To find the mean deviation from the median for the given data set, we will follow these steps:
### Step 1: Sort the Data
First, we need to arrange the data in ascending order.
**Given data:** 9, 12, 18, 3, 5, 3, 10, 12, 21, 12, 21
**Sorted data:** 3, 3, 5, 9, 10, 12, 12, 12, 18, 21, 21
### Step 2: Find the Median
Next, we will calculate the median of the sorted data.
- The number of observations (n) is 11 (which is odd).
- The median is given by the formula:
\[
\text{Median} = \text{Value at } \left(\frac{n + 1}{2}\right) \text{th position}
\]
Here, \( \frac{11 + 1}{2} = 6 \), so we take the 6th term in the sorted list.
**6th term:** 12
Thus, the median is **12**.
### Step 3: Calculate the Absolute Deviations
Now, we will calculate the absolute deviations from the median for each data point.
\[
\text{Absolute Deviation} = |x_i - \text{Median}|
\]
Where \( x_i \) is each data point.
- For 3: \( |3 - 12| = 9 \)
- For 3: \( |3 - 12| = 9 \)
- For 5: \( |5 - 12| = 7 \)
- For 9: \( |9 - 12| = 3 \)
- For 10: \( |10 - 12| = 2 \)
- For 12: \( |12 - 12| = 0 \)
- For 12: \( |12 - 12| = 0 \)
- For 12: \( |12 - 12| = 0 \)
- For 18: \( |18 - 12| = 6 \)
- For 21: \( |21 - 12| = 9 \)
- For 21: \( |21 - 12| = 9 \)
### Step 4: Sum of Absolute Deviations
Now, we will sum all the absolute deviations calculated above.
\[
\text{Total Absolute Deviation} = 9 + 9 + 7 + 3 + 2 + 0 + 0 + 0 + 6 + 9 + 9 = 54
\]
### Step 5: Calculate the Mean Deviation
Finally, we will calculate the mean deviation from the median.
\[
\text{Mean Deviation} = \frac{\text{Total Absolute Deviation}}{n}
\]
Where \( n \) is the number of observations.
\[
\text{Mean Deviation} = \frac{54}{11} \approx 4.909 \text{ (rounded to 4.9)}
\]
### Final Answer
The mean deviation from the median for the given data is approximately **4.9**.
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