Pie Charts

• The fundamental principle of a pie chart is proportionality: the area, arc length, and central angle of a sector are all proportional to the frequency of the category it represents.

• The relationship between the frequency of a category ($f$) and its central angle ($\theta$) is governed by the total frequency ($N$) and the total degrees in a circle ($360^\circ$).

• This relationship is expressed by the formula: $\theta = \frac{f}{N} \times 360^\circ$

Step-by-Step Construction

• Step 1: Calculate the Total: Sum the frequencies of all categories to find the total value ($N$).

• Step 2: Determine the Multiplier: Calculate the 'degrees per unit' by dividing $360^\circ$ by the total frequency ($360 / N$). This constant can then be multiplied by each individual frequency to find the required angles.

• Step 3: Calculate Angles: Multiply each category's frequency by the multiplier to find its specific central angle. Ensure the sum of all calculated angles is exactly $360^\circ$.

• Step 4: Draw and Measure: Draw a circle and a starting radius (usually vertical). Use a protractor to measure the first angle, draw the next radius, and repeat for all categories.

• The 'Not to Scale' Rule: If a diagram is labeled 'not to scale', do not use a protractor. Instead, use the ratio of a known angle to its value to find the total or other missing values.

• Unit Value Method: For complex problems, calculate what $1^\circ$ represents ($Total / 360$) or what $1$ unit of frequency represents ($360 / Total$). This makes scaling much simpler.

• Verification: Always check that your percentages sum to $100\%$ and your angles sum to $360^\circ$. If they do not, a calculation error has occurred.