Impact of Removing Outliers on Median: Practical Examples and Potential Pitfall

**Impact of Removing Outliers on Median: Practical Examples and Potential Pitfalls**

**1. Practical Example: Household Monthly Expenses**

**Scenario:**

You have monthly expenses data for a small group of households (in dollars):

`200, 220, 240, 250, 260, 5000`

**Steps:**

1. **Calculate the Median:**

   - Sort the data: `200, 220, 240, 250, 260, 5000`

   - Since there are 6 values, the median is the average of the 3rd and 4th values: `(240 + 250) / 2 = 245`

2. **Identify and Remove Outliers:**

   - Using the IQR method:

     - **Calculate Q1 and Q3:**

       - Lower half: `200, 220, 240` (Q1 = 220)

       - Upper half: `250, 260, 5000` (Q3 = 260)

       - IQR = Q3 - Q1 = 260 - 220 = 40

     - **Calculate Bounds:**

       - Lower bound: 220 - 1.5 * 40 = 220 - 60 = 160

       - Upper bound: 260 + 1.5 * 40 = 260 + 60 = 320

     - **Identify Outliers:** The value `5000` is above the upper bound of 320, so it is an outlier.

   - **Remove the Outlier:**

     - New data set: `200, 220, 240, 250, 260`

3. **Recalculate the Median:**

   - For the new data set `200, 220, 240, 250, 260` (5 values), the median is the middle value: `240`.

   - **Comparison:**

     - Original Median: `245`

     - New Median (after removing `5000`): `240`

   Removing the outlier `5000` caused the median to change from `245` to `240`. This demonstrates how extreme values can skew the median, affecting its representation of central tendency.

**2. Scenario: Analyzing Annual Salaries in a Small Company**

**Scenario:**

You have annual salaries (in thousands of dollars) for a small company:

`50, 52, 55, 60, 62, 65, 500`

**Steps:**

1. **Calculate the Median:**

   - Sort the data: `50, 52, 55, 60, 62, 65, 500`

   - With 7 values, the median is the 4th value: `60`

2. **Identify and Remove Outliers:**

   - Using the IQR method:

     - **Calculate Q1 and Q3:**

       - Lower half: `50, 52, 55` (Q1 = 52)

       - Upper half: `62, 65, 500` (Q3 = 65)

       - IQR = Q3 - Q1 = 65 - 52 = 13

     - **Calculate Bounds:**

       - Lower bound: 52 - 1.5 * 13 = 52 - 19.5 = 32.5

       - Upper bound: 65 + 1.5 * 13 = 65 + 19.5 = 84.5

     - **Identify Outliers:** The value `500` is above the upper bound of 84.5, so it is an outlier.

   - **Remove the Outlier:**

     - New data set: `50, 52, 55, 60, 62, 65`

3. **Recalculate the Median:**

   - For the new data set `50, 52, 55, 60, 62, 65` (6 values), the median is the average of the 3rd and 4th values: `(55 + 60) / 2 = 57.5`

   - **Comparison:**

     - Original Median: `60`

     - New Median (after removing `500`): `57.5`

   **Potential Negative Impact:**

   - **Misleading Trends:** Removing the high salary outlier might hide critical information about salary distribution and the presence of significant managerial roles.

   - **Loss of Insight:** Excluding the outlier could misrepresent the salary distribution, leading to incorrect assumptions about salary levels within the company.

   - **Decision Making:** For accurate budget planning or compensation strategies, knowing the full range of salaries, including outliers, is important. Removing outliers might lead to underestimating compensation needs or overlooking disparities.

In summary, while removing outliers can sometimes provide a clearer view of central tendencies, it can also obscure important data trends. Careful consideration is needed to balance the benefits of outlier removal with the potential loss of critical information.