To calculate the standard deviation to the fourth decimal place, follow these steps:

1. Find the Mean (Average): $\text{Mean} = \frac{11.8 + 10.3 + 10.7 + 10.6 + 11.5 + 8.3 + 10.5 + 10.9 + 10.7 + 11.2}{10}$

2. Find the Variance:

   • Subtract the mean from each data point.
   • Square each result.
   • Find the average of those squared differences.

3. Find the Standard Deviation:

   • Take the square root of the variance.

Let's calculate that step by step:

1. Calculate the mean:

$\text{Mean} = \frac{11.8 + 10.3 + 10.7 + 10.6 + 11.5 + 8.3 + 10.5 + 10.9 + 10.7 + 11.2}{10} = \frac{106.5}{10} = 10.65$

2. Calculate the squared differences from the mean:

(11.8 - 10.65)^2 &= (1.15)^2 = 1.3225 \\ (10.3 - 10.65)^2 &= (-0.35)^2 = 0.1225 \\ (10.7 - 10.65)^2 &= (0.05)^2 = 0.0025 \\ (10.6 - 10.65)^2 &= (-0.05)^2 = 0.0025 \\ (11.5 - 10.65)^2 &= (0.85)^2 = 0.7225 \\ (8.3 - 10.65)^2 &= (-2.35)^2 = 5.5225 \\ (10.5 - 10.65)^2 &= (-0.15)^2 = 0.0225 \\ (10.9 - 10.65)^2 &= (0.25)^2 = 0.0625 \\ (10.7 - 10.65)^2 &= (0.05)^2 = 0.0025 \\ (11.2 - 10.65)^2 &= (0.55)^2 = 0.3025 \\ \end{align*}$$ ### 3. Sum the squared differences: $$1.3225 + 0.1225 + 0.0025 + 0.0025 + 0.7225 + 5.5225 + 0.0225 + 0.0625 + 0.0025 + 0.3025 = 8.086$$ ### 4. Calculate the variance (divide by the number of data points, since we assume this is a population standard deviation): $$\text{Variance} = \frac{8.086}{10} = 0.8086$$ ### 5. Find the standard deviation (square root of the variance): $$\text{Standard Deviation} = \sqrt{0.8086} \approx 0.8992$$ Thus, the standard deviation is approximately **0.8992**. Would you like more details or have any questions? Here are 5 related questions you might find interesting: 1. How do you calculate the variance from a sample instead of a population? 2. What is the difference between variance and standard deviation? 3. How does removing an outlier affect the standard deviation? 4. Can you use standard deviation to compare two different datasets? 5. How would standard deviation change if all values in the dataset were increased by the same amount? **Tip:** For small datasets, it's important to distinguish whether you're calculating the standard deviation for a population or a sample, as this affects the formula slightly.