1. Given data

The observations are $2,3,3,4,5,7,a,b$ with:

• mean $=4$
• standard deviation $=\sqrt{2}$

There are $8$ observations.

2. Use the mean

Mean is $\bar x=\frac{2+3+3+4+5+7+a+b}{8}=4$ So, $2+3+3+4+5+7+a+b=32$ $24+a+b=32$ $a+b=8$

3. Use the standard deviation

For a set of observations, variance is $\sigma^2=\frac{1}{n}\sum x_i^2-\bar x^2$

Given $\sigma=\sqrt{2}$, so $\sigma^2=2$ Hence, $2=\frac{1}{8}\sum x_i^2-4^2$ $2=\frac{1}{8}\sum x_i^2-16$ $\frac{1}{8}\sum x_i^2=18$ $\sum x_i^2=144$

Now, $2^2+3^2+3^2+4^2+5^2+7^2+a^2+b^2=144$ $4+9+9+16+25+49+a^2+b^2=144$ $112+a^2+b^2=144$ $a^2+b^2=32$

4. Find $a$ and $b$

We have $a+b=8$ $a^2+b^2=32$

Using $(a+b)^2=a^2+b^2+2ab$ $64=32+2ab$ $2ab=32$ $ab=16$

Thus $a,b$ are roots of $x^2-8x+16=0$ $(x-4)^2=0$ So, $a=b=4$

Hence the observations are $2,3,3,4,4,4,5,7$

5. Find the mode

The mode is the value occurring most frequently.

Here, $4$ occurs $3$ times, so $\text{Mode}=4$

6. Mean deviation about the mode

Mean deviation about mode $=4$ is $\frac{1}{8}\sum |x_i-4|$

Compute absolute deviations:

• for $2$: $|2-4|=2$
• for $3$: $|3-4|=1$
• for $3$: $|3-4|=1$
• for $4$: $|4-4|=0$
• for $4$: $|4-4|=0$
• for $4$: $|4-4|=0$
• for $5$: $|5-4|=1$
• for $7$: $|7-4|=3$

Sum: $2+1+1+0+0+0+1+3=8$

Therefore, $\text{Mean deviation about mode}=\frac{8}{8}=1$

7. Check options

• A: $\frac12$ ❌
• B: $\frac34$ ❌
• C: $1$ ✅
• D: $2$ ❌

So the correct option is $\boxed{\text{C}}$

8. Comparison with stored answer

Stored correct answer: C

My derived answer: C

They agree.