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1 Graphical Descriptive Techniques Chapter 2

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2 2.1 Introduction Descriptive statistics involves the arrangement, summary, and presentation of data, to enable meaningful interpretation, and to support decision making. Descriptive statistics methods make use of graphical techniques numerical descriptive measures. The methods presented apply to both the entire population the population sample

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3 2.2Types of data and information A variable - a characteristic of population or sample that is of interest for us. Cereal choice Capital expenditure The waiting time for medical services Data - the actual values of variables Interval data are numerical observations Nominal data are categorical observations Ordinal data are ordered categorical observations

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4 Types of data - examples Interval data Age - income Age - income Weight gain Weight gain Nominal Person Marital status 1married 2single 3single.. Person Marital status 1married 2single 3single.. Computer Brand 1IBM 2Dell 3IBM.. Computer Brand 1IBM 2Dell 3IBM..

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5 Types of data - examples Interval data Age - income Age - income Nominal data With nominal data, all we can do is, calculate the proportion of data that falls into each category. IBM Dell Compaq OtherTotal % 22% 16% 12% IBM Dell Compaq OtherTotal % 22% 16% 12% Weight gain Weight gain

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6 Types of data – analysis Knowing the type of data is necessary to properly select the technique to be used when analyzing data. Type of analysis allowed for each type of data Interval data – arithmetic calculations Nominal data – counting the number of observation in each category Ordinal data - computations based on an ordering process

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7 Cross-Sectional/Time-Series Data Cross sectional data is collected at a certain point in time Marketing survey (observe preferences by gender, age) Test score in a statistics course Starting salaries of an MBA program graduates Time series data is collected over successive points in time Weekly closing price of gold Amount of crude oil imported monthly

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8 2.3 Graphical Techniques for Interval Data Example 2.1Example 2.1: Providing information concerning the monthly bills of new subscribers in the first month after signing on with a telephone company. Collect data Prepare a frequency distribution Draw a histogram

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9 Largest observation Collect data (There are 200 data points Prepare a frequency distribution How many classes to use? Number of observations Number of classes Less then , ,000 – 5, , , More than 50, Class width = [Range] / [# of classes] [ ] / [8] = Largest observation Largest observation Smallest observation Smallest observation Smallest observation Smallest observation Largest observation Example 2.1Example 2.1: Providing information

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10 Draw a Histogram Example 2.1Example 2.1: Providing information

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Bills Frequency What information can we extract from this histogram About half of all the bills are small 71+37= =32 A few bills are in the middle range Relatively, large number of large bills =60 Example 2.1Example 2.1: Providing information

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12 It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself. Relative frequencies should be used when the population relative frequencies are studied comparing two or more histograms the number of observations of the samples studied are different Class relative frequency = Class frequency Total number of observations Class frequency Total number of observations Relative frequency

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13 It is generally best to use equal class width, but sometimes unequal class width are called for. Unequal class width is used when the frequency associated with some classes is too low. Then, several classes are combined together to form a wider and more populated class. It is possible to form an open ended class at the higher end or lower end of the histogram. Class width

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14 There are four typical shape characteristics Shapes of histograms

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15 Positively skewed Negatively skewed Shapes of histograms

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16 A modal class is the one with the largest number of observations. A unimodal histogram The modal class Modal classes

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17 Modal classes A bimodal histogram A modal class

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18 Many statistical techniques require that the population be bell shaped. Drawing the histogram helps verify the shape of the population in question Bell shaped histograms

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19 Example 2.2Example 2.2: Selecting an investment An investor is considering investing in one out of two investments. The returns on these investments were recorded. From the two histograms, how can the investor interpret the Expected returns The spread of the return (the risk involved with each investment) Interpreting histograms

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20 Example 2.2 Example Histograms Return on investment AReturn on investment B The center of the returns of Investment A is slightly lower than that for Investment B Interpretation: The center of the returns of Investment A is slightly lower than that for Investment B The center for B The center for A

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The spread of returns for Investment A is less than that for investment B Interpretation: The spread of returns for Investment A is less than that for investment B Return on investment AReturn on investment B Sample size = Example 2.2 Example Histograms

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Return on investment AReturn on investment B Both histograms are slightly positively skewed. There is a possibility of large returns. Interpretation: Both histograms are slightly positively skewed. There is a possibility of large returns. Example 2.2 Example Histograms

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23 Example 2.2: Conclusion It seems that investment A is better, because: Its expected return is only slightly below that of investment B The risk from investing in A is smaller. The possibility of having a high rate of return exists for both investment. Providing information

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24 Example 2.3Example 2.3: Comparing students performance Students performance in two statistics classes were compared. The two classes differed in their teaching emphasis Class A – mathematical analysis and development of theory. Class B – applications and computer based analysis. The final mark for each student in each course was recorded. Draw histograms and interpret the results. Interpreting histograms

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25 Interpreting histograms The mathematical emphasis creates two groups, and a larger spread.

