Question

gemma is creating a histogram based on the table below.which scale can she use for the vertical axis such that the difference in the heights of the bars is maximized?0 - 500 - 4010 - 5025 - 40

Explanation:

Step1: Analyze each scale option

• For scale \(0 - 50\):

The values \(40\), \(30\), \(35\) are plotted as they are. The differences between the bars: \(|40 - 30|=10\), \(|40 - 35| = 5\), \(|35 - 30|=5\).

• For scale \(0 - 40\):

The value \(40\) is at the top of the scale. The value \(30\) is at \(\frac{30}{40}\) of the scale height, and \(35\) is at \(\frac{35}{40}\) of the scale height. The differences in heights are relatively smaller compared to a scale that can accommodate all values without compressing the largest value.

• For scale \(10 - 50\):

The value \(40\) is at \(40-10 = 30\) units from the start of the scale. The value \(30\) is at \(30 - 10=20\) units from the start of the scale. The value \(35\) is at \(35 - 10 = 25\) units from the start of the scale. The differences in heights: \(|30-20| = 10\), \(|30 - 25|=5\), \(|25 - 20|=5\).

• For scale \(25 - 40\):

The value \(40\) is at \(40 - 25=15\) units from the start of the scale. The value \(30\) is at \(30 - 25 = 5\) units from the start of the scale. The value \(35\) is at \(35 - 25=10\) units from the start of the scale. The differences in heights: \(|15 - 5|=10\), \(|15 - 10| = 5\), \(|10 - 5|=5\). But the scale \(25 - 40\) does not include the value \(40\) (if we assume non - inclusive upper bound in some interpretations, and also it has a smaller range compared to \(0 - 50\) which can represent the data without truncating or compressing the maximum value \(40\) in a less intuitive way.

Since the scale \(0 - 50\) allows the values \(40\), \(30\), \(35\) to be plotted without truncating the maximum value (\(40\)) and gives a clear visual representation of the differences between the frequencies (as the range is large enough to spread out the values), it maximizes the difference in the heights of the bars.

Answer:

\(0 - 50\)