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.

dotplot i, dotplot ii, dotplot iii graphs 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 the inter - quartile range

For Dotplot I:
First, find the median. The total number of data points is \(n_1\). Count the dots: \(n_1=(1 + 2+3 + 4+1+1+1+6 + 1)=20\). The median is the average of the 10th and 11th values. Ordering the data (by the number line), the 10th and 11th values are 4 and 5. \(Q_2=\frac{4 + 5}{2}=4.5\).
\(Q_1\) (the median of the lower half, \(n = 10\) data points in the lower half) is the 5th value. The 5th value is 3. \(Q_3\) (the median of the upper half, \(n = 10\) data points in the upper half) is the 15th value. The 15th value is 9. \(IQR_1=Q_3 - Q_1=9 - 3=6\).
For Dotplot II:
The total number of data points \(n_2=(4+2 + 2+3+3+5+3+2+1)=25\). The median (\(Q_2\)) is the 13th value. Counting the dots: \(Q_2 = 6\). \(Q_1\) (median of the first 12 values) is the 6th value. The 6th value is 3. \(Q_3\) (median of the last 12 values) is the 19th value. The 19th value is 7. \(IQR_2=Q_3 - Q_1=7 - 3 = 4\). So, statement 1 is false.

Step2: Calculate the range

The range is \(R=\text{Max}-\text{Min}\).
For Dotplot I: \(\text{Max}=10\), \(\text{Min}=1\), \(R_1=10 - 1=9\).
For Dotplot II: \(\text{Max}=10\), \(\text{Min}=1\), \(R_2=10 - 1=9\). So, statement 2 is true.

Step3: Calculate the median

For Dotplot I: \(n_1 = 20\), median is the average of the 10th and 11th values. As above, \(M_1=\frac{4 + 5}{2}=4.5\).
For Dotplot III: \(n_3=(2+4+4+6+2+2+2)=22\). The median is the average of the 11th and 12th values. Counting the dots (by the number line), the 11th and 12th values are 4. \(M_3 = 4\). So, \(M_1>M_3\), statement 3 is true.

Answer:

2 and 3 only