- The frequencies are \(30\), \(35\), and \(40\).
- To maximize the visual difference, choose an axis scale with the smallest range that covers all the data points.
- The scale \(25-40\) has the smallest range (\(15\) units) and frames the data most tightly.
- Final choice: \(\boxed{25-40}\).
Explanation
- Analyze the Data and Objective
To understand how the scale of the vertical axis affects the visual representation of data in a histogram, we first identify our data frequencies. The number of people in each salary range are 40, 30, and 35. These are the values that will determine the heights of the three bars in our histogram. We want to choose a vertical axis scale that makes the differences between these values (40, 35, and 30) appear as large as possible.
- Explain the Principle of Visual Maximization
The visual "impact" or apparent difference between bar heights in a histogram is maximized when the range of the axis is as small as possible and sits as close to the data as possible. If an axis includes a large portion of empty space (like starting from 0 when all data values are large), the bars will look more similar in height because the relative difference is squashed. By starting the axis closer to the minimum value (30) and ending it closer to the maximum value (40), the bars will span a larger percentage of the available vertical space, effectively magnifying the visual differences between them.
- Evaluate the Given Scales
Let's evaluate the suggested scales:
- \(0-50\): The range is \(50 - 0 = 50\). The values (30, 35, 40) occupy a relatively small portion (60% to 80%) of the height.
- \(0-40\): The range is \(40 - 0 = 40\). This is better, but still includes the space from 0 to 30.
- \(10-50\): The range is \(50 - 10 = 40\). Again, the data only occupies a portion of this space.
- \(25-40\): The range is \(40 - 25 = 15\). This is the smallest range, and it is tightly bound around our data values. By focusing the axis exclusively on the interval where our data actually exists, the relative change from 30 to 40 will appear much more dramatic than if the axis had to accommodate values down to zero.
- Conclusion and Final Selection
Comparing the options, the scale \(25-40\) has the narrowest range and fits the data most tightly. This choice forces the bars for 30, 35, and 40 to spread across the entire visual height, thereby maximizing the perceived differences between them. Therefore, the scale \(25-40\) is the best choice for Gemma's objective.
Examples
Imagine you are presenting data about how many students visited the library each week. If you have 95, 98, and 99 visitors, displaying them on an axis from 0 to 100 will make the bars look almost identical in height. However, using an axis from 90 to 100 will clearly show the trends and differences between those weeks, making your presentation much more informative and visually effective. Understanding how to scale axes is a fundamental skill in data visualization and statistics.