Given data

The data set is: $7,\ 8,\ 9,\ 7,\ 8,\ 7,\ \lambda,\ 8$

Number of observations: $n=8$

Use the mean condition

Mean is given as $8$.

So, $\frac{7+8+9+7+8+7+\lambda+8}{8}=8$

First add the known terms: $7+8+9+7+8+7+8=54$

Hence, $\frac{54+\lambda}{8}=8$

$54+\lambda=64$

$\lambda=10$

Write the complete data

The data becomes: $7,\ 8,\ 9,\ 7,\ 8,\ 7,\ 10,\ 8$

Find variance

Using the formula: $\text{Variance}=\frac{1}{n}\sum (x_i-\bar{x})^2$

Here, mean $\bar{x}=8$.

Now compute squared deviations:

• For $7$: $(7-8)^2=1$
• For $8$: $(8-8)^2=0$
• For $9$: $(9-8)^2=1$
• For $7$: $(7-8)^2=1$
• For $8$: $(8-8)^2=0$
• For $7$: $(7-8)^2=1$
• For $10$: $(10-8)^2=4$
• For $8$: $(8-8)^2=0$

Sum of squared deviations: $1+0+1+1+0+1+4+0=8$

Therefore, $\text{Variance}=\frac{8}{8}=1$

Check options

• A: $\frac{7}{8}$
• B: $1$ ✅
• C: $\frac{9}{8}$
• D: $2$

So the correct option is: $\boxed{\text{B}}$