Descriptive Statistics and Analytics: Tabular and Graphical Methods

Chapter 2: Business Statistics and Analytics in Practice

2.1: Graphically Summarizing Qualitative Data

• Learning Objective 2.1: Summarize qualitative data using frequency distributions, bar charts, and pie charts.
• Qualitative Data: Composed of categories identified by names. Can be summarized using frequency distributions.
• Frequency Distribution:
  • Definition: A table summarizing the number of items in several nonoverlapping classes.

Example 2.1: Describing Pizza Preferences

• Data Source: Table 2.1 lists pizza preferences from 50 college students.


• Frequency Distribution Table 2.2: Pizza Preferences
  

• Relative Frequency: Proportion of items in each class calculated as:
  • $ext{Relative Frequency} = \frac{\text{Frequency of the class}}{\text{Total number of observations}}$
• To convert it into percent frequency:
  • $ext{Percent Frequency} = ( ext{Relative Frequency}) \times 100$

Example Relative Frequency Distribution

• Frequency Terminology:
  • Relative Frequency Distribution: Table listing relative frequency for each class.
  • Percent Frequency Distribution: Table listing percent frequency for each class.

Bar Charts and Pie Charts

• Bar Chart: Rectangles representing frequency for each category; height can be frequency, relative frequency, or percent frequency.
• Pie Chart: Circle divided into slices, where the size of each slice represents its relative frequency or percent frequency.

The Pareto Chart

• Definition: A bar chart with defects listed on the horizontal scale; bar height indicates frequency.
  • Bars arranged in decreasing height from left to right; often shows cumulative percentage points.

2.2: Graphically Summarizing Quantitative Data

• Purpose: To summarize and describe the shape of distributions of population or samples by grouping measurements into classes.
• Frequency Distribution: List of data classes with counts of values belonging to each class represented in a table, often visualized as a histogram.

Constructing a Frequency Distribution

1. Determine the number of classes ($K$).
2. Calculate class length:
   • $ext{Class Length} = \frac{\text{Largest measurement - Smallest measurement}}{K}$
   • Round to a suitable number (e.g., days).
3. Create nonoverlapping classes of equal width.
4. Tally and count the number of measurements in each class.
5. Graph the histogram.

Example 2.2: The e-Billing Case

• Sample Payment Times (in days): 65 invoices ranging from 10 to 27 days.

Frequency Classes:

• Calculate $K$ as the smallest integer for which $2^K$ is at least 65; results in $K = 7$ classes.

Class Ranges Example:

• Class 1: 10 days to less than 13 days
• Class 2: 13 days to less than 16 days
• Class 3: 16 days to less than 19 days
• Class 4: 19 days to less than 22 days
• Class 5: 22 days to less than 25 days
• Class 6: 25 days to less than 28 days
• Class 7: 28 days to less than 31 days

Histogram Representation

• Rectangles represent classes; base corresponds to class length; height corresponds to frequency.

Common Distribution Shapes

• Right Skewed: Longer right tail in histogram.
• Left Skewed: Longer left tail in histogram.
• Symmetrical: Right and left tails are mirror images.

Frequency Polygons

• Points plotted above each class midpoint at height equal to the frequency. Useful for comparing distributions.

Cumulative Distributions

• Created from the same classes; counts measurements less than upper boundary of the class, forming a running total.

Example Table 2.10: Cumulative Distributions

1. (Class)
2. (Frequency)
3. (Cumulative Frequency)
4. (Cumulative Relative Frequency)
5. (Cumulative Percent Frequency)

• Displayed as cumulative totals alongside frequencies.

Ogive

• Graph of cumulative distribution; plots points above each upper-class boundary and connects with lines.

2.3: Dot Plots

• Definition: Simple graph summarizing datasets, useful to detect outliers.
• Example: Display exam scores across attendance policies via dot plots.

2.4: Stem-and-Leaf Displays

• Definition: Graph displaying measurements in order and allows visibility of all measurements and shape of data distribution.
• Best suited for small to moderate-sized distributions.

2.5: Contingency Tables

• Definition: Classifies data across two dimensions (rows and columns).
  • Needs a row variable, column variable, and counting variable.
• Example: A cross-tabulation of fund types versus client satisfaction levels.

Row and Column Percentages in Contingency Tables

• Calculation:
  • Row Percentages: Frequency of each cell divided by its row total.
  • Column Percentages: Frequency of each cell divided by its column total.

2.6: Misleading Graphs and Charts

• Examples showcasing the manipulation of vertical axes’ starting points to misrepresent data trends.
• Importance of Scale: Choosing where to start a vertical scale can significantly affect data interpretation.