Comprehensive Notes: Quantitative Methods I - Data Collection and Presentation

Meaning and Uses of Quantitative Methods By Businesses

• Definitions

  • It is a process of decision making which places emphasis on the quantification of variables that are of concern to authorities in decision making.

  • It is the process of applying mathematical and statistical models in complex business problems to simplify it for business policy formulation and strategy development.

• Uses of Quantitative Methods in Business

  • It is useful in the following areas of management decisions:

  • Planning

  • Forecasting

  • Controlling

  • Investment and Portfolio management

  • Financial and Economic analysis

Tools of Quantitative Methods

• Differentiation and Integration: essential for planning

• Break even Analysis: Useful for planning and control purposes

• Time Value of Money: Useful in investment decision making

• Matrices: Essentially useful in Portfolio and Risk Management

• Correlation and Regression: Essential for understanding relationships, predicting and forecasting

• Time Series Analysis: Used for forecasting

• Probability: Used to determine the likelihood of an event happening

• Linear Programming: Useful in finding optimal solutions to economic decisions pertaining to profitability and cost minimization

Topic 2: Data Collection and Presentation

Outline

• Bar Charts, Pie Charts, Pictograms

• Histograms

• Cumulative Frequency Curves

• Lorenz Curves

• Application in Practice

Data Collection

• Data: raw numbers or facts that are processed to produce information

• In other words data refers to unprocessed facts or numbers

• Data does not lend itself to understanding unless it is processed to become information

• Levels of Data Measurement

• Types of Data

• Primary Data

• Secondary Data

Data Collection: Levels of Data Measurement

• Nominal level of data

• Ordinal level of data

• Interval level of data

• Ratio level of data

Data Collection: Types of Data

1. Primary Data

   • Primary data refers to data collected by going to the field for a first-hand experience

   • Sources of Primary Data include:

     • Experiments

     • Survey

     • Questionnaire

     • Interview

     • Observations

2. Secondary Data

   • Data collected from a source that has already been published in any form

   • Sources of Secondary Data include:

     • Books

     • Magazines

     • Online database

     • Financial Statements

     • Etc.

Data Presentation

• Data presentation clearly depicts the characteristics of a set of raw facts and clearly identifies the pattern of information from the data to easily assist decision making by management.

• The forms of Data Presentation Include:

  • Bar Chart

  • Pie Chart

  • Histogram

  • Frequency Polygon

  • Cumulative Frequency Curve or Ogive

  • Lorenz Curve

Data Presentation: Bar Chart

• Bar Chart definition: It is a graphical method of presenting qualitative data that have been summarized in a frequency distribution or a relative frequency distribution.

• Illustration 1 (Example data):

  • Scores (Marks) | Frequency

  • 19 | 5

  • 13 | 9

  • 9 | 6

  • Total | 20

• Solution 1 (How to read the bar chart):

  • Bars of equal width are drawn to represent various classes (in this case, grades).

  • The height of each bar represents the frequencies of various classes.

• Illustration 2 (Dataset for 150 students):

  • Grade | No Of Students

  • A | 12

  • B+ | 25

  • B | 10

  • C+ | 35

  • C | 30

  • D+ | 16

  • D | 14

  • E | 8

Data Presentation: Pie Chart

• Pie chart is a graphical device for presenting qualitative data by subdividing a circle into sectors that correspond to the relative frequency of each class.

• Steps for constructing a pie chart:

  • Convert the frequency for each class into a proportional part of the circle using the formula $\text{Degrees} = 360 \times \frac{f}{n}$ where $f$ is the frequency for each class and $n$ is the sum of the frequencies.

  • Find the Degrees corresponding to each class.

  • Using a protractor, graph each section and write its name and corresponding percentage.

• Illustration 1 (Example dataset: Monthly Expenditure)

  • Items: Food = 360, Clothing = 162, Airtime = 198, Miscellaneous = 180; Total = 900

  • Degrees:

  • Food: $360/900 \times 360 = 144^{\circ}$

  • Clothing: $162/900 \times 360 = 64.8^{\circ}$

  • Airtime: $198/900 \times 360 = 79.2^{\circ}$

  • Miscellaneous: $180/900 \times 360 = 72^{\circ}$

  • Check: Sum of degrees = 360 degrees

Data Presentation: Histogram

• Definition: A graphical method of presenting a frequency or a relative frequency distribution of quantitative data. The histogram uses contiguous vertical bars to display the frequency of the data (except where the frequency is 0).

• Steps for constructing a histogram:

  • Draw and label the x (horizontal) and the y (vertical) axes.

  • Represent the frequencies on the y axis and the class boundaries on the x axis.

  • Use the frequencies as the heights and draw vertical bars for each class.

  • Note: For the histogram we need the frequencies and the class boundaries.

