For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. (Prices are in thousands of dollars)

[461, 639, 525, 659, 230, 140, 592, 502, 393, 5500, 387, 85, 853]

a) The IQR is:
Number
(in thousands of dollars)

(Round your answer to a whole number)

b) The potential outlier/s is/are:
Number
(in thousands of dollars)

(Round your answer to a whole number)

To calculate the Interquartile Range (IQR) and determine potential outliers, we need to follow these steps:

1. **Order the data set from smallest to largest.**
2. **Determine the first quartile (Q1) and the third quartile (Q3).**
3. **Calculate the IQR as \( IQR = Q3 - Q1 \).**
4. **Identify potential outliers using the rule \( \text{Potential Outliers} < Q1 - 1.5 \times IQR \) or \( \text{Potential Outliers} > Q3 + 1.5 \times IQR \).**

Let's begin with the given data set:

\[ [461, 639, 525, 659, 230, 140, 592, 502, 393, 5500, 387, 85, 853] \]

### Step 1: Order the data set
\[ [85, 140, 230, 387, 393, 461, 502, 525, 592, 639, 659, 853, 5500] \]

### Step 2: Determine Q1 and Q3

Since there are 13 values, the median is the 7th value:

\[ \text{Median} = 502 \]

Q1 is the median of the first half of the data (lower half):

\[ [85, 140, 230, 387, 393, 461] \]

Median of the lower half:

\[ Q1 = \frac{230 + 387}{2} = \frac{617}{2} = 308.5 \]

Q3 is the median of the second half of the data (upper half):

\[ [525, 592, 639, 659, 853, 5500] \]

Median of the upper half:

\[ Q3 = \frac{639 + 659}{2} = \frac{1298}{2} = 649 \]

### Step 3: Calculate the IQR

\[ IQR = Q3 - Q1 = 649 - 308.5 = 340.5 \]

Rounded to the nearest whole number:

\[ IQR \approx 341 \]

### Step 4: Identify potential outliers

\[ Q1 - 1.5 \times IQR = 308.5 - 1.5 \times 340.5 = 308.5 - 510.75 = -202.25 \]

\[ Q3 + 1.5 \times IQR = 649 + 1.5 \times 340.5 = 649 + 510.75 = 1159.75 \]

Potential outliers are any values less than \(-202.25\) or greater than \(1159.75\).

In this data set, the values \(85, 140, 230, 387, 393, 461, 502, 525, 592, 639, 659, 853\) are all within the range, but \(5500\) is greater than \(1159.75\).

### Answers:

a) The IQR is: 
\[ 341 \]

b) The potential outlier/s is/are:
\[ 5500 \]

Do you want details or have any questions? Here are five numbered questions you can ask:

1. How do you calculate the median of a data set with an even number of values?
2. Can you explain the concept of quartiles in more detail?
3. What is the significance of the IQR in a data set?
4. How do you identify outliers using the IQR?
5. How does the presence of outliers affect statistical analysis?

**Tip:** Always check for outliers in your data set, as they can significantly impact the results of your statistical analysis.

Learn how to calculate the Interquartile Range (IQR) and identify potential outliers in a set of real estate prices. This problem involves ordering data, determining quartiles, calculating IQR, and applying rules to detect outliers. Suitable for advanced high school or college-level students.

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