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**Graphical Descriptive Techniques**

Chapter 2 Graphical Descriptive Techniques

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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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**2.2 Types 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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**Types of data - examples**

Interval data Nominal Age - income . . Person Marital status 1 married 2 single 3 single . . Weight gain +10 +5 . Computer Brand 1 IBM 2 Dell 3 IBM . .

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**Types of data - examples**

Interval data Nominal data With nominal data, all we can do is, calculate the proportion of data that falls into each category. Age - income . . Weight gain +10 +5 . IBM Dell Compaq Other Total 50% % % %

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**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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**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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**2.3 Graphical Techniques for Interval Data**

Example 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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**Example 2.1: Providing information**

Collect data Prepare a frequency distribution How many classes to use? Number of observations Number of classes Less then , 1,000 – 5, 5, , More than 50, Class width = [Range] / [# of classes] [ ] / [8] = (There are 200 data points Largest observation Largest observation Largest observation Largest observation Smallest observation Smallest observation Smallest observation Smallest observation

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**Example 2.1: Providing information**

Draw a Histogram

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**Example 2.1: Providing information**

What information can we extract from this histogram Relatively, large number of large bills About half of all the bills are small A few bills are in the middle range 80 71+37=108 =32 =60 60 Frequency 40 20 15 30 45 60 75 90 105 120 Bills

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Relative frequency 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

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Class width 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.

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**Shapes of histograms Symmetry**

There are four typical shape characteristics

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Shapes of histograms Skewness Negatively skewed Positively skewed

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**Modal classes A unimodal histogram**

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

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Modal classes A bimodal histogram A modal class A modal class

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**Bell shaped histograms**

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

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**Interpreting histograms**

Example 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)

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**Example 2.2 - Histograms Return on investment A Return on investment B**

The center for B The center for A 18- 16- 14- 12- 10- 8- 6- 4- 2- 0- 18- 16- 14- 12- 10- 8- 6- 4- 2- 0- Return on investment A Return on investment B Interpretation: The center of the returns of Investment A is slightly lower than that for Investment B

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**Example 2.2 - Histograms Return on investment A Return on investment B**

Sample size =50 Sample size =50 18- 16- 14- 12- 10- 8- 6- 4- 2- 0- 18- 16- 14- 12- 10- 8- 6- 4- 2- 0- 17 16 34 26 46 43 Return on investment A Return on investment B Interpretation: The spread of returns for Investment A is less than that for investment B

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**Example 2.2 - Histograms Return on investment A Return on investment B**

18- 16- 14- 12- 10- 8- 6- 4- 2- 0- 18- 16- 14- 12- 10- 8- 6- 4- 2- 0- Return on investment A Return on investment B Interpretation: Both histograms are slightly positively skewed. There is a possibility of large returns.

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**Providing information**

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.

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**Interpreting histograms**

Example 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.

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**Interpreting histograms**

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

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Stem and Leaf Display 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).

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**Stem and Leaf Display Split each observation into two parts.**

There are several ways of doing that: Observation: Stem Leaf 42 19 Stem Leaf 4 2 A stem and leaf display for Example 2.1 will use this method next.

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

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**} Ogives Ogives are cumulative relative frequency distributions.**

Example continued 120 1.000 105 .930 90 .790 } } 75 .700 60 .650 .540 .605 .355 15 30 45

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**2.4 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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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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The Pie Chart 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.

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**The Pie Chart (28.9 /100)(3600) = 1040 Other 11.1% Accounting 28.9%**

General management 14.2% Finance 20.6% Marketing 25.3%

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**The Bar Chart Rectangles represent each category.**

The height of the rectangle represents the frequency. The base of the rectangle is arbitrary 73 64 52 36 28

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The Bar Chart Use bar charts also when the order in which nominal data are presented is meaningful. Total number of new products introduced in North America in the years 1989,…,1994 20,000 15,000 10,000 5,000 ‘ ‘ ‘ ‘ ‘ ‘94

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**2.5 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 315 229 335 261 ……………..

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**2.5 Describing the Relationship Between Two Variables**

Solution The size (independent variable, X) affects the price (dependent variable, Y) We use Excel to create a scatter diagram Y The greater the house size, the greater the price X

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**Typical Patterns of Scatter Diagrams**

Positive linear relationship No relationship Negative linear relationship Negative nonlinear relationship Nonlinear (concave) relationship This is a weak linear relationship. A non linear relationship seems to fit the data better.

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**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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**Contingency table 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)

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**Contingency table 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.

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**Bar charts for a contingency table**

Blue-collar workers prefer the “Star” and the “Sun”. White-collar workers and professionals mostly read the “Post” and the “Globe and Mail”

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**2.6 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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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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**Line Chart 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.

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