• Illustration 1 (Example: 50 states high temperatures):

  • Class limits (Degrees) | Number of states

  • 100–104 | 2

  • 105–109 | 8

  • 110–114 | 18

  • 115–119 | 13

  • 120–124 | 7

  • 125–129 | 1

  • 130–134 | 1

• Data Presentation Solution 1 (Class boundaries and frequencies)

  • Class Limits -> Class Boundaries -> Frequency -> Cumulative Frequency

  • 100–104 -> 99.5–104.5 -> 2 -> 2

  • 105–109 -> 104.5–109.5 -> 8 -> 10

  • 110–114 -> 109.5–114.5 -> 18 -> 28

  • 115–119 -> 114.5–119.5 -> 13 -> 41

  • 120–124 -> 119.5–124.5 -> 7 -> 48

  • 125–129 -> 124.5–129.5 -> 1 -> 49

  • 130–134 -> 129.5–134.5 -> 1 -> 50

• Illustration 2 (Histogram for QM quiz marks):

  • Data: Marks (0–5, 5–10, 10–15, 15–20, 20–25, 25–35, 35–50) with frequencies (25, 80, 120, 160, 130, 96, 60)

Data Presentation: Frequency Polygon

• Definition: A graph that displays the data by using lines that connect points plotted for the frequencies at the midpoints of the classes.

• Illustration 1 (Example): Minutes used for studies by 50 students

  • Class Intervals and Frequencies (example from slides):

  • 100–104 | 2

  • 105–109 | 8

  • 110–114 | 18

  • 115–119 | 13

  • 120–124 | 7

  • 125–129 | 1

  • 130–134 | 1

• Solution 1: The frequency polygon connects midpoints of each class with the corresponding frequencies.

• Class boundaries and midpoints table (from slides):

  • 99.5–104.5 Midpoint = 102 Frequency = 2

  • 104.5–109.5 Midpoint = 107 Frequency = 8

  • 109.5–114.5 Midpoint = 112 Frequency = 18

  • 114.5–119.5 Midpoint = 117 Frequency = 13

  • 119.5–124.5 Midpoint = 122 Frequency = 7

  • 124.5–129.5 Midpoint = 127 Frequency = 1

  • 129.5–134.5 Midpoint = 132 Frequency = 1

• Solution 1 (continued): Frequency polygon construction uses these midpoints and frequencies to plot and connect points.

Data Presentation: Cumulative Frequency Curve (Ogive)

• Steps for constructing an ogive:

  • Draw and label the x (horizontal) and the y (vertical) axes.

  • Represent the cumulative frequencies on the y axis and the upper class boundaries / Class Midpoint on the x axis.

  • Plot the cumulative frequencies against their respective upper class boundary / Class Midpoint.

  • Join the points plotted by a smooth curve.

• Illustration 1: Given data, construct an ogive

  • Marks: 0–10, 11–20, 21–30, 31–40, 41–50, 51–60, 61–70, 71–80

  • Frequencies: 2, 8, 12, 18, 28, 22, 6, 4

• Data Presentation Solution 1 (Ogive data):

  • Class Boundaries -> Frequency -> Cumulative Frequency

  • 0.5–10.5 -> 2 -> 2

  • 10.5–20.5 -> 8 -> 10

  • 20.5–30.5 -> 12 -> 22

  • 30.5–40.5 -> 18 -> 40

  • 40.5–50.5 -> 28 -> 68

  • 50.5–60.5 -> 22 -> 90

  • 60.5–70.5 -> 6 -> 96

  • 70.5–80.5 -> 4 -> 100

• Solution 1 continued: Graphical ogive with upper class boundaries on the x-axis and cumulative frequency on the y-axis.

• Trial 2: Draw an Ogive for marks scored by QM quiz candidates (total 100). Data: 10–20: 21; 20–30: 19; 30–40: 60; 40–50: 42; 50–60: 24; 60–70: 18; 70–80: 17

Data Presentation: Lorenz Curve

• Definition: The Lorenz curve is an extension of the ogive used in economics to show the distribution of income or wealth among the population. It is a graph of cumulative percentage wealth, income, or some other measure of wealth, against cumulative percentage of population. Main use: determine the fairness of income distribution.

• Steps to Draw a Lorenz Curve

  • Convert the population into percentages and rank them (ascending/descending).

  • Convert the wealth/income to percentages and write them by their respective percentage population.

  • Accumulate the population percentages to get the cumulative percentage of population.

  • Accumulate the income/wealth percentages to get the cumulative percentage of income/wealth.

  • Represent cumulative percentage of population on the X axis and cumulative percentage of wealth/income on the Y axis.

  • Plot the cumulative percentage of population against the cumulative percentage of wealth and join the points with a smooth curve.

  • Draw the line of equity from (0%, 0%) to (100%, 100%).

• Illustration 1: Revenue offices in Astoi District presented the following table on the percentages of total wealth before and after tax owned by various percentages of population. Task: Draw a Lorenz curve and comment on the effectiveness of the tax imposition.

  • Data table (percentages): Population vs Income before Tax vs Income After Tax (values shown in slides)

• Data Presentation Solution 1: The Lorenz curve is used to compare distributions before and after tax; the line of equity is the 45-degree line from (0,0) to (100,100).

• Using Lorenz curves to assess tax: after-tax curve moving closer to the line of equality indicates a more fair distribution of wealth.

• Illustration 2: Trial 3 – Population and wealth distribution of traders in a region before and after an E-levy. Task: Draw Lorenz curve and comment on effectiveness of E-levy.

• Trial 4: What are the uses of a Lorenz curve?

  • Uses include assessing inequality, comparing distributions across groups, visualizing changes in distribution over time, and informing policy discussions.

End of Lecture

• This concludes the data collection and presentation content for Quantitative Methods I.