Graphical Descriptive Techniques Chapter 2 Graphical Descriptive Techniques

2.1 Introduction Descriptive statistics summary and presentation of data, to support interpretation and decision making. Methods graphical techniques numerical descriptive measures. Apply to both the entire population the sample

2.2 Types of data and information A variable - a characteristic of population or sample. 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

Types of data - examples Interval data Nominal Age - income 55 75000 42 68000 . . Person Marital status 1 married 2 single 3 single . . Weight gain +10 +5 . Computer Brand 1 IBM 2 Dell 3 IBM . .

Types of data – analysis 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

Cross-Sectional/Time-Series Data Cross sectional data is collected at a certain point in time 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

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

Example 2.1: Providing information Collect data Prepare a frequency distribution How many classes to use? Number of observations Number of classes Less then 50 5-7 50 - 200 7-9 200 - 500 9-10 500 - 1,000 10-11 1,000 – 5,000 11-13 5,000- 50,000 13-17 More than 50,000 17-20 Class width = [Range] / [# of classes] [119.63 - 0] / [8] = 14.95 15 (There are 200 data points Largest observation Largest observation Largest observation Largest observation Smallest observation Smallest observation Smallest observation Smallest observation

Example 2.1: Providing information Draw a Histogram

Relative frequency Relative frequencies should be used when 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 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. several classes are combined together to form a wider class. It is possible to form an open ended class at the higher end or lower end of the histogram.

Shapes of histograms There are four typical shape characteristics Symmetric distribution

Shapes of histograms Skewness Negatively skewed Positively skewed

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

Modal classes A bimodal histogram A modal class A modal class

Bell shaped histograms Drawing the histogram helps verify the shape of the population in question

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

Stem and Leaf Display Stem and leaf diagrams use the actual value of the original observations (whereas, the histogram does not).

Stem and Leaf Display Split each observation into two parts. There are several ways of doing that: 42.19 42.19 Observation: Stem Leaf 42 19 Stem Leaf 4 2 A stem and leaf display for Example 2.1 will use this method next.

Stem and Leaf Display A stem and leaf display for Example 2.1 Stem Leaf 0 0000000000111112222223333345555556666666778888999999 1 000001111233333334455555667889999 2 0000111112344666778999 3 001335589 4 124445589 5 33566 6 3458 7 022224556789 8 334457889999 9 00112222233344555999 10 001344446699 11 124557889 The length of each line represents the frequency of the class defined by the stem.

} Ogives Ogives are cumulative relative frequency distributions. Example 2.1 - continued 120 1.000 105 .930 90 .790 } } 75 .700 60 .650 .540 .605 .355 15 30 45

2.4 Graphical Techniques for Nominal data The raw data can be categorized and we can display frequencies by Bar charts Pie chart

The Pie Chart (28.9 /100)(3600) = 1040 Other 11.1% Accounting 28.9% General management 14.2% Finance 20.6% Marketing 25.3%

The Bar Chart 73 64 52 36 28

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 ‘89 ‘90 ‘91 ‘92 ‘93 ‘94

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 Use graphical technique to describe the relationship between size and price. Size Price 315 229 335 261 ……………..

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

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.

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.

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)

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.

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”

2.6 Describing Time-Series Data Time-series data is often depicted on a line chart (a plot of the variable over time).

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

Line Chart For the first five years – total tax was relatively flat From 1993 there was a rapid increase in tax revenues.