Lecture1-2 - Descriptive Statistics-Tabular and Graphical Representation

Lecture1-2 - Descriptive Statistics-Tabular and Graphical Representation

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Unformatted text preview: EM 521 Applied Statistics Descriptive Statistics: Tabular and Graphical Presentations AS&W – Chapter 2 1 Descriptive Statistics: Tabular and Graphical Presentations Summarizing Qualitative Data Summarizing Quantitative Data 2 Summarizing Qualitative Data Frequency Distribution Relative Frequency Distribution Percent Frequency Distribution Bar Graph Pie Chart 3 Frequency Distribution A A frequency frequency distribution distribution is is aa tabular tabular summary summary of of data data showing showing the the frequency frequency (or (or number) number) of of items items in in each each of of several several nonoverlapping nonoverlapping classes. classes. The The objective objective is is to to provide provide insights insights about about the the data data that that cannot cannot be be quickly quickly obtained obtained by by looking looking only only at at the the original original data. data. 4 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 guestsAverage are: Below Average Above Average Above Average Above Average Average Above Average Average Above Average Above Average Below Average Poor Excellent Above Average Average Above Average Below Average Poor Above Average Average 5 Frequency Distribution Rating Frequency 2 Poor 3 Below Average 5 Average 9 Above Average 1 Excellent Total 20 6 Relative Frequency Distribution The The relative relative frequency frequency of of aa class class is is the the fraction fraction or or proportion proportion of of the the total total number number of of data data items items belonging belonging to to the the class. class. A A relative relative frequency frequency distribution distribution is is aa tabular tabular summary summary of of aa set set of of data data showing showing the the relative relative frequency frequency for for each each class. class. 7 Percent Frequency Distribution The The percent percent frequency frequency of of aa class class is is the the relative relative frequency frequency multiplied multiplied by by 100. 100. A A percent percent frequency frequency distribution distribution is is aa tabular tabular summary summary of of aa set set of of data data showing showing the the percent percent frequency frequency for for each each class. class. 8 Relative Frequency and Percent Frequency Distributions Relative Percent Rating Frequency Frequency .10 10 Poor .15 15 Below Average .25 25 .10(100) = Average 10 .45 45 Above Average .05 5 Excellent Total 1.00 100 1/20 = . 05 9 Bar Graph A bar graph is a graphical device for depicting 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 frequenc 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 e class is a separate category. 10 Bar Graph Frequency 10 9 Marada Inn Quality Ratings 8 7 6 5 4 3 2 1 Poor Below Average Above Excellent Average Average Rating 11 Pie Chart The pie chart is a commonly used graphical device for presenting relative frequency distributions for qualitative data. First draw a circle; then use the relative frequencies to subdivide the circle into sectors that correspond to the relative frequency for each class. 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. 12 Pie Chart Marada Inn Quality Ratings Excellent 5% Poor 10% Above Average 45% Below Average 15% Average 25% 13 Example: Marada Inn Insights Gained from the Preceding Pie Chart • One-half of the customers surveyed gave Marada a quality rating of “above average” or “excellen (looking at the left side of the pie). This might please the manager. • For each customer who gave an “excellent” ratin there were two customers who gave a “poor” rating (looking at the top of the pie). This should shoul displease the manager. 14 Summarizing Quantitative Data Frequency Distribution Relative Frequency and Percent Frequency Distributions Dot Plot Histogram Cumulative Distributions Ogive 15 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 tune-ups performed in the shop. She examines 50 customer invoices for tune-ups. The costs of parts, rounded to the nearest dollar, are listed on the next slide. 