To determine if there are any possible outliers in the data using a modified box plot, we can follow these steps:

1. Calculate the quartiles (Q1, Q2, Q3):

   • Q1 (first quartile): This is the median of the lower half of the data.
   • Q2 (second quartile or median): This is the median of the entire dataset.
   • Q3 (third quartile): This is the median of the upper half of the data.

2. Find the Interquartile Range (IQR):

   • $IQR = Q3 - Q1$

3. Determine the lower and upper bounds:

   • Lower bound: $Q1 - 1.5 \times IQR$
   • Upper bound: $Q3 + 1.5 \times IQR$

4. Identify outliers:

   • Any data point below the lower bound or above the upper bound is considered a potential outlier.

Step 1: Calculate Quartiles

Given dataTo determine if there are any possible outliers in the data using a modified box plot, we can follow these steps:

1. Calculate the quartiles (Q1, Q2, Q3):

   • Q1 (first quartile): This is the median of the lower half of the data.
   • Q2 (second quartile or median): This is the median of the entire dataset.
   • Q3 (third quartile): This is the median of the upper half of the data.

2. Find the Interquartile Range (IQR):

   • $IQR = Q3 - Q1$

3. Determine the lower and upper bounds:

   • Lower bound: $Q1 - 1.5 \times IQR$
   • Upper bound: $Q3 + 1.5 \times IQR$

4. Identify outliers:

   • Any data point below the lower bound or above the upper bound is considered a potential outlier.

Step 1: Calculate Quartiles

Given 12, 15, 17, 21, 21, 22, 23, 23, 23, 23, 24, 25, 28, 30

• Q2 (Median): 23 (since it is the average of the 7th and 8th values in the sorted list)
• Q1: Median of the first half (12, 15, 17, 21, 21, 22, 23) is 21.
• Q3: Median of the second half (23, 23, 23, 24, 25, 28, 30) is 24.

Step 2: Calculate IQR

$IQR = Q3 - Q1 = 24 - 21 = 3$

Step 3: Determine the Lower and Upper Bounds

• Lower bound: $Q1 - 1.5 \times IQR = 21 - 1.5 \times 3 = 21 - 4.5 = 16.5$
• Upper bound: $Q3 + 1.5 \times IQR = 24 + 1.5 \times 3 = 24 + 4.5 = 28.5$

Step 4: Identify Outliers

• Any value below 16.5 or above 28.5 is considered a possible outlier.

Values in the data:

• 12, 15, and 17 are below 16.5, so they are possible outliers.
• 30 is above 28.5, so it is a possible outlier.

Conclusion: The possible outliers in the data set are 12, 15, 17, and 30.

Answer:

• 12
• 15
• 17
• 30

Would you like further details or have any questions?

Follow-up Questions:

1. What are the key differences between a modified box plot and a regular box plot?
2. How does the choice of the multiplier (1.5) affect the detection of outliers in box plots?
3. Can a value exactly at the boundary (like 16.5 or 28.5) be considered an outlier?
4. How would you interpret outliers in the context of a real-world data set?
5. What are some alternative methods to detect outliers in a data set?

Tip:

When identifying outliers, it's helpful to plot the data using a box plot, as it visually represents the spread and identifies outliers easily.