Chapter 2 - Graphical and Tabular Descriptive Techniques

Chapter 2 - Graphical and Tabular Descriptive Techniques

Chapter Three Graphical Descriptive Techniques II 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.1 Example 3.1 Following deregulation of telephone service, several new companies were created to compete in the business of providing long-distance telephone service. In almost all cases these companies competed on price since the service each offered is similar. Pricing a service or product in the face of stiff competition is very difficult. Factors to be considered include supply, demand, price elasticity, and the actions of competitors. Long-distance packages may employ per-minute charges, a flat monthly rate, or some combination

of the two. Determining the appropriate rate structure is facilitated by acquiring information about the behaviors of customers and in particular the size of monthly long-distance bills. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.2 Example 3.1 As part of a larger study, a long-distance company wanted to acquire information about the monthly bills of new subscribers in the first month after signing with the company. The companys marketing manager conducted a survey of 200 new residential subscribers wherein the first months bills were recorded. These data are stored in file Xm03-01. The general manager planned to present his findings to senior executives. What information can be extracted from these data?

2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.3 Example 3.1 We have chosen eight classes defined in such a way that each observation falls into one and only one class. These classes are defined as follows: Classes Amounts that are less than or equal to 15 Amounts that are more than 15 but less than or equal to 30 Amounts that are more than 30 but less than or equal to 45 Amounts that are more than 45 but less than or equal to 60 Amounts that are more than 60 but less than or equal to 75 Amounts that are more than 75 but less than or equal to 90 Amounts that are more than 90 but less than or equal to 105 Amounts that are more than 105 but less than or equal to 120

2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.4 Example 3.1 Histogram 80 70 Frequency 60 50 40 30 20 10 0 15

30 45 60 75 90 105 120 Bills 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.5 Interpret about half (71+37=108) of the bills are small, i.e. less than $30 (18+28+14=60)200 = 30% i.e. nearly a third of the phone bills are $90 or more. There are only a few telephone bills in the middle range. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.6

Building a Histogram 1) Collect the Data 2) Create a frequency distribution for the data How? a) Determine the number of classes to use How? Refer to table 3.2: With 200 observations, we should have between 7 & 10 classes Alternative, we could use Sturges formula: Number of class intervals = 1 + 3.3 log (n) 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.7 Building a Histogram 1) Collect the Data 2) Create a frequency distribution for the data How? a) Determine the number of classes to use. [8] b) Determine how large to make each class How? Look at the range of the data, that is, Range = Largest Observation Smallest Observation Range = $119.63 $0 = $119.63 Then each class width becomes: Range (# classes) = 119.63 8 15 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.8

Building a Histogram 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.9 Building a Histogram 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.10 Shapes of Histograms Variable Frequency

Frequency Frequency Symmetry A histogram is said to be symmetric if, when we draw a vertical line down the center of the histogram, the two sides are identical in shape and size: Variable Variable 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.11 Shapes of Histograms

Frequency Frequency Skewness A skewed histogram is one with a long tail extending to either the right or the left: Variable Positively Skewed Variable Negatively Skewed 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.12 Shapes of Histograms Modality A unimodal histogram is one with a single peak, while a bimodal histogram is one with two peaks: Bimodal Frequency Frequency Unimodal Variable Variable

A modal class is the class with the largest number of observations 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.13 Shapes of Histograms Many statistical techniques require that the population be bell shaped. Drawing the histogram helps verify the shape of the population in question. Frequency

Bell Shape A special type of symmetric unimodal histogram is one that is bell shaped: Variable Bell Shaped 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.14 Histogram Comparison Compare & contrast the following histograms based on data from Ex. 3.3 & Ex. 3.4: unimodal vs. bimodal

The two courses, Business Statistics and Mathematical Statistics have very different histograms 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.15 Stem & Leaf Display Retains information about individual observations that would normally be lost in the creation of a histogram. Split each observation into two parts, a stem and a leaf: e.g. Observation value: 42.19 There are several ways to split it up We could split it at the decimal point: Stem

Leaf 42 19 4 2 Or split it at the tens position (while rounding to the nearest integer in the ones position) 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.16 Stem & Leaf Display Continue this process for all the observations. Then, use

the stems for the classes and each leaf becomes part of the histogram (based on Example 3.1 data) as follows Stem Leaf 0 1 2 3 4 5 6 7 8 9 10 11 0000000000111112222223333345555556666666778888999999 000001111233333334455555667889999

0000111112344666778999 001335589 124445589 33566 3458 022224556789 Thus, we still have access to our 334457889999 00112222233344555999 original data points value! 001344446699 124557889 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.17 Histogram and Stem & Leaf

Compare the overall shapes of the figures 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.18 Ogive (pronounced Oh-jive) is a graph of a cumulative frequency distribution. We create an ogive in three steps First, from the frequency distribution created earlier, calculate relative frequencies: Relative Frequency = # of observations in a class Total # of observations 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.19 Relative Frequencies For example, we had 71 observations in our first class (telephone bills from $0.00 to $15.00). Thus, the relative frequency for this class is 71 200 (the total # of phone bills) = 0.355 (or 35.5%) 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.20 Ogive Is a graph of a cumulative frequency distribution. We create an ogive in three steps 1) Calculate relative frequencies. 2) Calculate cumulative relative frequencies by adding the current class relative frequency to the previous class

cumulative relative frequency. (For the first class, its cumulative relative frequency is just its relative frequency) 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.21 Cumulative Relative Frequencies first class : : 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.22

