Calculate the Standard Deviation (to 4 decimal places) from the following list of values:

12.15, 17.34, 18.48, 12.15, 24.61, 24.51, 18.11, 21.05, 29.81, 27.39, 28.98, 11.69, 17.38, 7.2, 18.38, 12.62, 26.18, 17.27

To calculate the standard deviation, follow these steps:

1. **Find the mean** of the data set.
2. **Subtract the mean** from each data point and **square the result**.
3. **Find the average** of these squared differences.
4. **Take the square root** of this average.

Let's go through the steps:

### Step 1: Find the Mean
\[ \text{Mean} = \frac{\sum \text{values}}{n} \]

Where:
- \(\sum \text{values}\) is the sum of all the data points.
- \(n\) is the number of data points.

### Step 2: Subtract the Mean and Square the Results

### Step 3: Calculate the Variance
\[ \text{Variance} = \frac{\sum (\text{data point} - \text{mean})^2}{n} \]

### Step 4: Take the Square Root of the Variance to get the Standard Deviation

Now let's compute these values.

First, sum up the values:

\[ \text{Sum of values} = 12.15 + 17.34 + 18.48 + 12.15 + 24.61 + 24.51 + 18.11 + 21.05 + 29.81 + 27.39 + 28.98 + 11.69 + 17.38 + 7.2 + 18.38 + 12.62 + 26.18 + 17.27 \]

\[ \text{Sum of values} = 347.90 \]

Next, calculate the mean:

\[ \text{Mean} = \frac{347.90}{18} = 19.3278 \]

Now, subtract the mean from each value, square the result, and sum these squared differences:

For example, for the first value:
\[ (12.15 - 19.3278)^2 = (-7.1778)^2 = 51.5173 \]

Repeat this for all values and sum them up.

Finally, find the variance:

\[ \text{Variance} = \frac{\sum (\text{data point} - \text{mean})^2}{18} \]

Take the square root of the variance to find the standard deviation.

I'll perform these calculations for you.The standard deviation of the given list of values is **6.4156** (rounded to four decimal places).

Would you like more details on the calculation process or have any other questions?

Here are 5 related questions to expand your understanding:
1. How does standard deviation relate to variance?
2. What is the difference between population and sample standard deviation?
3. How can standard deviation be used to understand the spread of data?
4. What are the implications of a high versus low standard deviation?
5. How do you calculate standard deviation by hand for small data sets?

**Tip:** Standard deviation is a key concept in statistics, useful for understanding the spread and variability of a data set relative to its mean.

Learn how to calculate the standard deviation from a list of values with a detailed step-by-step guide. This problem covers key statistical concepts, including mean, variance, and standard deviation. Suitable for advanced high school students and above.

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