Mean deviation of 39,40,41,41,42,42,43,43,44,44, 45 through median is

Options

A. 15
B. 1.5
C. 43
D. 35

Answer

To find the mean deviation of the given data set \(39, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45\) through the median, we will follow these steps:

### Step 1: Arrange the Data
The data is already arranged in ascending order:
\[ 39, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45 \]

### Step 2: Find the Median
Since there are 11 observations (which is odd), the median is the value at the \(\frac{n + 1}{2}\)th position.
- Here, \(n = 11\).
- The median position is \(\frac{11 + 1}{2} = 6\)th term.

The 6th term in the ordered list is \(42\).
Thus, the median \(M = 42\).

### Step 3: Calculate the Mean Deviation
The formula for mean deviation through the median is:
\[ \text{Mean Deviation} = \frac{\sum |x_i - M|}{n} \]
Where \(x_i\) are the data points, \(M\) is the median, and \(n\) is the number of observations.

### Step 4: Calculate \(|x_i - M|\) for Each Data Point
Now we will calculate \(|x_i - 42|\) for each data point:
- For \(39\): \(|39 - 42| = | -3 | = 3\)
- For \(40\): \(|40 - 42| = | -2 | = 2\)
- For \(41\): \(|41 - 42| = | -1 | = 1\)
- For \(41\): \(|41 - 42| = | -1 | = 1\)
- For \(42\): \(|42 - 42| = | 0 | = 0\)
- For \(42\): \(|42 - 42| = | 0 | = 0\)
- For \(43\): \(|43 - 42| = | 1 | = 1\)
- For \(43\): \(|43 - 42| = | 1 | = 1\)
- For \(44\): \(|44 - 42| = | 2 | = 2\)
- For \(44\): \(|44 - 42| = | 2 | = 2\)
- For \(45\): \(|45 - 42| = | 3 | = 3\)

### Step 5: Sum of Deviations
Now, we sum these absolute deviations:
\[ 3 + 2 + 1 + 1 + 0 + 0 + 1 + 1 + 2 + 2 + 3 = 16 \]

### Step 6: Calculate the Mean Deviation
Now we divide the total sum of deviations by the number of observations:
\[ \text{Mean Deviation} = \frac{16}{11} \approx 1.4545 \]

### Final Answer
Thus, the mean deviation of the given data through the median is approximately \(1.45\).

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