To find the Inter Quartile Range (IQR) of the given data set, we need to follow these steps:

1. List the data in ascending order.
2. Determine the quartiles (Q1, Q2, and Q3).
3. Calculate the IQR.

Step 1: List the Data

First, we interpret the stem and leaf plot and list out all the ratings in ascending order.

• From stem 4: 44
• From stem 5: 52, 53, 56
• From stem 6: 68
• From stem 8: 80, 81, 83, 89, 89
• From stem 9: 90

Now, put these in ascending order:

$44, 52, 53, 56, 68, 80, 81, 83, 89, 89, 90$

Step 2: Determine the Quartiles

To find the quartiles, we use the following method:

• $Q1$ (first quartile) is the median of the lower half of the data (excluding the median if the number of data points is odd).
• $Q2$ (second quartile) is the median of the data set.
• $Q3$ (third quartile) is the median of the upper half of the data (excluding the median if the number of data points is odd).

The data set has 11 values. The median (Q2) will be the 6th value:

$Q2 = 68$

For $Q1$, consider the first five numbers:

$44, 52, 53, 56, 68$

The median of these values is the 3rd value:

$Q1 = 53$

For $Q3$, consider the last five numbers:

$80, 81, 83, 89, 89, 90$

The median of these values is the 3rd value:

$Q3 = 89$

Step 3: Calculate the IQR

The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

$IQR = Q3 - Q1$

Substitute the values:

$IQR = 89 - 53 = 36$

Final Answer

The Inter Quartile Range (IQR) is $36.0$.

Do you want details or have any questions?

Here are 8 related questions that expand on this information:

1. What is the median (Q2) of the given data set?
2. How do you find the quartiles of a data set?
3. Why is the IQR important in statistical analysis?
4. What is a stem and leaf plot, and how is it used in data representation?
5. Can you calculate the range of the given data set?
6. What does the term "outlier" mean in statistics, and how does IQR help in detecting them?
7. How do you interpret the values in a stem and leaf plot?
8. What other statistical measures can be derived from the given data?

Tip: Understanding the Inter Quartile Range (IQR) is crucial as it helps to measure the spread of the middle 50% of a data set, reducing the effect of outliers.