Business Statistics

Pass path: Construct frequency distribution tables, draw Histogram and Ogive curves, and calculate pie chart sector angles. Top path: Differentiate inclusive vs exclusive class intervals with true class boundaries, and determine Median and Quartiles graphically from intersecting Ogives.

Part 1: Core Concepts (2 Marks)

• ★ Frequency Distribution: A tabular arrangement of data that groups raw measurements into mutually exclusive and exhaustive numerical class intervals, displaying the count of observations (frequency) falling into each class.
• ★ Ogive Curve (Cumulative Frequency Curve): A continuous line graph plotting cumulative frequencies against class boundaries:
  • Less-than Ogive: Rising curve plotting cumulative frequency against upper class boundaries.
  • More-than Ogive: Falling curve plotting cumulative frequency against lower class boundaries.
  • Intersection Property: The $X$-coordinate of the intersection point of Less-than and More-than Ogives corresponds exactly to the Median.
• ★ Pie Chart Sector Angle Formula:

  Swipe to see the full formula↔

  $$\text{Sector Angle } \theta = \frac{\text{Component Value}}{\text{Total Value}} \times 360^\circ$$

Part 2: Structural Frameworks (10 Marks)

1. Graphical Estimation of Partition Values from Ogives

When required to estimate the Median, Lower Quartile ($Q_1$), and Upper Quartile ($Q_3$) graphically:

1. Construct Cumulative Frequency Table:
   • Compute cumulative frequency ($cf$) for all classes.
2. Plot Less-Than Ogive:
   • $X$-axis: Upper class boundaries.
   • $Y$-axis: Cumulative frequencies ($cf$).
3. Locate Partition Position Markers on $Y$-axis:
   • For Median ($Q_2$): Locate $Y = \frac{N}{2}$.
   • For Lower Quartile ($Q_1$): Locate $Y = \frac{N}{4}$.
   • For Upper Quartile ($Q_3$): Locate $Y = \frac{3N}{4}$.
4. Project to $X$-axis:
   • Draw a horizontal line from the calculated $Y$-value to intersect the Ogive curve.
   • Drop a vertical perpendicular down to the $X$-axis.
   • The value read from the $X$-axis is the estimated partition value.

Part 3: Distinction Tables

Histogram vs. Bar Diagram

Part 4: Worked Problems & Exam Traps

Problem 1 (Standard): Pie Chart Angle Calculation

A company’s quarterly budget allocation is: Production NPR 500,000, Marketing NPR 250,000, R&D NPR 150,000, Administration NPR 100,000. Calculate sector angles for a Pie Chart.

• Total Budget $= 500{,}000 + 250{,}000 + 150{,}000 + 100{,}000 = \text{NPR } 1{,}000{,}000$.
• Production: $\frac{500{,}000}{1{,}000{,}000} \times 360^\circ = 180^\circ$.
• Marketing: $\frac{250{,}000}{1{,}000{,}000} \times 360^\circ = 90^\circ$.
• R&D: $\frac{150{,}000}{1{,}000{,}000} \times 360^\circ = 54^\circ$.
• Administration: $\frac{100{,}000}{1{,}000{,}000} \times 360^\circ = 36^\circ$.
• Verification: $180 + 90 + 54 + 36 = 360^\circ$.

Students lose marks here because when plotting Histograms for inclusive data (e.g., $10-19, 20-29$), they plot using class limits directly instead of converting to continuous true class boundaries ($9.5-19.5, 19.5-29.5$).