To find the standard deviation for the given data set:

Step 1: Find the mean (average)

The mean is calculated by summing all the numbers and dividing by the total count of numbers.

$\text{Mean} = \frac{\sum x_i}{N}$

Where $\sum x_i$ is the sum of all data points, and $N$ is the number of data points.

The sum of the data points: $38 + 50 + 50 + 65 + 65 + 75 + 90 + 160 + 185 + 207 + 270 + 300 + 500 + 2000 + 3000 = 5605$

Number of data points, $N = 15$.

$\text{Mean} = \frac{5605}{15} = 373.67$

Step 2: Find the squared differences from the mean

For each data point, subtract the mean and square the result:

$(x_i - \mu)^2$

$(38 - 373.67)^2 = (-335.67)^2 = 112417.68$ $(50 - 373.67)^2 = (-323.67)^2 = 104847.02$ $(50 - 373.67)^2 = (-323.67)^2 = 104847.02$ $(65 - 373.67)^2 = (-308.67)^2 = 95122.16$ $(65 - 373.67)^2 = (-308.67)^2 = 95122.16$ $(75 - 373.67)^2 = (-298.67)^2 = 89219.37$ $(90 - 373.67)^2 = (-283.67)^2 = 80466.52$ $(160 - 373.67)^2 = (-213.67)^2 = 45650.99$ $(185 - 373.67)^2 = (-188.67)^2 = 35682.33$ $(207 - 373.67)^2 = (-166.67)^2 = 27708.89$ $(270 - 373.67)^2 = (-103.67)^2 = 10748.81$ $(300 - 373.67)^2 = (-73.67)^2 = 5427.35$ $(500 - 373.67)^2 = (126.33)^2 = 15947.15$ $(2000 - 373.67)^2 = (1626.33)^2 = 2641569.91$ $(3000 - 373.67)^2 = (2626.33)^2 = 6894769.07$

Step 3: Calculate the variance

The variance is the average of these squared differences:

$\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}$

Summing up the squared differences:

$\sum (x_i - \mu)^2 = 112417.68 + 104847.02 + 104847.02 + 95122.16 + 95122.16 + 89219.37 + 80466.52 + 45650.99 + 35682.33 + 27708.89 + 10748.81 + 5427.35 + 15947.15 + 2641569.91 + 6894769.07 = 11054728.85$

Now, divide by $N$:

$\text{Variance} = \frac{11054728.85}{15} = 738315.25$

Step 4: Find the standard deviation

The standard deviation is the square root of the variance:

$\text{Standard Deviation} = \sqrt{738315.25} \approx 858.71$

Final Answer:

The standard deviation of the given data set is approximately 858.71.

Would you like further details on how this is calculated, or do you have any questions?

Here are some related questions for you:

1. How do you calculate the sample standard deviation instead of population standard deviation?
2. What does the standard deviation tell us about a data set?
3. How can the presence of outliers affect the standard deviation?
4. What is the difference between variance and standard deviation?
5. How does the range of data points influence the standard deviation?

Tip: When working with data sets that include outliers, the standard deviation might be heavily influenced, so always check for extreme values before concluding.