Bar & pie charts

Quantitative Aptitude – Bar & Pie Charts

Advanced, MIT‑Level Study Notes
(For Delhi TET – Teacher Eligibility Test)

These notes assume you already know the basics of data representation.
They take you from first principles to a rigorous, statistical view of bar and pie charts, discuss edge‑cases, and finish with a challenging problem that tests both conceptual understanding and computational skill.

1. What Is a Bar / Pie Chart?

2. From Raw Data to a Chart – The First Principles

2.1 Frequency Table

Given data ( {x_i}_{i=1}^n ) belonging to categories (C_1,\dots,C_k),

1. Count occurrences:
   [ f_i = |{x_j : x_j \in C_i}| ]
2. Total: ( N = \sum_{i=1}^k f_i ).

2.2 Relative Frequency / Proportion

[ p_i = \frac{f_i}{N}\quad\text{(0 ≤ }p_i\le 1) ]

2.3 Bar Chart Construction

• Vertical bars: Height (= f_i) or (= p_i \times H) where (H) is a chosen scale.
• Horizontal bars: Width (= f_i) or (= p_i \times W).
• Stacked bars: Add cumulative heights: the top of bar (i) is ( \sum_{j=1}^{i} f_j).
• Grouped bars: For sub‑categories within each main category.

2.4 Pie Chart Construction

• Angle of slice (i):
  [ \theta_i = p_i \times 360^\circ ]
• Area of slice (i) (if drawing by area, not just angle):
  [ A_i = p_i \times A_{\text{total}} ] where (A_{\text{total}}) is area of the whole circle.

Why angles?
The fraction of the circle’s circumference that slice (i) occupies is exactly (p_i).
Since the total circumference is (2\pi r), the arc length (s_i = p_i \cdot 2\pi r) and the corresponding central angle (\theta_i = s_i/r = p_i \cdot 360^\circ).

3. Rigorous Theoretical Underpinnings

3.1 Scale Types & Measurement Theory

Key point: Bar charts preserve quantitative relationships; pie charts only preserve proportional relationships.

3.2 Human Visual Perception

• **Bar heights