The question asks to calculate the standard deviation for a set of data points. Here’s a step-by-step solution:

Given Data

The data values are: $1.73, 2.03, 1.89, 2.79, 2.38, 1.08, 1.36, 0.85, 1.77, 2.13$

Step 1: Find the Mean (Average)

To calculate the mean $\bar{x}$: $\bar{x} = \frac{\sum x}{n}$ where $n = 10$ (since there are 10 data points).

1. Calculate the sum of all values: $1.73 + 2.03 + 1.89 + 2.79 + 2.38 + 1.08 + 1.36 + 0.85 + 1.77 + 2.13 = 17.01$

2. Now, calculate the mean: $\bar{x} = \frac{17.01}{10} = 1.701$

Step 2: Find the Variance

Variance $\sigma^2$ is the average of the squared differences from the mean.

1. Subtract the mean from each value, square the result, and sum all squared differences: $(1.73 - 1.701)^2 + (2.03 - 1.701)^2 + (1.89 - 1.701)^2 + (2.79 - 1.701)^2 + (2.38 - 1.701)^2 + (1.08 - 1.701)^2 + (1.36 - 1.701)^2 + (0.85 - 1.701)^2 + (1.77 - 1.701)^2 + (2.13 - 1.701)^2$

   Calculating each term:

   • $(1.73 - 1.701)^2 = 0.000841$
   • $(2.03 - 1.701)^2 = 0.107929$
   • $(1.89 - 1.701)^2 = 0.035721$
   • $(2.79 - 1.701)^2 = 1.185481$
   • $(2.38 - 1.701)^2 = 0.461041$
   • $(1.08 - 1.701)^2 = 0.384361$
   • $(1.36 - 1.701)^2 = 0.116641$
   • $(0.85 - 1.701)^2 = 0.722521$
   • $(1.77 - 1.701)^2 = 0.004761$
   • $(2.13 - 1.701)^2 = 0.184041$

2. Sum these squared differences: $0.000841 + 0.107929 + 0.035721 + 1.185481 + 0.461041 + 0.384361 + 0.116641 + 0.722521 + 0.004761 + 0.184041 = 3.203338$

3. Divide by $n$ (for population standard deviation) or $n-1$ (for sample standard deviation):

   • Since we have a sample, divide by $n - 1 = 9$: $\sigma^2 = \frac{3.203338}{9} = 0.355926$

Step 3: Find the Standard Deviation

The standard deviation $\sigma$ is the square root of the variance: $\sigma = \sqrt{0.355926} \approx 0.5966$

Final Answer

The standard deviation for the given data is approximately: $\sigma \approx 0.60 \text{ (rounded to two decimal places)}$

Would you like a further breakdown or any clarifications on specific steps?

Here are related questions you might find helpful:

1. How is the sample mean different from the population mean?
2. What are the differences between population and sample standard deviation?
3. How can standard deviation help in understanding data distribution?
4. What does a higher or lower standard deviation indicate?
5. How would the calculation change if this were population data?

Tip: When calculating variance and standard deviation, keep your calculations organized to avoid errors, especially in large datasets.