Visual Presentation and Numerical Measures: Describing Data in Business Statistics

Describing Data

Types of Variables

In statistics, variables are classified based on their nature and measurement scale. Understanding variable types is essential for selecting appropriate methods of data presentation and analysis.

• Qualitative Variables: Describe categories or qualities (e.g., class standing, gender).

• Quantitative Variables: Represent numerical values and can be further divided into:

  • Discrete: Countable values (e.g., number of siblings).

  • Continuous: Measurable values within a range (e.g., height, time spent).

Key Terms

• Class: One of the categories for qualitative data.

• Class Frequency: Number of observations in a particular class.

• Class Relative Frequency: The frequency of the class divided by the total number of observations.

• Class Percentage: The relative frequency multiplied by 100 to express as a percentage.

Data Presentation

Methods for Qualitative and Quantitative Data

Data can be presented visually to facilitate understanding and comparison. The choice of method depends on the type of data.

• Qualitative Data:

  • Summary Table

  • Bar Graph

  • Pie Chart

  • Pareto Diagram

• Quantitative Data:

  • Dot Plot

  • Stem-and-Leaf Display

  • Frequency Distribution

  • Histogram

Example Table: Frequency and Relative Frequency of Class Standing

Class	Frequency	Relative Frequency
Freshman	8	8/29 = 0.28
Sophomore	13	13/29 = 0.45
Junior	5	5/29 = 0.17
Senior	3	3/29 = 0.10
Total	29	1.00

Bar Chart

• Displays frequencies of categories using bars.

• Useful for comparing different groups.

• Example: Bar chart of class standing for ST2113.

Pie Chart

• Shows breakdown of total quantity into categories.

• Useful for showing relative differences.

• Angle size is proportional to percentage: $360^ ext{o} imes ext{percent}$

Pareto Diagram

• Similar to a bar graph, but categories are arranged in descending order of frequency.

• Highlights the most significant categories.

Graphical Methods for Describing Quantitative Data

Dot Plot

• Each observation is plotted as a dot above a horizontal axis.

• Stacked dots represent repeated values.

• Example: Prices of DVD players.

Stem-and-Leaf Display

• Divides each observation into a stem (leading digit(s)) and leaf (trailing digit).

• Stems are listed in a column; leaves are placed in corresponding rows.

• Preserves original data values and shows distribution.

Frequency Distribution and Histogram

• Groups quantitative data into intervals (classes).

• Frequency distribution table shows tally and relative frequency.

• Histogram displays frequency or relative frequency for each interval as bars.

Example Table: Days to Maturity for 40 Short-Term Investments

Days to Maturity	Tally	Frequency	Relative Frequency
30-39	|||	3	0.075
40-49	|||||	5	0.125
50-59	|||||||||	9	0.225
60-69	|||||||||	10	0.25
70-79	|||||	7	0.175
80-89	|||	4	0.1
90-99	|	2	0.05
Total		40	1.00

Numerical Measures of Central Tendency

Mean

The mean is the arithmetic average of a data set.

• Formula: $\text{mean} = \frac{\sum x_i}{n}$

• Represents the center of the data.

• Symbols: Greek letters (e.g., $\mu$) for population mean, Roman letters (e.g., $\bar{x}$) for sample mean.

Median

• Middle value in an ordered sequence.

• Not affected by extreme values (outliers).

• For odd-sized samples: median is the middle value.

• For even-sized samples: median is the average of the two middle values.

Mode

• Value that occurs most often in the data set.

• May be no mode, one mode, or multiple modes.

• Not affected by extreme values.

• Applicable to both quantitative and qualitative data.

Numerical Measures of Variability

Range

• Difference between largest and smallest observations.

• Formula: $\text{Range} = \text{Maximum} - \text{Minimum}$

• Simple measure of dispersion; does not consider distribution of data.

Variance and Standard Deviation

• Measure how data are distributed around the mean.

• Most common measures of dispersion.

• Show variation about the mean ($\bar{x}$ for sample, $\mu$ for population).

Sample Variance Formula

$s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}$

Sample Standard Deviation Formula

$s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}}$

Alternative Formula for Sample Variance

$s^2 = \frac{\sum x^2 - (\sum x)^2 / n}{n-1}$

Symbols for Sample and Population Variance and Standard Deviation

• Population variance: $\sigma^2$

• Sample variance: $s^2$

• Population standard deviation: $\sigma$

• Sample standard deviation: $s$

Summary Table: Measures of Central Tendency and Variability

Measure	Definition	Formula
Mean	Arithmetic average	$\bar{x} = \frac{\sum x_i}{n}$
Median	Middle value	Depends on ordered data
Mode	Most frequent value	None
Range	Max - Min	$\text{Range} = \text{Max} - \text{Min}$
Variance	Average squared deviation from mean	$s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$
Standard Deviation	Square root of variance	$s = \sqrt{s^2}$

Additional info:

• These notes cover the essential methods for describing sets of data, including both visual and numerical techniques, as outlined in Chapter 2 of a typical Statistics for Business course.

• Examples and tables are based on class standing and investment maturity data, which are common applications in business statistics.