To calculate the standard deviation, we need to follow these steps:

Steps:

1. Find the mean (average): The mean is the sum of all the values divided by the number of values.

   Mean $\mu$ = $\frac{\sum x_i}{n}$, where $x_i$ are the data points and $n$ is the total number of data points.

2. Subtract the mean from each data point and square the result.

3. Find the average of the squared differences.

4. Take the square root of that average to get the standard deviation.

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Step 1: Calculate the mean

The data points are:
155, 40, 260, 345, 345, 40, 99, 291, 260, 104, 165, 465.

$\text{Sum} = 155 + 40 + 260 + 345 + 345 + 40 + 99 + 291 + 260 + 104 + 165 + 465 = 2579$

There are 12 values, so the mean is:

$\mu = \frac{2579}{12} = 214.92$

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Step 2: Calculate the squared differences from the mean

Now subtract the mean (214.92) from each data point, square the result, and sum them up:

$(155 - 214.92)^2 = (-59.92)^2 = 3589.79$ $(40 - 214.92)^2 = (-174.92)^2 = 30585.47$ $(260 - 214.92)^2 = (45.08)^2 = 2032.76$ $(345 - 214.92)^2 = (130.08)^2 = 16920.57$ $(345 - 214.92)^2 = (130.08)^2 = 16920.57$ $(40 - 214.92)^2 = (-174.92)^2 = 30585.47$ $(99 - 214.92)^2 = (-115.92)^2 = 13447.26$ $(291 - 214.92)^2 = (76.08)^2 = 5786.69$ $(260 - 214.92)^2 = (45.08)^2 = 2032.76$ $(104 - 214.92)^2 = (-110.92)^2 = 12303.94$ $(165 - 214.92)^2 = (-49.92)^2 = 2492.18$ $(465 - 214.92)^2 = (250.08)^2 = 62540.73$

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Step 3: Find the average of the squared differences

Now, add up all the squared differences:

$\text{Sum of squared differences} = 3589.79 + 30585.47 + 2032.76 + 16920.57 + 16920.57 + 30585.47 + 13447.26 + 5786.69 + 2032.76 + 12303.94 + 2492.18 + 62540.73 = 186626.23$

Now, divide by the number of values (12) to get the variance:

$\text{Variance} = \frac{186626.23}{12} = 15552.19$

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Step 4: Take the square root of the variance to find the standard deviation

Finally, take the square root of the variance:

$\text{Standard Deviation} = \sqrt{15552.19} = 124.89$

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Final Answer:

The standard deviation of the data set is approximately 124.89.

Would you like more details on any of the steps, or do you have any further questions?

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Related Questions:

1. How does variance differ from standard deviation?
2. What is the difference between population and sample standard deviation?
3. How can I calculate the standard deviation of a sample?
4. Why is standard deviation a good measure of spread in data?
5. Can you explain the concept of outliers and how they affect standard deviation?

Tip: The standard deviation can give you a good sense of how spread out the values are in a data set. If the standard deviation is large, the values are more spread out from the mean. If it's small, the values are closer to the mean.