Find the mean deviation from the median for the data : 9,12,18,3,5,3,10,12,21,12,21

Answer

To find the mean deviation from the median for the data set: 9, 12, 18, 3, 5, 3, 10, 12, 21, 12, 21, we will follow these steps:

### Step 1: Arrange the data in increasing order.
The given data is:
\[ 9, 12, 18, 3, 5, 3, 10, 12, 21, 12, 21 \]

Arranging it in increasing order:
\[ 3, 3, 5, 9, 10, 12, 12, 12, 18, 21, 21 \]

### Step 2: Find the median.
Since there are 11 observations (which is odd), the median is the middle value. The median can be found using the formula:
\[ \text{Median} = \text{Value at } \left( \frac{n + 1}{2} \right) \text{th position} \]
where \( n \) is the number of observations.

Calculating:
\[ \frac{11 + 1}{2} = 6 \]
The 6th term in the ordered list is 12. Therefore, the median \( m \) is:
\[ m = 12 \]

### Step 3: Calculate the absolute deviations from the median.
We will now calculate the absolute deviation of each data point from the median (12):
- 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 the absolute deviations.
Now, we will sum all the absolute deviations calculated:
\[ \text{Sum} = 9 + 9 + 7 + 3 + 2 + 0 + 0 + 0 + 6 + 9 + 9 = 54 \]

### Step 5: Calculate the mean deviation.
The mean deviation is calculated using the formula:
\[ \text{Mean Deviation} = \frac{\text{Sum of absolute deviations}}{n} \]
where \( n \) is the number of observations.

Calculating:
\[ \text{Mean Deviation} = \frac{54}{11} \approx 4.91 \]

### Final Answer:
The mean deviation from the median for the given data is approximately \( 4.91 \).

---