To identify outliers in the data set $\{840, 296, 157, 750, 5,580, 400\}$, we will use the Interquartile Range (IQR) method.

Steps to Identify Outliers:

1. Sort the Data: Arrange the data in ascending order. $\{5, 157, 296, 400, 750, 840\}$

2. Calculate the Quartiles:

   • Median (Q2): The middle value of the sorted data: $Q2 = \frac{296 + 400}{2} = 348$
   • Lower Quartile (Q1): Median of the lower half: $Q1 = \frac{5 + 157}{2} = 81$
   • Upper Quartile (Q3): Median of the upper half: $Q3 = \frac{750 + 840}{2} = 795$

3. Calculate the IQR: $\text{IQR} = Q3 - Q1 = 795 - 81 = 714$

4. Determine Outlier Bounds:

   • Lower Bound: $Q1 - 1.5 \times \text{IQR} = 81 - 1.5 \times 714 = -990$ (no lower outliers as data can't go below 0).
   • Upper Bound: $Q3 + 1.5 \times \text{IQR} = 795 + 1.5 \times 714 = 1866$.

5. Identify Outliers:

   • Any data point less than the lower bound or greater than the upper bound is an outlier.
   • Observing the data: $\{840, 296, 157, 750, 5, 580, 400\}$ All values except $5,580$ are within the range $[0, 1866]$. Therefore, $5,580$ is an outlier.

Conclusion:

The only outlier in the data is $5,580$.

Would you like further details or have any questions?

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Related Questions:

1. What is the significance of IQR in outlier detection?
2. How would the presence of multiple outliers affect statistical measures like mean and variance?
3. Could the Z-score method detect $5,580$ as an outlier in this data set?
4. How can removing outliers impact the analysis of a data set?
5. What are other methods for identifying outliers besides IQR and Z-scores?

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Tip:

Outlier detection is vital for improving data quality and ensuring statistical analyses are not skewed by extreme values.