16 Example: Hudson Auto Repair Sample of Parts Cost for 50 Tune-ups 91 71 104 85 62 78 69 74 97 82 93 57 72 89 62 68 88 68 98 101 75 52 66 75 97 105 83 68 79 105 99 79 77 71 79 80 75 65 69 69 97 62 72 76 80 109 67 74 62 73 17 Frequency Distribution 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 18 L a rg e s tD lN V u e S m a l e s t D a V l u e a m b ro fC Frequency Distribution Guidelines for Selecting Width of Classes •Use classes of equal width. •Approximate Class Width = 19 Frequency Distribution For Hudson Auto Repair, if we choose six classes: Approximate Class Width = (109 - 52)/6 = 9.5 1 Parts Cost ($) Frequency 50-59 2 60-69 13 70-79 16 80-89 7 90-99 7 100-109 5 Total 50 20 Relative Frequency and Percent Frequency Distributions Parts Relative Percent Cost ($) Frequency Frequency 50-59 .04 4 60-69 .26 2/50 26 . 70-79 .32 32 04(100 ) 80-89 .14 14 90-99 .14 14 100-109 .10 10 Total 1.00 100 21 Relative Frequency and Percent Frequency Distributions Insights Gained from the Percent Frequency Distribution • Only 4% of the parts costs are in the $50-59 class. • • 30% of the parts costs are under $70. The greatest percentage (32% or almost one-third of the parts costs are in the $70-79 class. • 10% of the parts costs are $100 or more. 22 Dot Plot One of the simplest graphical summaries of data is a dot plot. A horizontal axis shows the range of data values. Then each data value is represented by a dot placed above the axis. 23 Dot Plot Tune-up Parts Cost . 50 . . .. . . . . .. .. .. .. . . ..... .......... .. . .. . . .. . 60 70 80 90 100 Cost ($) 24 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. 25 Histogram Tune-up Parts Cost 18 Frequency 16 14 12 10 8 6 4 2 Parts Cost ($) 5059 6069 7079 8089 9099 100-110 26 Histogram Symmetric • Left tail is the mirror image of the right tail • Examples: heights and weights of people Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 27 Histogram Moderately Skewed Left • A longer tail to the left • Example: exam scores Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 28 Histogram Moderately Right Skewed • A Longer tail to the right • Example: housing values Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 29 Histogram Highly Skewed Right • A very long tail to the right • Example: executive salaries Relative Frequency .35 .30 .25 .20 .15 .10 .05 0 30 Cumulative Distributions Cumulative Cumulative frequency frequency distribution distribution shows shows the the number number of of items items with with values values less less than than or or equal equal to to the the upper upper limit limit of of each each class.. class.. Cumulative Cumulative relative relative frequency frequency distribution distribution –– shows shows the the proportion proportion of of items items with with values values less less than than or or equal equal to to the the upper upper limit limit of of each each class. class. Cumulative Cumulative percent percent frequency frequency distribution distribution –– shows shows the the percentage percentage of of items items with with values values less less than than or or equal equal to to the the upper upper limit limit of of each each class. class. 31 Cumulative Distributions Hudson Auto Repair Cumulative Cumulative Cumulative Relative Percent Cost ($) Frequency Frequency Frequency 2 .04 < 59 4 15 .30 < 69 30 31 < 79 2 + .62 15/50 62 . 13 38 .76 < 89 76 30(100 ) 45 .90 < 99 90 50 1.00 < 109 100 32 Ogive An ogive is a graph of a cumulative distribution. The data values are shown on the horizontal axis. Shown on the vertical axis are the: • cumulative frequencies, or • cumulative relative frequencies, or • cumulative percent frequencies The frequency (one of the above) of each class is plotted as a point. The plotted points are connected by straight lines. 33 Ogive Hudson Auto Repair • Because the class limits for the parts-cost data are 50-59, 60-69, and so on, there appear to be one-unit 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 50-59 class, 69.5 is used for the 60-69 class, and so on. 34 Cumulative Percent Frequency Ogive with Cumulative Percent Frequencies Tune-up Tune-up Parts Parts Cost Cost 100 80 60 (89.5, 76) 40 20 50 60 70 80 90 100 Parts Cost ($) 110 35 ...
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