Let the median and the mean deviation about the median of 7 observation 170, 125, 230, 190, 210, a, b be 170 and `frac{205}{7}` respectively. Then the mean deviation about the mean of these 7 observations is :

Options

A. 31
B. 28
C. 30
D. 32

Answer

To solve the problem step by step, we will follow the given information about the observations, median, and mean deviation.

### Step 1: Understanding the Observations
We are given 7 observations: 170, 125, 230, 190, 210, a, b. We need to find the values of a and b based on the median and mean deviation.

### Step 2: Finding the Median
The median of a set of observations is the middle value when they are arranged in ascending order. Given that the median is 170, we can conclude that at least one of the unknowns (a or b) must be 170, because the median is the fourth term in a sorted list of 7 numbers.

### Step 3: Arranging Observations
Let's arrange the observations in ascending order. We can assume:
- a ≤ b
- The sorted order will be: a, b, 125, 170, 190, 210, 230

### Step 4: Mean Deviation about the Median
The mean deviation about the median is given as \( \frac{205}{7} \). The formula for mean deviation about the median is:

\[ MD = \frac{1}{n} \sum |x_i - \text{median}| \]

Where \( x_i \) are the observations. Thus, we have:

\[ \frac{1}{7} \left( |a - 170| + |b - 170| + |125 - 170| + |190 - 170| + |210 - 170| + |230 - 170| \right) = \frac{205}{7} \]

### Step 5: Simplifying the Equation
Now, we can simplify the equation:

\[ |a - 170| + |b - 170| + |125 - 170| + |190 - 170| + |210 - 170| + |230 - 170| = 205 \]

Calculating the absolute differences:
- \( |125 - 170| = 45 \)
- \( |190 - 170| = 20 \)
- \( |210 - 170| = 40 \)
- \( |230 - 170| = 60 \)

Substituting these values into the equation gives:

\[ |a - 170| + |b - 170| + 45 + 20 + 40 + 60 = 205 \]

This simplifies to:

\[ |a - 170| + |b - 170| + 165 = 205 \]

Thus,

\[ |a - 170| + |b - 170| = 40 \]

### Step 6: Solving for a and b
Now we have two equations:
1. \( |a - 170| + |b - 170| = 40 \)
2. \( a + b = 300 \) (from the earlier calculations)

Assuming \( a \leq b \), we can express \( b \) in terms of \( a \):

Let \( a = 170 - x \) and \( b = 170 + y \) where \( x + y = 40 \).

Substituting into the second equation:

\[ (170 - x) + (170 + y) = 300 \]

This simplifies to:

\[ 340 + (y - x) = 300 \implies y - x = -40 \]

Now we have:
1. \( x + y = 40 \)
2. \( y - x = -40 \)

Solving these two equations simultaneously gives:

Adding them:
\[ 2y = 0 \implies y = 0 \implies x = 40 \]

Thus, we find \( a = 130 \) and \( b = 210 \).

### Step 7: Finding the Mean
Now we can calculate the mean of the observations:

\[ \text{Mean} = \frac{a + b + 125 + 170 + 190 + 210 + 230}{7} = \frac{130 + 210 + 125 + 170 + 190 + 210 + 230}{7} \]

Calculating the sum:

\[ = \frac{130 + 210 + 125 + 170 + 190 + 210 + 230}{7} = \frac{1075}{7} = 153.57 \]

### Step 8: Mean Deviation about the Mean
Now we find the mean deviation about the mean:

\[ MD = \frac{1}{7} \left( |130 - 153.57| + |210 - 153.57| + |125 - 153.57| + |170 - 153.57| + |190 - 153.57| + |210 - 153.57| + |230 - 153.57| \right) \]

Calculating each term:

\[ = \frac{1}{7} \left( 23.57 + 56.43 + 28.57 + 16.43 + 36.43 + 56.43 + 76.43 \right) \]

Summing these values gives:

\[ = \frac{1}{7} \left( 298.28 \right) \approx 42.61 \]

### Final Answer
Thus, the mean deviation about the mean is approximately 42.61.