How to find upper and lower outlier boundaries?

Understand the Problem

The question is asking how to determine the upper and lower boundaries for outliers in a given data set. This typically involves calculating the interquartile range (IQR) and applying it to identify values that fall outside of specified thresholds.

Answer

The boundaries for outliers are calculated using $Q1 - 1.5 \times IQR$ for the lower boundary and $Q3 + 1.5 \times IQR$ for the upper boundary.

Steps to Solve

Calculate the first and third quartiles (Q1 and Q3)

To find the boundaries for outliers, we first need to determine the first quartile (Q1) and the third quartile (Q3) of the data set. Q1 is the value at the 25th percentile and Q3 is the value at the 75th percentile.

Find the interquartile range (IQR)

The interquartile range (IQR) is calculated by subtracting Q1 from Q3. This gives us the spread of the middle 50% of the data. $$ IQR = Q3 - Q1 $$

Calculate the lower and upper boundaries for outliers

To determine the boundaries for outliers, use the following formulas:

Lower boundary: $$ \text{Lower Boundary} = Q1 - 1.5 \times IQR $$
Upper boundary: $$ \text{Upper Boundary} = Q3 + 1.5 \times IQR $$

Identify the outliers

Any data points that fall below the lower boundary or above the upper boundary are considered outliers.

The lower boundary for outliers is $Q1 - 1.5 \times IQR$ and the upper boundary for outliers is $Q3 + 1.5 \times IQR$.

More Information

Outliers are significant because they can affect statistical analyses and interpretations. Identifying them helps ensure that the analysis of the data set is robust and reflective of the underlying trends without distortion from extreme values.

Tips

Failing to correctly calculate Q1 and Q3, which can lead to incorrect boundaries. Ensure you know how to find these quartiles.
Not using the 1.5 factor when calculating boundaries; this is crucial for identifying outliers.

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