Linear Interpolation

• The mathematical foundation of linear interpolation is based on Similar Triangles or the concept of a constant slope ($m = \frac{\Delta y}{\Delta x}$).

• It assumes that the ratio of the distance from the lower boundary to the target value is proportional to the ratio of the frequency covered to the total frequency of that interval.

• This proportionality can be expressed as: $\frac{\text{Value} - L}{w} = \frac{\text{Target Position} - C}{f}$ where $L$ is the lower boundary, $w$ is the class width, $C$ is the cumulative frequency before the interval, and $f$ is the frequency of the interval.

Step-by-Step Estimation

• Step 1: Identify the Interval: Determine which class interval contains the target value (e.g., for the median, find the interval where the $\frac{n}{2}$th value lies using cumulative frequencies).
• Step 2: Calculate the Fraction: Find how far into the interval the target value is by subtracting the previous cumulative frequency from the target position, then dividing by the frequency of the current interval.
• Step 3: Scale by Width: Multiply this fraction by the class width ($w = \text{Upper Bound} - \text{Lower Bound}$) to find the physical distance from the start of the interval.
• Step 4: Final Adjustment: Add this distance to the Lower Boundary ($L$) of the interval to get the estimated value.

The General Formula: $Estimated Value = L + \left( \frac{\text{Target Position} - C}{f} \right) \times w$

• Interpolation vs. Extrapolation: Interpolation estimates values within the range of known data points, whereas extrapolation attempts to predict values outside the known range, which is significantly less reliable.

• Check Boundaries: Always use the exact boundaries of the class interval. If data is given as $10-19$ and $20-29$, the boundary between them is $19.5$.

• Sanity Check: Ensure your final answer actually falls within the class interval you identified in Step 1. If your estimate is $22.5$ but the interval is $30 \le x < 40$, a calculation error has occurred.

• Units: Always include the units of measurement (e.g., kg, cm, seconds) in your final answer to avoid losing marks.

• Formula vs. Logic: While the formula is useful, understanding the 'fraction of the way through' logic is safer for exams as it is harder to misremember than a complex algebraic string.

• Wrong Target Position: For the median in interpolation, use $\frac{n}{2}$. Do not use $\frac{n+1}{2}$, which is reserved for small, discrete data sets.

• Cumulative Frequency Errors: Students often use the cumulative frequency of the current interval instead of the previous interval ($C$) when calculating how far 'into' the class the value is.

• Class Width Confusion: Ensure you use the width of the specific interval containing the median, not the width of the first interval or an average width.