A set of five numbers is 8, 9, x, (x+1), 15. If the mean of these numbers is 11, find the value of x, the variance, and the standard deviation.

Multiply both sides by 5:

$$11 \times 5 = 8 + 9 + x + x + 1 + 15$$ $$55 = 33 + 2x$$

Subtract 33 from both sides:

$$55 - 33 = 2x$$ $$22 = 2x$$

Divide by 2:

$$x = \frac{22}{2}$$ $$\boxed{x = 11}$$

The formula for variance is $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$.
First, calculate the deviations from the mean $(x_i - \mu)$ and their squares $(x_i - \mu)^2$:

• For 8: $(8 - 11)^2 = (-3)^2 = 9$
• For 9: $(9 - 11)^2 = (-2)^2 = 4$
• For 11: $(11 - 11)^2 = (0)^2 = 0$
• For 12: $(12 - 11)^2 = (1)^2 = 1$
• For 15: $(15 - 11)^2 = (4)^2 = 16$

Sum of the squared deviations:

$$\sum (x_i - \mu)^2 = 9 + 4 + 0 + 1 + 16 = 30$$

Now, calculate the variance:

$$\sigma^2 = \frac{30}{5} = 6$$

The standard deviation is the square root of the variance.

$$\sigma = \sqrt{\sigma^2}$$ $$\sigma = \sqrt{6}$$ $$\boxed{\sigma \approx 2.449}$$