156	41	261	346	346	41	100	292	261	105	166	468 find the standard deviation

To find the standard deviation of the given data set:

**Data set:**
156, 41, 261, 346, 346, 41, 100, 292, 261, 105, 166, 468

### Steps to calculate standard deviation:

1. **Find the mean (average):**
   \[
   \text{Mean} = \frac{\sum X}{n}
   \]
   Where \( \sum X \) is the sum of all data points, and \( n \) is the number of data points.

   First, add all the numbers:

   \[
   156 + 41 + 261 + 346 + 346 + 41 + 100 + 292 + 261 + 105 + 166 + 468 = 2582
   \]

   There are 12 numbers in the data set, so:

   \[
   \text{Mean} = \frac{2582}{12} = 215.17
   \]

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

   \[
   (156 - 215.17)^2 = (-59.17)^2 = 3504.73
   \]
   \[
   (41 - 215.17)^2 = (-174.17)^2 = 30323.17
   \]
   \[
   (261 - 215.17)^2 = (45.83)^2 = 2109.47
   \]
   \[
   (346 - 215.17)^2 = (130.83)^2 = 17117.51
   \]
   \[
   (346 - 215.17)^2 = (130.83)^2 = 17117.51
   \]
   \[
   (41 - 215.17)^2 = (-174.17)^2 = 30323.17
   \]
   \[
   (100 - 215.17)^2 = (-115.17)^2 = 13250.33
   \]
   \[
   (292 - 215.17)^2 = (76.83)^2 = 5908.57
   \]
   \[
   (261 - 215.17)^2 = (45.83)^2 = 2109.47
   \]
   \[
   (105 - 215.17)^2 = (-110.17)^2 = 12137.89
   \]
   \[
   (166 - 215.17)^2 = (-49.17)^2 = 2417.87
   \]
   \[
   (468 - 215.17)^2 = (252.83)^2 = 63929.86
   \]

3. **Find the average of these squared differences:**
   Add up all the squared differences:

   \[
   3504.73 + 30323.17 + 2109.47 + 17117.51 + 17117.51 + 30323.17 + 13250.33 + 5908.57 + 2109.47 + 12137.89 + 2417.87 + 63929.86 = 213749.78
   \]

   Divide by the number of data points (12) to get the variance:

   \[
   \text{Variance} = \frac{213749.78}{12} = 17812.48
   \]

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

   \[
   \text{Standard deviation} = \sqrt{17812.48} = 133.43
   \]

### Final Answer:
The standard deviation of the data set is approximately **133.43**.

Would you like more details on the process or have any questions about the calculation? Here are some related questions you could explore:

1. How does the variance relate to the standard deviation?
2. What is the difference between population and sample standard deviation?
3. How do outliers affect the standard deviation?
4. How would you calculate the standard deviation if some data were missing?
5. What does a high standard deviation indicate about a data set?

**Tip:** When calculating standard deviation, always double-check whether your data represents a sample or a full population, as the formulas for variance and standard deviation differ slightly depending on that.

156 41 261 346 346 41 100 292 261 105 166 468 find the standard deviation

Learn how to calculate the standard deviation for a given data set: 156, 41, 261, 346, 346, 41, 100, 292, 261, 105, 166, 468. This problem demonstrates how to calculate the mean, variance, and finally the standard deviation, with a result of 133.43. Ideal for students in Grades 9-12.

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