1. Arrange the known observations in ascending order

The given observations are: $170,125,230,190,210,a,b$

The known numbers in ascending order are: $125,170,190,210,230$

There are 7 observations, and the median is given as $170$.

For 7 observations, the median is the 4th term after arranging in ascending order.

So in the full ordered list, the 4th term must be $170$.

Since among the known values only $125$ is less than $170$, we need two more values less than or equal to $170$ so that $170$ becomes the 4th term.

Thus both $a$ and $b$ must be less than or equal to $170$.

2. Use the mean deviation about the median

Mean deviation about median is given by $\frac{1}{7}\sum |x_i-170|=\frac{205}{7}$

Hence, $\sum |x_i-170|=205$

Now compute contributions from known observations:

$|170-170|=0$ $|125-170|=45$ $|230-170|=60$ $|190-170|=20$ $|210-170|=40$

Sum of these is: $0+45+60+20+40=165$

Therefore, $|a-170|+|b-170|=205-165=40$

Since both $a,b\le 170$, we have $|a-170|=170-a, \quad |b-170|=170-b$

So, $(170-a)+(170-b)=40$ $340-(a+b)=40$ $a+b=300$

3. Find the mean of all 7 observations

Sum of known 5 observations: $170+125+230+190+210=925$

Including $a+b=300$: $\text{Total sum}=925+300=1225$

Thus mean is $\bar{x}=\frac{1225}{7}=175$

4. Compute mean deviation about the mean

We need $\frac{1}{7}\sum |x_i-175|$

For known observations: $|170-175|=5$ $|125-175|=50$ $|230-175|=55$ $|190-175|=15$ $|210-175|=35$

Their sum is: $5+50+55+15+35=160$

Now consider $a$ and $b$.

We know:

• $a,b\le 170$
• $a+b=300$

So both are less than $175$, hence $|a-175|=175-a, \quad |b-175|=175-b$

Therefore, $|a-175|+|b-175|=(175-a)+(175-b)=350-(a+b)=350-300=50$

Thus total absolute deviation about mean: $160+50=210$

Hence mean deviation about mean is $\frac{210}{7}=30$

5. Match with the options

$30$ corresponds to Option C.

6. Comparison with stored answer

Stored correct answer: C

Derived answer: C

So the derived answer agrees with the stored answer.