or the proportion of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Cumulative percent frequency distribution – shows Cumulative percent frequency distribution – shows the percentage of items with values less than or the percentage of items with values less than or equal to the upper limit of each class. equal to the upper limit of each class. Slide 36 Cumulative Distributions s Example: Hudson Auto Repair Cumulative Cumulative Cumulative Relative Percent Frequency Frequency Cost ($) Frequency 2 .04 < 59 4 15 .30 < 69 30 31 2 + 13 .62 < 79 15/50 62 .30(100) 38 .76 < 89 76 45 .90 < 99 90 1.00 < 109 50 100 Slide 37 Ogive s An ogive is a graph of a cumulative distribution. s The data values are shown on the horizontal axis. s Shown on the vertical axis are the: • cumulative frequencies, or • cumulative relative frequencies, or • cumulative percent frequencies s The frequency (one of the above) of each class is plotted as a point. s The plotted points are connected by straight lines. Slide 38 Ogive s Example: Hudson Auto Repair • Because the class limits for the parts­cost data are 50­ 59, 60­69, and so on, there appear to be one­unit gaps from 59 to 60, 69 to 70, and so on. • These gaps are eliminated by plotting points halfway between the class limits. • Thus, 59.5 is used for the 50­59 class, 69.5 is used for the 60­69 class, and so on. Slide 39 Ogive with Cumulative Percent Frequencies Cumulative Percent Frequency Tune­up Parts Cost Tune­up Parts Cost 100 80 60 (89.5, 76) 40 20 Parts 50 60 70 80 90 100 110 Cost ($) Slide 40 Frequency Distribution Table Another Example Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Class Frequency 15 but < 25 3 25 but < 35 5 35 but < 45 2 Slide 41 Frequency Distribution Table Example (Continued) Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 24 26, 24 21 41, Class Midpoint Frequency 15 but < 25 Width 20 3 25 but < 35 30 5 35 but < 45 40 2 Boundaries (Upper + Lower Boundaries) / 2 Slide 42 Stated and True (or Real) Class Limits True Classes: Are those classes such that the upper true limit of a class is the same as the lower true limit of the next class. For comparison, the stated class limits and true class limits are given in the following table—next slide: Slide 43 Stated and True (or Real) Class Limits Stated $600­$799 $800­$999 True $599.50 up to but not including $799.50 $799.50 up to but not including $999.50 In the first column of the above table the data were rounded to the nearest dollar. For example, $799.50 was rounded up to $800 and tailed in the second class. Any amount over $799 but under 799.50 was rounded down to $799 and included in the first class. Thus, the $600­$799 class actually includes all data from $599.50 inclusive up to but not including $799.50. Slide 44 Relative Frequency & % Distribution Tables Example (Continued) The relative frequency of a class is obtained by dividing the class frequency by the total frequency, which in the following problem = 10. Relative Frequency Relative Distribution Distribution Percentage Percentage Distribution Distribution Class Prop. Class % 15 but < 25 .3 15 but < 25 30.0 25 but < 35 .5 25 but < 35 50.0 35 but < 45 .2 35 but < 45 20.0 Slide 45 Cumulative Percentage Distribution Table Example (Continued) Raw Data: 24, 26, 24, 21, 27, 27, 30, 41, 32, 38 Raw Class Cumulative Percentage Percentage Percentage less than lower class boundary 15 but < 25 Lower Lower class boundary 0.0 25 but < 35 30.0 35 but < 45 80.0 30% + 50% 30% 45 but < 55 100.0 80% + 20% Slide 46 End of Chapters 2 & 3, Part A Slide 47 ...
View Full Document

Ask a homework question - tutors are online