The mean deviation of the data 4,5,7,8,9,10,6 from the median is

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find the mean deviation of the data set \(4, 5, 7, 8, 9, 10, 6\) from the median, we will follow these steps:

### Step 1: Arrange the data in ascending order
The first step is to arrange the given data in ascending order:
\[ 4, 5, 6, 7, 8, 9, 10 \]

### Step 2: Count the number of observations
Next, we count the number of observations in the data set:
- The observations are: \(4, 5, 6, 7, 8, 9, 10\)
- Total number of observations \(n = 7\)

### Step 3: Find the median
Since the number of observations \(n\) is odd, the median is the middle value. The formula for the median in this case is:
\[ \text{Median} = \text{Value at position} \left(\frac{n + 1}{2}\right) \]
Calculating this gives:
\[ \text{Median} = \text{Value at position} \left(\frac{7 + 1}{2}\right) = \text{Value at position} 4 \]
The 4th value in the ordered list \(4, 5, 6, 7, 8, 9, 10\) is \(7\). Thus, the median is:
\[ \text{Median} = 7 \]

### Step 4: Calculate the absolute deviations from the median
Now, we calculate the absolute deviations of each observation from the median:
- \( |4 - 7| = 3 \)
- \( |5 - 7| = 2 \)
- \( |6 - 7| = 1 \)
- \( |7 - 7| = 0 \)
- \( |8 - 7| = 1 \)
- \( |9 - 7| = 2 \)
- \( |10 - 7| = 3 \)

### Step 5: Sum the absolute deviations
Next, we sum all the absolute deviations calculated in the previous step:
\[ 3 + 2 + 1 + 0 + 1 + 2 + 3 = 12 \]

### Step 6: Calculate the mean deviation
Finally, we calculate the mean deviation by dividing the total absolute deviation by the number of observations:
\[ \text{Mean Deviation} = \frac{\text{Sum of absolute deviations}}{n} = \frac{12}{7} \approx 1.714285714 \]

Thus, the mean deviation of the data from the median is approximately \(1.71\).

### Final Answer
The mean deviation of the data \(4, 5, 7, 8, 9, 10, 6\) from the median is approximately \(1.71\).

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