To find the Interquartile Range (IQR) and identify any outliers for the given data, we will proceed as follows:

Step 1: Order the Data

First, we need to order the data from smallest to largest.

Given dataTo find the Interquartile Range (IQR) and identify any outliers for the given data, we will proceed as follows:

Step 1: Order the Data

First, we need to order the data from smallest to largest.

Given
$42, 52, 36, 44, 40, 37, 41, 46, 43, 38, 36, 55, 43, 35, 15, 53, 37, 50, 30, 29$

Ordered dataTo find the Interquartile Range (IQR) and identify any outliers for the given data, we will proceed as follows:

Step 1: Order the Data

First, we need to order the data from smallest to largest.

Given
$42, 52, 36, 44, 40, 37, 41, 46, 43, 38, 36, 55, 43, 35, 15, 53, 37, 50, 30, 29$

Ordered $15, 29, 30, 35, 36, 36, 37, 37, 38, 40, 41, 42, 43, 43, 44, 46, 50, 52, 53, 55$

Step 2: Find the Quartiles

a. Find the Quartiles (Q1, Q2, Q3):

• Median (Q2):
  Since there are 20 data points, the median will be the average of the 10th and 11th values in the ordered list.
  $Q2 = \frac{40 + 41}{2} = 40.5$

• First Quartile (Q1):
  Q1 is the median of the first half of the data (before the median).
  Data points for Q1: $15, 29, 30, 35, 36, 36, 37, 37, 38, 40$
  $Q1 = \frac{36 + 36}{2} = 36$

• Third Quartile (Q3):
  Q3 is the median of the second half of the data (after the median).
  Data points for Q3: $41, 42, 43, 43, 44, 46, 50, 52, 53, 55$
  $Q3 = \frac{44 + 46}{2} = 45$

Step 3: Find the Interquartile Range (IQR)

b. Interquartile Range (IQR):
$IQR = Q3 - Q1 = 45 - 36 = 9$

Step 4: Identify Any Outliers

c. Identify Any Outliers:

Outliers are typically identified as any data points below $Q1 - 1.5 \times IQR$ or above $Q3 + 1.5 \times IQR$.

• Lower Bound:
  $Q1 - 1.5 \times IQR = 36 - 1.5 \times 9 = 36 - 13.5 = 22.5$

• Upper Bound:
  $Q3 + 1.5 \times IQR = 45 + 1.5 \times 9 = 45 + 13.5 = 58.5$

Any data points below 22.5 or above 58.5 are considered outliers.

Outliers:

• Below 22.5: $15$
• Above 58.5: None

Thus, the only outlier is $15$.

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Would you like more details or have any questions?

Related Questions:

1. How do you interpret the significance of the IQR in a data set?
2. What are other methods to detect outliers apart from the IQR method?
3. Can there be more than one outlier in a data set, and how do you handle them?
4. What is the effect of outliers on the mean and standard deviation?
5. How does the IQR change if an outlier is removed from the data set?

Tip:

Always visualize your data using a box plot to easily identify the quartiles, IQR, and any potential outliers.