To find the mean deviation about the median for the given data set \( 11, 3, 8, 7, 5, 14, 10, 2, 9 \), we will follow these steps:
### Step 1: Arrange the Data
First, we need to arrange the data in ascending order:
- The data in ascending order is: \( 2, 3, 5, 7, 8, 9, 10, 11, 14 \)
### Step 2: Find the Median
Since there are 9 data points (which is odd), the median is the middle value. The median can be calculated using the formula:
\[
\text{Median} = \text{Value at position } \frac{n + 1}{2}
\]
where \( n \) is the number of data points. Here, \( n = 9 \):
\[
\text{Median} = \text{Value at position } \frac{9 + 1}{2} = \text{Value at position } 5
\]
The 5th value in the ordered list is \( 8 \). Thus, the median is \( 8 \).
### Step 3: Calculate Deviations from the Median
Next, we calculate the absolute deviations of each data point from the median:
- For \( 2 \): \( |2 - 8| = 6 \)
- For \( 3 \): \( |3 - 8| = 5 \)
- For \( 5 \): \( |5 - 8| = 3 \)
- For \( 7 \): \( |7 - 8| = 1 \)
- For \( 8 \): \( |8 - 8| = 0 \)
- For \( 9 \): \( |9 - 8| = 1 \)
- For \( 10 \): \( |10 - 8| = 2 \)
- For \( 11 \): \( |11 - 8| = 3 \)
- For \( 14 \): \( |14 - 8| = 6 \)
### Step 4: Sum of Deviations
Now, we sum all the absolute deviations:
\[
\text{Sum of deviations} = 6 + 5 + 3 + 1 + 0 + 1 + 2 + 3 + 6 = 27
\]
### Step 5: Calculate Mean Deviation
Finally, we calculate the mean deviation by dividing the total sum of deviations by the number of data points:
\[
\text{Mean Deviation} = \frac{\text{Sum of deviations}}{n} = \frac{27}{9} = 3
\]
### Conclusion
The mean deviation about the median for the given data is \( 3 \).
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