1. Use the given mean

For the 6 observations $-3,\;4,\;7,\;-6,\;\alpha,\;\beta$ the mean is $2$.

So, $\frac{-3+4+7-6+\alpha+\beta}{6}=2$

First simplify the known sum: $-3+4+7-6=2$ Hence, $\frac{2+\alpha+\beta}{6}=2$ $2+\alpha+\beta=12$ $\alpha+\beta=10 \quad ...(1)$

2. Use the given variance

Variance about the mean $2$ is $23$, so $\frac{1}{6}\sum (x_i-2)^2=23$ Thus, $\sum (x_i-2)^2=138$

Now compute for the known observations:

• For $-3$: $(-3-2)^2=25$
• For $4$: $(4-2)^2=4$
• For $7$: $(7-2)^2=25$
• For $-6$: $(-6-2)^2=64$

Their sum is $25+4+25+64=118$ So, $(\alpha-2)^2+(\beta-2)^2=138-118=20 \quad ...(2)$

Expand: $(\alpha-2)^2+(\beta-2)^2=\alpha^2+\beta^2-4(\alpha+\beta)+8=20$ Using $(1)$, $\alpha+\beta=10$: $\alpha^2+\beta^2-40+8=20$ $\alpha^2+\beta^2=52 \quad ...(3)$

Now use $(\alpha+\beta)^2=\alpha^2+\beta^2+2\alpha\beta$ $10^2=52+2\alpha\beta$ $100=52+2\alpha\beta$ $2\alpha\beta=48$ $\alpha\beta=24$

So $\alpha,\beta$ are roots of $t^2-10t+24=0$ $t^2-10t+24=(t-4)(t-6)=0$ Hence, $\alpha,\beta=4,6$

3. Now find the mean deviation about the mean

The mean is $2$, so mean deviation about the mean is $\frac{1}{6}\sum |x_i-2|$

Observations are now: $-3,\;4,\;7,\;-6,\;4,\;6$

Compute absolute deviations from $2$:

• For $-3$: $|-3-2|=5$
• For $4$: $|4-2|=2$
• For $7$: $|7-2|=5$
• For $-6$: $|-6-2|=8$
• For $4$: $|4-2|=2$
• For $6$: $|6-2|=4$

Sum: $5+2+5+8+2+4=26$

Therefore, $\text{Mean deviation about mean} = \frac{26}{6}=\frac{13}{3}$

4. Compare with the given options

Thus the correct option is $\boxed{\frac{13}{3}}$ which is Option D.

5. Verification with stored correct answer

Stored correct answer: D

Our derived answer: D

So they agree.