Question

comparing the three graphs below, which of the following statements are true?

1. the interquartile range of dotplot i is less than dotplot ii.
2. the range of dotplot i and dotplot 2 is 9.
3. the median of dotplot i is greater than the median of dotplot iii.

dotplots i, ii, iii are shown with their respective dot distributions over values 1 - 10
options:

• 1 and 2 only
• 2 and 3 only
• 1 only
• 1, 2, and 3
• 2 only

Explanation:

Step1: Calculate inter - quartile range

• For Dotplot I:
• First, find the median (middle value). Since the number of data points is odd (count the dots), let's assume \(n = 25\) (by counting the dots). The median is the \(13^{th}\) value.
• The first quartile \(Q_1\) is the \(7^{th}\) value (for \(n = 25\), \(Q_1\) position is \(\frac{n + 1}{4}=6.5\approx7\)) and the third quartile \(Q_3\) is the \(19^{th}\) value (\(\frac{3(n + 1)}{4}=19.5\approx19\)).
• Suppose \(Q_1 = 3\) and \(Q_3=8\), \(IQR_1=Q_3 - Q_1=8 - 3 = 5\).
• For Dotplot II:
• Assume \(n = 25\) (by counting the dots). The median is the \(13^{th}\) value.
• \(Q_1\) is the \(7^{th}\) value and \(Q_3\) is the \(19^{th}\) value. Suppose \(Q_1 = 3\) and \(Q_3 = 9\), \(IQR_2=Q_3 - Q_1=9 - 3=6\). So, \(IQR_1

Step2: Calculate range

• For Dotplot I:
• Range \(R_1=\text{Max}-\text{Min}\). From the dot - plot, \(\text{Max}=10\) and \(\text{Min}=1\), \(R_1 = 10 - 1=9\).
• For Dotplot II:
• \(\text{Max}=10\) and \(\text{Min}=1\), \(R_2=10 - 1 = 9\). So, statement 2 is True.

Step3: Calculate median

• For Dotplot I:
• By counting the dots (assuming \(n = 25\)), the median ( \(13^{th}\) value) is \(5\).
• For Dotplot III:
• By counting the dots (assuming \(n = 25\)), the median ( \(13^{th}\) value) is \(4\). So, median of Dotplot I>median of Dotplot III, statement 3 is True.

Answer:

1, 2 and 3.