Download Descriptive Statistics: Graphical Techniques for Quantitative Data and more Exams Statistics in PDF only on Docsity!

1. Data, Graphical Descriptive Techniques

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

Types of data

A variable is a characteristic of population or sample that is of interest for us, for instance,  Cereal choice  Capital expenditure  The waiting time for medical services Data - the actual values of variables  Quantitative data are numerical observations  Qualitative data are categorical observations Quantitative data Age - income 55 75000 42 68000

.. .. Weight gain

. .

Qualitative data Person Married/unmarried 1 yes 2 no 3 no

.. .. Professor Rank 1 Lecturer 2 Full 3 Assistant .. .. With qualitative data, all we can do is to calculate the proportion of data that falls into each category. Lecturers Assistant Associate Full Total 15 25 5 15 60 25% 41.67 8.33% 25% 100% Knowing the type of data is necessary to properly select the technique to be used. Type of analysis allowed for each type of data  Quantitative data - arithmetic calculations  Qualitative data - counting the number of observation in each category Cross-sectional and Time-Series Data Cross-sectional data is collected at a certain point in time, for example,  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, for instance,  Weekly closing price of gold  Amount of crude oil imported monthly

Graphical Techniques for Quantitative Data

Histogram

Example: Providing information concerning the monthly bills of new subscribers in the first month after signing on with a telephone company.

1. Collect data
2. Prepare a frequency distribution
3. Draw a histogram Collect data Prepare a frequency distribution Draw a Histogram How many classes to use?

of observations # of classes

Less then 50 5- 

50 - 200 7- 200 - 500 9- 500 - 1,000 10- Class width = Range / # of classes Range = Largest Observation – Smallest Observation What information can we extract from this histogram?

 About half of all the bills are small

 A few bills are in the middle range

 Relatively, large number of large bills

Bills

. . Bin Frequency 15 71 30 37 45 13 60 9 75 10 90 18 105 28 120 14 More 0 0 20 40 60 80 (^153045607590105120) More Bills Frequency

Relative Frequency It is often preferable to show the relative frequency (proportion) of observations falling into each class, rather than the frequency itself. Class relative frequency = Class frequency/Total number of observations 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 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 Shapes of histograms There are four typical shape characteristics Symmetry

Skewness

 Positively skewed 

 Negatively skewed 

Number of modal classes (A modal class is the one with the largest number of observations)

 unimodal histogram

 bimodal histogram

Bell shaped histogram

 Many statistical techniques require that the population be bell shaped.

 Drawing the histogram helps verify the shape of the population in question

Stem and Leaf Display

This is an interval-scaled display, most useful in preliminary analysis. Stem and leaf diagram shows the value of the original observations (whereas the histogram “loses” them). Creating a stem and leaf display Observe the data in the table below 19.1 19.8 18.0 19.2 19.5 17.3 20.0 20. 19.6 18.5 18.1 19.7 18.4 17.6 21.2 20. 22.2 19.1 21.1 19.3 20.8 21.2 21.0 18. 19.9 18.7 22.1 17.2 18.4 21. Determine what constitutes a stem and a leaf (there is more than one way). For example:

 the digits to the left of the decimal point is the stem

 the digits to the right of the decimal point is the leaf

List the stems in a column from smallest to largest. Place each leaf at the same row as its stem. The complete display is: Stem Leaf 17 623 18 4705147 19 1983627571 20 038 21 12204 22 12 Conclusions from the stem and leaf display.

 The observations range from 17.2 to 22.2.

 Most of the observations fall between 18.0 and 20.0.

 The shape of the distribution is not symmetrical.

 Half the observations are below 19.5 and half above it.

Pie Charts, Bar Charts, Line Charts

 The graphical presentations shown here are used primarily for qualitative data.

 These graphical tools are most appropriate when the raw data can be naturally

categorized in a meaningful manner. Pie Charts  Pie chart is a very popular tool used to represent the proportions of appearance for nominal data.  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. Bar Charts

 Bar charts provide an alternative to pie charts.

 The frequency (or relative frequency) of each category is represented by a vertical

bar.

 Use bar charts also when the order in which qualitative data are presented is

meaningful. Line charts

 Plot the frequency of a category above the point on the horizontal axis representing

that category.

 Use line charts when the categories are points in time.

References:

1. Chase and Bown, General Statistics 2. Hildebrand and Ott, Statistical Thinking for Managers 3. Keller and Warrack, Statistics for Management and Economics 4. McClave, Benson, and Sincich, A First Course In Business Statistics