To calculate the standard deviation of the given data set, follow these steps:

Data Set:

$8, 6, 3, 5, 3, 4, 2, 9$

Step 1: Calculate the Mean

The mean ($\mu$) is the sum of all the data points divided by the number of data points.

$\mu = \frac{8 + 6 + 3 + 5 + 3 + 4 + 2 + 9}{8} = \frac{40}{8} = 5$

Step 2: Find the Differences from the Mean

Subtract the mean from each data point to get the deviation of each point:

$\text{Deviations}: (8 - 5), (6 - 5), (3 - 5), (5 - 5), (3 - 5), (4 - 5), (2 - 5), (9 - 5)$ $= 3, 1, -2, 0, -2, -1, -3, 4$

Step 3: Square Each Deviation

Now, square each of these deviations:

$(3^2), (1^2), (-2^2), (0^2), (-2^2), (-1^2), (-3^2), (4^2)$ $= 9, 1, 4, 0, 4, 1, 9, 16$

Step 4: Calculate the Mean of the Squared Deviations (Variance)

Find the mean of these squared deviations:

$\text{Variance} = \frac{9 + 1 + 4 + 0 + 4 + 1 + 9 + 16}{8} = \frac{44}{8} = 5.5$

Step 5: Take the Square Root (Standard Deviation)

The standard deviation ($\sigma$) is the square root of the variance:

$\sigma = \sqrt{5.5} \approx 2.35$

Answer:

The standard deviation of the data set is approximately 2.35.

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Would you like more details on this process or have any questions?

Here are some additional questions related to this topic:

1. How does the standard deviation change if you add more data points?
2. What is the difference between standard deviation and variance?
3. How does the range of a data set relate to its standard deviation?
4. What does a low standard deviation indicate about a data set?
5. How would the standard deviation change if all the scores were increased by the same amount?

Tip: The standard deviation measures the spread of data around the mean. The larger the value, the more spread out the data points are from the mean.