The problem in the image asks to identify the correct interquartile range (IQR) from a box plot and determine the mistake made by someone who incorrectly stated that the IQR is 10.

To solve this:

1. Interquartile Range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). Looking at the box plot:

   • Q1 is approximately at 10.
   • Q3 is approximately at 20.

   So, the interquartile range (IQR) is: $\text{IQR} = Q3 - Q1 = 20 - 10 = 10$

   However, it seems your friend might have made a mistake interpreting what the IQR actually represents.

Now, let's consider the choices:

• Option A: "Your friend found the difference between the maximum and the minimum."

  • This would give the range of the data, not the IQR. The maximum is at 30, and the minimum is at 0, so the total range is $30 - 0 = 30$, which is not relevant to the mistake your friend made.

• Option B: "Your friend found the difference between the third quartile and the median."

  • This refers to calculating $Q3 - Q2$, but that would not result in 10 because the median (Q2) is at 15, so $Q3 - Q2 = 20 - 15 = 5$. This is not relevant either.

• Option C: "Your friend found the difference between the median and the first quartile."

  • This would be $Q2 - Q1$, which is $15 - 10 = 5$, so this is not likely what your friend calculated.

• Option D: "Your friend found the difference between the maximum and the median."

  • This difference would be $30 - 15 = 15$, which does not match the friend's claim of 10.

Conclusion:

Based on the options, Option A seems most likely, as it suggests your friend may have mistakenly found the range (difference between the maximum and minimum), rather than the interquartile range.

Would you like further details or have any other questions?

Here are five related questions:

1. What is the full definition of the interquartile range (IQR)?
2. How do you find the median from a box plot?
3. What is the difference between range and interquartile range?
4. How can extreme values (outliers) affect a box plot?
5. What does each part of a box plot represent?

Tip: Always remember, the IQR only measures the spread of the middle 50% of data, making it less sensitive to extreme values or outliers.