Problem 91 - Algebra and Trigonometry

Question

Difficulty of a Task The difficulty in "acquiring a target" such as using your mouse to click on an icon on your computer screen) depends on the distance to the target and the size of the target. According to Fits's Law, the index of difficulty (ID) is given by $$ \mathrm{ID}=\frac{\log (2 A / W)}{\log 2} $$ where \(W\) is the width of the target and \(A\) is the distance to the center of the target. Compare the difficulty of clicking on an icon that is 5 \(\mathrm{mm}\) wide to clicking on one that is 10 \(\mathrm{mm}\) wide. In each case, assume that the mouse is 100 \(\mathrm{mm}\) from the icon.

Step-by-Step Solution

Answer

The icon that is 10 mm wide is easier to click on than the 5 mm wide icon.

Step 1: Understand the Formula

The formula given is for Fitts's Law, expressed as \( \mathrm{ID} = \frac{\log (2 A / W)}{\log 2} \). Here, \( A \) is the distance to the target, and \( W \) is the width of the target. The formula calculates the index of difficulty (ID) for acquiring a target.

Step 2: Input Values for First Icon

For the first icon, \( W = 5 \text{ mm} \) and \( A = 100 \text{ mm} \). Substitute these values into the formula: \( \mathrm{ID}_1 = \frac{\log (2 \times 100 / 5)}{\log 2} \).

Step 3: Calculate ID for First Icon

Compute \( \mathrm{ID}_1 \):1. Calculate \( 2 \times 100 = 200 \).2. Divide by 5 to get \( \frac{200}{5} = 40 \).3. Find \( \log 40 \) (base 10 is used for simplicity): \( \log 40 \approx 1.602 \).4. Calculate \( \log 2 \approx 0.301 \).5. Compute \( \mathrm{ID}_1 = \frac{1.602}{0.301} \approx 5.32 \).

Step 4: Input Values for Second Icon

For the second icon, \( W = 10 \text{ mm} \) and \( A = 100 \text{ mm} \). Substitute these values into the formula: \( \mathrm{ID}_2 = \frac{\log (2 \times 100 / 10)}{\log 2} \).

Step 5: Calculate ID for Second Icon

Compute \( \mathrm{ID}_2 \):1. Calculate \( 2 \times 100 = 200 \).2. Divide by 10 to get \( \frac{200}{10} = 20 \).3. Find \( \log 20 \approx 1.301 \).4. Compute \( \mathrm{ID}_2 = \frac{1.301}{0.301} \approx 4.32 \).

Step 6: Compare the IDs

The index of difficulty for the first icon is approximately 5.32, while for the second icon, it is approximately 4.32. Therefore, the second icon (10 mm wide) is easier to click on than the first icon (5 mm wide).

Key Concepts

Index of Difficulty

The Index of Difficulty (ID) in Fitts's Law quantifies how challenging it is to "acquire a target" or, in simpler terms, how hard it is to click on a particular area using a pointing device like a mouse. Fitts's Law uses a mathematical formula to measure this difficulty. Notably, it ties the target's distance and size into the calculation. The mathematical equation looks like this: \[ \mathrm{ID} = \frac{\log (2A / W)}{\log 2} \]Let’s break this down: - **A** represents the distance from the starting point to the center of the target.- **W** is the target's width. The ID rises if the distance is greater or if the target is smaller, meaning it is more difficult to click accurately. The formula uses a base-2 logarithm to manage how these parameters grow, reflecting real-world intuitions about task difficulty.

Target Acquisition

Target acquisition describes the process of selecting a target area on a digital interface, like an icon on your computer screen. This task's difficulty is impacted by numerous factors, but the primary considerations according to Fitts's Law are distance to the target and the target's size. Here's a simple way to understand why these factors matter:

• **Distance to the Target (A):** Imagine stretching your arm to grab something far away—the further it is, the harder it becomes.
• **Size of the Target (W):** Picture trying to select a small button versus a large one on your touchscreen. The larger button lets you be more relaxed with your aim, hence reducing the difficulty.

Together, these elements determine how efficiently and accurately a user can complete the task of target acquisition. The concept is a cornerstone in designing user interfaces where fast and easy interaction is key.

Logarithmic Function

In the equation for the Index of Difficulty, the term \( \log (2A/W) \) plays a crucial role. Understanding logarithms can demystify how Fitts's Law calculates difficulty.A logarithm helps to express how our perception of difficulty scales non-linearly with increasing ratios of distance to width. In practical terms:

• A logarithmic function grows slower than a linear function as inputs increase. This reflects our experience of tasks becoming harder, but not at the same rapid rate as the values themselves grow.
• The specific base of the logarithm in Fitts's Law is 2, which means it measures complexity in binary terms, aligning with computer science principles.

By using a logarithmic function, Fitts's Law captures the diminishing returns of increasing distance or reducing target size, effectively mirroring how users actually experience task difficulty.