To calculate the IQR (Interquartile Range), let's work step by step based on the data provided in the image.

1. Sort the Data

The given data is already sorted:

$1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9$

2. Determine the Quartiles (Q1 and Q3)

Step 2.1: Find Q1 (Lower Quartile)

• $Q1$ is the median of the lower half of the data (the data before the overall median).
• Lower half:
  $1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 4$
• Median of this lower half:
  $2, 2, \mathbf{2}, 2, 2, 3, 3, 4$
  The median lies between the 6th and 7th numbers:
  $Q1 = \frac{2 + 3}{2} = 2.5$

Step 2.2: Find Q3 (Upper Quartile)

• $Q3$ is the median of the upper half of the data (the data after the overall median).
• Upper half:
  $5, 5, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9$
• Median of this upper half:
  $7, 7, \mathbf{7}, 7, 8, 8, 8, 9$
  The median lies between the 6th and 7th numbers:
  $Q3 = \frac{7 + 8}{2} = 7.5$

3. Calculate the Interquartile Range (IQR)

The formula for $IQR$ is: $IQR = Q3 - Q1$ Substitute the values: $IQR = 7.5 - 2.5 = 5$

Final Answer:

$\boxed{IQR = 5}$

Would you like further clarification or a breakdown of a specific part? 😊

Related Questions:

1. What is the significance of the interquartile range in statistics?
2. How is the IQR used to detect outliers in a dataset?
3. What is the relationship between IQR and the overall range of a dataset?
4. How can you visually represent Q1, Q3, and IQR using a boxplot?
5. How does the size of the IQR indicate data dispersion?

Tip:

When calculating medians for quartiles, always double-check if the dataset contains an even or odd number of values in each half! This ensures accurate calculations.