Question

maria was asked to find the range to the box plot below. her work is shown in the table. there is a mistake. identify it and correct the mistake.
a. what would be maria’s solution if the question is to find the interquartile range of the box plot?
b. what is the difference between range and interquartile range?
turn and talk:
evelyn said that the median of the dot plot below is 6 by using the number from 0 to 12. explain why her answer can not be correct using evidence from the dot plot to support your answer

Explanation:

Part a (Interquartile Range Calculation)

Step1: Recall IQR formula

The interquartile range (IQR) is calculated as \( \text{IQR} = Q_3 - Q_1 \), where \( Q_1 \) is the first quartile (25th percentile) and \( Q_3 \) is the third quartile (75th percentile). From the box - plot, assume the lower quartile (\( Q_1 \)) is at 3 and the upper quartile (\( Q_3 \)) is at 7 (since the box plot's left edge is at 3 and right edge at 7, and the median is in between).

Step2: Calculate IQR

Using the formula \( \text{IQR}=Q_3 - Q_1 \), substitute \( Q_3 = 7 \) and \( Q_1 = 3 \). So \( \text{IQR}=7 - 3=4 \). But if Maria made a mistake in calculating range (which is \( \text{Range}=\text{Max}-\text{Min} \)) and used that formula for IQR, but if we assume the correct IQR calculation from the box - plot (assuming the box is from 3 to 7 and whiskers from, say, 0 to 10), the IQR is \( 7 - 3 = 4 \). But if Maria's work was \( 9 - 4=5 \) (wrongly using max - min for IQR), but the correct IQR calculation:
First, identify \( Q_1 \) (25% of data below) and \( Q_3 \) (75% of data below). From the box - plot, if the box starts at 3 (\( Q_1 \)) and ends at 7 (\( Q_3 \)), then \( \text{IQR}=7 - 3 = 4 \).

• Range: It is calculated as \( \text{Range}=\text{Maximum value}-\text{Minimum value} \). It measures the spread of all the data points from the smallest to the largest value. It is highly affected by outliers (extreme values) because it depends on the minimum and maximum values.
• Interquartile Range (IQR): It is calculated as \( \text{IQR}=Q_3 - Q_1 \), where \( Q_1 \) is the first quartile (25th percentile) and \( Q_3 \) is the third quartile (75th percentile). It measures the spread of the middle 50% of the data. It is not affected by outliers because it only considers the data between the 25th and 75th percentiles.

To find the median, we need to order the data and find the middle value (or the average of two middle values for even number of data points). First, we count the number of dots (data points) in the dot - plot. Let's assume the number of dots:

• For \( x = 0 \): Let's say 5 dots
• \( x = 1 \): 2 dots
• \( x = 2 \): 3 dots
• \( x = 3 \): 6 dots
• \( x = 4 \): 2 dots
• \( x = 5 \): 5 dots
• \( x = 8 \): 2 dots
• \( x = 9 \): 3 dots
• \( x = 10 \): 7 dots
• \( x = 11 \): 4 dots
• \( x = 12 \): 1 dot

Now, sum the number of dots: \( 5 + 2+3 + 6+2 + 5+2 + 3+7 + 4+1=38 \) (even number of data points). The median is the average of the 19th and 20th values when ordered. Let's cumulative count:

• Up to \( x = 0 \): 5 (1 - 5)
• Up to \( x = 1 \): \( 5 + 2=7 \) (6 - 7)
• Up to \( x = 2 \): \( 7+3 = 10 \) (8 - 10)
• Up to \( x = 3 \): \( 10 + 6=16 \) (11 - 16)
• Up to \( x = 4 \): \( 16+2 = 18 \) (17 - 18)
• Up to \( x = 5 \): \( 18+5 = 23 \) (19 - 23)

So the 19th and 20th values are at \( x = 5 \). So the median should be 5 (or average of 5 and 5, which is 5), not 6. So Evelyn's answer is wrong because when we count the cumulative frequency of the dot - plot, the middle values (19th and 20th) fall at \( x = 5 \), not 6.

Answer:

If we assume the box - plot has \( Q_1 = 3 \) and \( Q_3 = 7 \), the interquartile range is \( 4 \). If Maria made a mistake (like using range formula for IQR), but the correct IQR calculation gives \( 4 \) (depending on box - plot values, here we assume standard box - plot with \( Q_1 = 3 \), \( Q_3 = 7 \)).

Part b (Difference between Range and Interquartile Range)