Ogive Is a graph of a cumulative frequency distribution. 1) Calculate relative frequencies. 2) Calculate cumulative relative frequencies. 3) Graph the cumulative relative frequencies 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.23 Ogive The ogive can be used to answer questions like: What telephone bill value is at the 50th percentile?

around $35 (Refer also to Fig. 2.13 in your textbook) 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.24 Describing Time Series Data Observations measured at the same point in time are called cross-sectional data. Observations measured at successive points in time are called time-series data. Time-series data graphed on a line chart, which plots the value of the variable on the vertical axis against the time periods on the horizontal axis. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.25 Example 3.5 We recorded the monthly average retail price of gasoline since 1976. Xm03-05 Draw a line chart to describe these data and briefly describe the results. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.26 Average price of gasoline Example 3.5 450

400 350 300 Line Chart 250 200 150 100 50 0 1 25 49 73 97 121 145 169 193 217 241 265 289 313 337 361 385 Month 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.27 Example 3.6 Price of Gasoline in 1982-84 Constant Dollars Xm03-06 Remove the effect of inflation in Example 3.5 to determine whether gasoline prices are higher than they have been in the past after removing the effect of inflation. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.28 Ajusted price of gasoline Example 3.6

200 180 160 140 120 100 80 60 40 20 0 1 25 49 73 97 121 145 169 193 217 241 265 289 313 337 361 385 Month 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.29 Example 3.6 Using constant 1982-1984 dollars, we can see that the average price of a gallon of gasoline hit its peak in the middle of 2008 (month 390). From there it dropped rapidly and in late 2009 it was about equal to the adjusted price in 1976. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.30 Graphing the Relationship Between Two Interval Variables Moving from nominal data to interval data, we are frequently interested in how two interval variables are related.

To explore this relationship, we employ a scatter diagram, which plots two variables against one another. The independent variable is labeled X and is usually placed on the horizontal axis, while the other, dependent variable, Y, is mapped to the vertical axis. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.31 Example 3.7 A real estate agent wanted to know to what extent the selling price of a home is related to its size. To acquire this information he took a sample of 12 homes that had recently sold, recording the price in thousands of dollars and the size in hundreds of square feet. These data are listed in the accompanying table. Use a graphical technique to describe the relationship between size and price. Xm03-07

Size 2354 1807 2637 2024 2241 1489 3377 2825 2302 2068 2715 1833 Price 315 229 355 261 234 216 308 306 289 204 265 195 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.32 Example 3.7 It appears that in fact there is a relationship, that is, the greater the house size the greater the selling price Price $1,000s) Scatte r Diagram 400 350 300 250

200 150 100 50 0 0 500 1000 1500 2000 2500 3000

3500 4000 House size 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.33 Patterns of Scatter Diagrams Linearity and Direction are two concepts we are interested in Positive Linear Relationship Negative Linear Relationship

Weak or Non-Linear Relationship 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.34 Chapter-Opening Example WERE OIL COMPANIES GOUGING CUSTOMERS 2000-2009: SOLUTION In January 2000 the average retail price of gasoline was $01.301 per gallon and the price of oil (West Texas intermediate crude) was $27.18 per barrel. Over the next 10 years the price of both substantially increased. Many drivers complained that the oil companies were guilty of price gouging. That is, they believed that when the price of oil increased the price of gas also increased, but when the price of oil decreased, the decrease in the price of gasoline seemed to lag behind. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

2.35 Chapter-Opening Example WERE OIL COMPANIES GOUGING CUSTOMERS 1999-2006: SOLUTION To determine whether this perception is accurate we determined the monthly figures for both commodities. Xm03-00 Graphically depict these data and describe the findings. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.36 Price of gasoline Chapter-Opening Example 450

400 350 300 250 200 150 100 50 0 Scatte r Diagram 0 20 40

60 80 100 120 140 160 Price of oil 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.37

Chapter-Opening Example The scatter diagram reveals that the two prices are strongly related linearly. When the price of oil was below $40 the relationship between the two was stronger than when the price of oil exceeded $40. 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.38 Summary I Factors That Identify When to Use Frequency and Relative Frequency Tables, Bar and Pie Charts 1. Objective: Describe a single set of data. 2. Data type: Nominal Factors That Identify When to Use a Histogram, Ogive, or Stem-and-Leaf Display 1. Objective: Describe a single set of data. 2. Data type: Interval

Factors that Identify When to Use a Cross-classification Table 1. Objective: Describe the relationship between two variables. 2. Data type: Nominal Factors that Identify When to Use a Scatter Diagram 1. Objective: Describe the relationship between two variables. 2. Data type: Interval 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.39 Summary II Interval Data Histogram Single Set of Data Relationship Scatter Diagram Between

Two Variables Nominal Data Frequency and Relative Frequency Tables, Bar and Pie Charts Cross-classification Table, Bar Charts 2012 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 2.40

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