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26 This is a graphical technique most often used in a preliminary analysis. Stem and leaf diagrams use the actual value of the original observations (whereas, the histogram does not). Stem and Leaf Display

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27 Split each observation into two parts. There are several ways of doing that: Stem Leaf 4219 Stem Leaf 42 A stem and leaf display for Example 2.1 will use this method next. Stem and Leaf Display Observation:

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28 A stem and leaf display for Example 2.1 StemLeaf Example 2.1 The length of each line represents the frequency of the class defined by the stem. Stem and Leaf Display

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29 Ogives } } Ogives are cumulative relative frequency distributions. Example continued

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Graphical Techniques for Nominal data The only allowable calculation on nominal data is to count the frequency of each value of a variable. When the raw data can be naturally categorized in a meaningful manner, we can display frequencies by Bar charts – emphasize frequency of occurrences of the different categories. Pie chart – emphasize the proportion of occurrences of each category.

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31 The Pie Chart The pie chart is a circle, subdivided into a number of slices that represent the various categories. The size of each slice is proportional to the percentage corresponding to the category it represents.

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32 Example 2.4 The student placement office at a university wanted to determine the general areas of employment of last year school graduates. Data was collected, and the count of the occurrences was recorded for each area. These counts were converted to proportions and the results were presented as a pie chart and a bar chart. The Pie Chart

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33 Marketing 25.3% Finance 20.6% General management 14.2% Other 11.1% Accounting 28.9% (28.9 /100)(360 0 ) = The Pie Chart

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34 Rectangles represent each category. The height of the rectangle represents the frequency. The base of the rectangle is arbitrary The Bar Chart

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35 Use bar charts also when the order in which nominal data are presented is meaningful. 0 5,000 10,000 15,000 20, Total number of new products introduced in North America in the years 1989,…,1994 The Bar Chart

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Describing the Relationship Between Two Variables We are interested in the relationship between two interval variables. Example 2.7 A real estate agent wants to study the relationship between house price and house size Twelve houses recently sold are sampled and there size and price recorded Use graphical technique to describe the relationship between size and price. Size Price ……………..

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37 Solution The size (independent variable, X) affects the price (dependent variable, Y) We use Excel to create a scatter diagram 2.5 Describing the Relationship Between Two Variables Y X The greater the house size, the greater the price

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38 Typical Patterns of Scatter Diagrams Positive linear relationship Negative linear relationship No relationship Negative nonlinear relationship This is a weak linear relationship. A non linear relationship seems to fit the data better. Nonlinear (concave) relationship

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39 Graphing the Relationship Between Two Nominal Variables We create a contingency table. This table lists the frequency for each combination of values of the two variables. We can create a bar chart that represent the frequency of occurrence of each combination of values.

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40 Example 2.8 To conduct an efficient advertisement campaign the relationship between occupation and newspapers readership is studied. The following table was created (To see the data click Xm02-08a)Xm02-08a Contingency table

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41 Solution If there is no relationship between occupation and newspaper read, the bar charts describing the frequency of readership of newspapers should look similar across occupations. Contingency table

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42 Blue-collar workers prefer the Star and the Sun. White-collar workers and professionals mostly read the Post and the Globe and Mail Bar charts for a contingency table

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Describing Time-Series Data Data can be classified according to the time it is collected. Cross-sectional data are all collected at the same time. Time-series data are collected at successive points in time. Time-series data is often depicted on a line chart (a plot of the variable over time).

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44 Line Chart Example 2.9 The total amount of income tax paid by individuals in 1987 through 1999 are listed below. Draw a graph of this data and describe the information produced

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45 For the first five years – total tax was relatively flat From 1993 there was a rapid increase in tax revenues. Line charts can be used to describe nominal data time series. Line Chart

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