To find the standard deviation of the given data set $\{y+1, y+2, y+5, y+9, y+8\}$, we follow these steps:

Step 1: Calculate the mean ($\mu$) of the data set.

$\mu = \frac{(y+1) + (y+2) + (y+5) + (y+9) + (y+8)}{5}$ Simplifying the sum: $\mu = \frac{5y + (1+2+5+9+8)}{5} = \frac{5y + 25}{5} = y + 5$

Step 2: Calculate the squared deviations from the mean.

For each value in the data set: $(y+1) - (y+5) = -4, \quad \text{so} \quad (-4)^2 = 16$ $(y+2) - (y+5) = -3, \quad \text{so} \quad (-3)^2 = 9$ $(y+5) - (y+5) = 0, \quad \text{so} \quad 0^2 = 0$ $(y+9) - (y+5) = 4, \quad \text{so} \quad 4^2 = 16$ $(y+8) - (y+5) = 3, \quad \text{so} \quad 3^2 = 9$

Step 3: Calculate the variance ($\sigma^2$).

The variance is the average of these squared deviations: $\sigma^2 = \frac{16 + 9 + 0 + 16 + 9}{5} = \frac{50}{5} = 10$

Step 4: Calculate the standard deviation ($\sigma$).

$\sigma = \sqrt{10} \approx 3.162$

Final Answer:

The standard deviation of the data set, correct to three significant figures, is approximately $\boxed{3.16}$.

Would you like more details or have any questions? Here are some related questions:

1. How does the standard deviation change if a constant is added to all data points?
2. What is the relationship between variance and standard deviation?
3. How do you calculate the standard deviation for a sample, as opposed to a population?
4. What is the significance of the standard deviation in statistical analysis?
5. How does standard deviation help in understanding the spread of data?

Tip: Remember that standard deviation gives a sense of how spread out the values in a data set are around the mean.