To identify unusual values in a dataset, we typically use the IQR (Interquartile Range) method. Here’s a step-by-step breakdown of the process to detect outliers:

1. Sort the Data

Sorted dataTo identify unusual values in a dataset, we typically use the IQR (Interquartile Range) method. Here’s a step-by-step breakdown of the process to detect outliers:

1. Sort the Data

Sorted
0.79, 1.38, 1.73, 1.93, 2.23, 2.49, 3.26, 4.05, 4.12, 4.46, 4.53, 4.63, 5.35, 5.42, 5.56, 5.94, 5.98, 6.16, 6.38, 7.60, 8.19, 8.88, 8.89, 9.86, 10.32, 12.83, 13.00, 14.57, 15.58, 16.07, 17.55, 19.41, 20.16, 23.62, 24.93, 26.67, 27.08, 31.47, 82.81, 192.00

2. Calculate the Quartiles

• Q1 (25th percentile): Average of 10th and 11th values = (4.46 + 4.53) / 2 = 4.50
• Q3 (75th percentile): Average of 30th and 31st values = (16.07 + 17.55) / 2 = 16.81

3. Calculate the IQR (Interquartile Range)

$\text{IQR} = Q3 - Q1 = 16.81 - 4.50 = 12.31$

4. Find the Lower and Upper Fences

Using the standard rule for outliers:

• Lower fence: $Q1 - 1.5 \times \text{IQR} = 4.50 - 1.5 \times 12.31 = 4.50 - 18.47 = -13.97$
• Upper fence: $Q3 + 1.5 \times \text{IQR} = 16.81 + 1.5 \times 12.31 = 16.81 + 18.47 = 35.28$

5. Identify Unusual Values (Outliers)

• Since the lower fence is negative, there are no low outliers.
• Any value above 35.28 is considered an outlier.

From the data, the following values are above 35.28:

• 82.81
• 192.00

6. Conclusion

The unusual (outlier) values in the dataset are:

• 82.81 million
• 192.00 million

These values are much higher than the rest, consistent with the distribution being skewed to the right.

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Do you have any questions or need further details?
Here are five related questions to explore:

1. How does the skewness affect the mean and median in this dataset?
2. What is the impact of outliers on the standard deviation?
3. How would the dataset change if we removed the outliers?
4. Could a log transformation reduce skewness in this distribution?
5. How do quartiles change if additional extreme outliers are introduced?

Tip: When working with skewed data, the median often provides a more representative measure of central tendency than the mean.