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``CHAPTER TWO
Graphs, Charts, and Tables
Many business periodicals, such as Fortune and Business Week, use graphs and charts extensively in conjunction with their article to help readers better understand key concepts. Also, many advertisements use graphs and charts to effectively convey their message. Because of that, this chapter introduces some of the most frequently used tools and techniques for describing data with graphs, charts, and tables.
2-1 Frequency Distributions and Histograms
Objectives
To construct, both manually and with a computer, frequency distributions.
To construct and interpret a frequency histogram.
To develop and interpret joint frequency tables.
What is a frequency distribution? And what is discrete data?
If we have two frequency distributions with different number of total observations, it is difficult to compare the frequencies of each category directly. So, comparisons are done using the relative frequencies which are computed by; EMBED Equation.3 where;
fi = frequency of the ith value of the discrete variable.
n = EMBED Equation.3 and
k = the number of different values for the discrete variable.
To develop a discrete data frequency distribution, perform the following steps:
List all possible values of the variable.
List the number of occurrences corresponding to each value.
And to develop a relative frequency distribution, do the following:
Divide each frequency count by the total number of data values.
In many cases the variable of interest is continuous (e.g. weight, time, length) or discrete and have many possible outcomes (e.g. age, income, stock prices). In such cases data are described using a grouped (continuous) frequency distribution.
Even when sorted, in ascending or descending order forming an array, the continuous data provide little information about the variable of interest. A discrete frequency distribution is not useful for such a data type. Instead we need to group the data into classes and count the number of observations in each class.
The classes should meet 4 criteria;
They must be mutually exclusive, that is non-overlapping classes so that the data can be placed in only one class.
They must be all inclusive, that is containing all the possible data values.
If at all possible they should be of equal length, that is the distance between the lowest and the highest possible values in each class is equal for all classes.
Avoid empty classes if possible.
A formula known as Sturgess Rule is often used to provide a guideline for determining the number of classes for a given size, n, of data. It is given by the formula;
# of Classes = 1 + 3.322 [log10(n)].
There are six steps for constructing a grouped frequency distribution. Relative frequencies can be transformed into percentages if multiplied by 100. Another extension can be constructed is the cumulative frequency distribution, listing the frequencies less than or equal to the upper limit of each class, and the cumulative relative frequency distribution, listing the relative frequencies less than or equal to the upper limit of each class.
Although frequency distributions are useful in analyzing large sets of data, they are in table format and may not be visually informative as a graph. What is a histogram?
A histogram shows three general types of information;
It provides a visual indication of where the approximate center of the data is.
The degree of spread (or variation) in the data is determined.
The shape of the distribution can be observed.
To construct an ogive (cumulative relative frequency polygon) perform the following steps;
Follow the steps of constructing a cumulative relative frequency distribution.
Use the horizontal axis to represent the upper class limits, and the vertical axis to represent the cumulative relative frequency.
Plot points above each class upper limit at a height equal to its corresponding cumulative relative frequency.
Connect the plotted points by line segments to form a polygon.
When studying the relation between two, quantitative or qualitative, variables, it is often reasonable to construct a joint frequency distribution for the two variables following the given steps;
Obtain a set of data consisting of paired responses for two variables.
Construct a table with r rows for one variable and c columns for the other.
Count the number of joint occurrences at each row level and each column level for all combinations of the variables.
Compute the row and column totals, which are called the marginal frequencies.
If a joined relative freq. dist. is required, divide each cell freq. by the total number of observations.
2-2 Bar & Pie Charts and Stem-and-Leaf Diagrams
Objectives
To construct and interpret various types of bar charts.
To construct and interpret a pie chart.
To construct and interpret a stem-and-leaf diagram.
What is a bar chart? And for what type of data it is used?
A bar chart is a graphical representation of a categorical data set in which a rectangle or bar is drawn over each category or class. Additionally, multiple variables can be depicted on the same bar chart.
A bar chart is constructed using the following steps:
Define the categories of the variable of interest.
For each category determine the appropriate measure or value.
For a vertical bar chart, locate the categories on the horizontal axis and the freq. or percentage on the vertical. The bar height is equal to the freq. or the percentage.
What is a pie chart? And for what type of data it is used?
A pie chart is a graph, of a categorical (qualitative) data set, in the shape of a circle. The circle is divided into slices corresponding to the categories or classes to be displayed. The size of each slice is proportional to the magnitude of the displayed variable (percentage or freq.) associated with each category.
A pie chart is constructed using the following steps:
Define the categories of the variable of interest (relative frequency).
Construct the pie chart by displaying one slice for each category that is proportional in size to the proportion of that category.
The stem-and-leaf diagram is similar to the histogram in that it displays the distribution for the quantitative variable. However, unlike the histogram, in which the individual values of the data are lost when the variable of interest is broken into classes, the stem-and-leaf diagram shows the individual data values.
To construct a stem-and-leaf diagram for a quantitative data set follow the steps:
Split the data into two parts the smallest significant digit is represented in the leaf part and, the rest of the digits are represented in the stem part.
Sort the data increasingly.
List all possible stems in a single column increasingly.
For each stem, list all the leaves associated with the stem in a row increasingly.
2-3 Line Chart & Scatter Diagrams
Objectives
To create a line chart and interpret the trend in data.
To construct a scatter plot and interpret it.
Most of the examples that have been presented thus far have involved cross-sectional data, or data gathered from many observations taken at the same time. However, if we have data that are measured over time (a time series), an effective tool for presenting such data is a line chart.
A line chart is a two-dimensional chart showing time on the horizontal axis and the variable of interest on the vertical axis.
Another tool used to study the relation between two variables simultaneously is the scatter diagram or the scatter plot.
A scatter plot is a two-dimensional graph of plotted points in which the vertical axis represents one of the variables (dependent or response) and the horizontal axis represents the other (independent or predictor). Each plotted point has coordinates whose values are obtained from the respective variables.
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