Find the mean deviation from the median for the following data :
13,17,16,14,11,13,10,16,11,18,12,17

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 Ascending Order
The given data is:
\[ 13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17 \]

Arranging this data in ascending order, we get:
\[ 10, 11, 11, 12, 13, 13, 14, 16, 16, 17, 17, 18 \]

### Step 2: Calculate the Median
Since the number of observations \( N = 12 \) (which is even), the median is calculated as the average of the \( \frac{N}{2} \)th term and the \( \left(\frac{N}{2} + 1\right) \)th term.

- The \( \frac{N}{2} \)th term (6th term) is \( 13 \).
- The \( \left(\frac{N}{2} + 1\right) \)th term (7th term) is \( 14 \).

Calculating the median:
\[ \text{Median} = \frac{13 + 14}{2} = \frac{27}{2} = 13.5 \]

### Step 3: Calculate the Deviations from the Median
Next, we calculate the absolute deviations of each observation from the median \( 13.5 \):

\[ \begin{align*} |10 - 13.5| & = 3.5 \\ |11 - 13.5| & = 2.5 \\ |11 - 13.5| & = 2.5 \\ |12 - 13.5| & = 1.5 \\ |13 - 13.5| & = 0.5 \\ |13 - 13.5| & = 0.5 \\ |14 - 13.5| & = 0.5 \\ |16 - 13.5| & = 2.5 \\ |16 - 13.5| & = 2.5 \\ |17 - 13.5| & = 3.5 \\ |17 - 13.5| & = 3.5 \\ |18 - 13.5| & = 4.5 \\ \end{align*} \]

### Step 4: Sum of Absolute Deviations
Now, we sum all the absolute deviations calculated:
\[ 3.5 + 2.5 + 2.5 + 1.5 + 0.5 + 0.5 + 0.5 + 2.5 + 2.5 + 3.5 + 3.5 + 4.5 = 28 \]

### Step 5: Calculate the Mean Deviation
Finally, we calculate the mean deviation from the median by dividing the total sum of absolute deviations by the number of observations \( N \):
\[ \text{Mean Deviation} = \frac{\text{Sum of Absolute Deviations}}{N} = \frac{28}{12} \approx 2.33 \]

### Final Answer
The mean deviation from the median is approximately \( 2.33 \).

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