A sample was done, collecting the data below. Calculate the standard deviation, to one decimal place.

x
13
8
3
21
19


To calculate the standard deviation of a set of numbers, follow these steps:

1. **Find the mean (average)** 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 each step with your data.

### Step 1: Calculate the mean
\[
\text{Mean} = \frac{\sum x}{n}
\]
Where \( \sum x \) is the sum of all data points, and \( n \) is the number of data points.

Given the data points:
\[
x = \{13, 8, 3, 21, 19\}
\]

\[
\sum x = 13 + 8 + 3 + 21 + 19 = 64
\]

\[
n = 5
\]

\[
\text{Mean} = \frac{64}{5} = 12.8
\]

### Step 2: Calculate the squared differences from the mean
\[
(x_i - \text{mean})^2
\]

\[
\begin{align*}
(13 - 12.8)^2 &= 0.04 \\
(8 - 12.8)^2 &= 23.04 \\
(3 - 12.8)^2 &= 96.04 \\
(21 - 12.8)^2 &= 67.24 \\
(19 - 12.8)^2 &= 37.44 \\
\end{align*}
\]

### Step 3: Calculate the mean of these squared differences (variance)
\[
\text{Variance} = \frac{\sum (x_i - \text{mean})^2}{n}
\]

\[
\sum (x_i - \text{mean})^2 = 0.04 + 23.04 + 96.04 + 67.24 + 37.44 = 223.8
\]

\[
\text{Variance} = \frac{223.8}{5} = 44.76
\]

### Step 4: Calculate the square root of the variance (standard deviation)
\[
\text{Standard Deviation} = \sqrt{44.76} \approx 6.7
\]

So, the standard deviation of the data set is approximately **6.7** (to one decimal place).

Would you like more details or have any questions?

Here are 8 related questions:
1. What is the significance of standard deviation in statistics?
2. How does standard deviation differ from variance?
3. Can standard deviation be negative?
4. What are the properties of a data set with a high standard deviation?
5. How would you calculate the standard deviation for a larger data set?
6. How is standard deviation used in real-world applications?
7. What are the limitations of using standard deviation?
8. How does the standard deviation change if a data point is significantly altered?

**Tip:** Always check for outliers in your data set, as they can significantly affect the standard deviation.

Learn how to calculate the standard deviation of a data set with a step-by-step guide. This example includes data points 13, 8, 3, 21, and 19, and covers key statistical concepts including standard deviation, mean, and variance calculation. Suitable for advanced high school students and learners interested in statistics.

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