july, 2000guang jin statistics in applied science and technology chapter 3 organizing and displaying...

July, 2000 Guang Jin

Statistics in Applied Science and Technology

Chapter 3

Organizing and Displaying Data


Key Concepts in this Chapter

• Scale of measurement: nominal, ordinal, and interval-ratio

• qualitative and quantitative variables

• discrete and continuous variables

• frequency distribution

• symmetrical, bimodal and skewed distributions

• positively and negatively skewed distributions

• frequency polygon, bar chart, pie chart, box and Whisker Plots


Scale of Measurement (Section 3.1)• Differentiates how variables are measured by the researcher

• Four scales:• Nominal• Ordinal• Interval• Ratio

• Through the researcher’s operational definitions of the variables, the scale are defined. These along with the research question, will determine the appropriate statistical analysis


Scale of Measurement: Nominal• Nominal: categorical/classificatory


Scale of Measurement: Ordinal

• Ordinal: rank-order

1st 2nd 3rd


• Interval: equal units, arbitrary zero point.

Scale of Measurement: Interval

30F 90F


Scale of Measurement: Ratio

• Ratio: interval but, in addition, includes zero and can be meaningful.


Qualitative & Quantitative Variables

Qualitative Quantitative

Nominal Interval

Ordinal Ratio


Quantitative Variables

• Quantitative variable can be classified further as discrete and continuous

• Discrete variables must always be integers (e.g., 0, 1, 2, etc.)

• Continuous variables may take on fractional values (e.g., 37.8, 138.2, etc.)


• In your job, what types of data are typically collected and used?

• For each type of data, specify the scale of measurement.

• For each type of data, specify whether it is quantitative or qualitative.

STOP THINK APPLY


Frequency Distribution

• A table (or a graph or an equation) that includes a set of intervals and displays their frequency (numbers of cases or occurrences) in each intervals.

Example: – Frequency Table for Systolic Blood Pressure

of Nonsmokers from Table 3.1 (Pg. 28)


Some Important features of Frequency Distribution• Class frequency - The number of observations falling

into any given interval is called the class frequency.

• Relative frequency - represents the relative percentage of one particular class interval to total cases of any class intervals (total frequency).

• Cumulative relative frequency (cumulative percentage or percentile) - gives that percentage of individuals having a measurement less than or equal to the upper boundary of the class interval.


• Unit of Measurement - The smallest possible difference between observations.

• Class Interval Width - The distance between the two tabled boundaries, after each boundary has been expanded by one-half of one unit of measurement.

Frequency Distribution (Continued)


Basic Guidelines for Frequency Distribution of grouped quantitative data

• Each observation should be included in one, and only one, class.

• List all classes, even those with zero frequencies.

• All classes (with both upper and lower boundaries) should be equal in width.


Usefulness of Tables

• Demonstrate patterns, trends and other kinds of relationship

• Serve as the basis for more visual displays of data such as graphs and charts

• Not overuse it.


Essential Components of tables

• Title– What are the data, e.g., percentages,

proportions, frequency distribution?– Who do the data represent, e.g., college

students, Health Sciences students?– Where are the data from, e.g., Illinois State

University, University of Illinois?


• Boxhead - column headings/captions– as few words as possible, yet precise

• Stub - row headings/captions– appropriate grouping

• Cell - the box formed by the intersection of columns and rows



• Footnote– Definitions– Abbreviations– explanation for any unusual numbers

• Source– If data are used from a source outside your

research, the exact reference to the source should be given.



Graphing Data(Section 3.4)

0

10

20

30

40

50

60

70

80

90

100

1st Qtr 2ndQtr

3rd Qtr 4th Qtr

East

West

North


Histogram

• Horizontal axis - depicts the class boundaries (not limits)

• Vertical axis - depicts the frequency (or relative frequency)

• Frequencies are represented not only by height but also by the area of each bar.


Frequency Polygon

• Frequency polygon uses the same axes as the histogram and is constructed by marking a point (at the same height as the histogram’s bar) at the midpoint of the class interval.


Typical Shapes of Frequency Polygons• Symmetrical -such as classic bell-shaped

• Bimodel - two peak frequencies

• Rectangular distribution - each class interval is equally represented.

• Positively skewed - a few extreme observations with relatively large values in the positive direction (tapers off in the positive direction).

• Negatively skewed - a few extreme observations with relatively small values in the negative direction (tapers off in the negative direction)


Cumulative Frequency Polygons (Ogive)• Horizontal scale is the same as that used for

a histogram

• Vertical scale indicates cumulative frequency or cumulative relative frequency.

• Percentiles can be obtained from an ogive.


Stem-and-leaf Displays

• “Stem” represent the class intervals

• “Leaves” are the strings of values within each class interval.

• Stem-and-leaf displayed all observations and provided a visual description of the shape of the distribution.


Bar Charts

• Particularly useful for displaying nominal or ordinal data

• Horizontal axis - represents various categories• Vertical axis - represents frequency or relative

frequency• In bar chart, relative frequencies are shown by

heights, but in a histogram, relative frequencies are shown by the areas within the bars.


Pie Chart

• In a pie chart, a circle is divided into wedges that correspond to the percentage frequencies of the distribution.

• Pie char is often used for displaying nominal or ordinal data with a small number of categories.


Box and Whisker Plots

• Box and Whisker Plots displays the median and quartile statistics in the same plot (Figure 3.10, Pg. 40)

• Median is the score that divides a ranked series of scores into two equal halves.

• Quartiles divide the scores into four equal groups.

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