Find the mean deviation from the median for the following data :
10,3,12,5,9,3,18,21,21,8,12,12

Answer

To find the mean deviation from the median for the given data set, we will follow these steps:

### Step 1: Arrange the data in increasing order
The given data is:
10, 3, 12, 5, 9, 3, 18, 21, 21, 8, 12, 12.

Arranging it in increasing order:
3, 3, 5, 8, 9, 10, 12, 12, 12, 18, 21, 21.

### Step 2: Calculate the median
Since there are 12 observations (even number), the median is calculated as the average of the 6th and 7th terms.

- 6th term = 10
- 7th term = 12

So, the median is:
\[ \text{Median} = \frac{10 + 12}{2} = \frac{22}{2} = 11. \]

### Step 3: Calculate the mean deviation from the median
The formula for mean deviation from the median is:
\[ \text{Mean Deviation} = \frac{\sum |x_i - \text{Median}|}{N} \]
where \(N\) is the total number of observations.

Calculating the absolute deviations from the median (11):
- |3 - 11| = 8
- |3 - 11| = 8
- |5 - 11| = 6
- |8 - 11| = 3
- |9 - 11| = 2
- |10 - 11| = 1
- |12 - 11| = 1
- |12 - 11| = 1
- |12 - 11| = 1
- |18 - 11| = 7
- |21 - 11| = 10
- |21 - 11| = 10

Now, summing these absolute deviations:
\[ 8 + 8 + 6 + 3 + 2 + 1 + 1 + 1 + 1 + 7 + 10 + 10 = 58. \]

### Step 4: Divide by the number of observations
Now, we divide the total absolute deviation by the number of observations (12):
\[ \text{Mean Deviation} = \frac{58}{12} \approx 4.83. \]

### Final Answer
The mean deviation from the median for the given data is approximately **4.83**.

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