The **standard deviation** is a measure of the amount of variation or dispersion in a set of data values. In simple terms, it quantifies how much the values in a dataset differ from the mean (average) of the dataset. The larger the standard deviation, the more spread out the values are. A smaller standard deviation indicates that the data points tend to be closer to the mean.

### Formula:
For a dataset with \( n \) values \( x_1, x_2, x_3, \dots, x_n \), the steps to calculate the standard deviation are:

1. **Calculate the mean** \( \mu \):
   \[
   \mu = \frac{x_1 + x_2 + \dots + x_n}{n}
   \]
   
2. **Subtract the mean** from each value and square the result to find the squared deviations.

3. **Find the average of these squared deviations**. This is called the **variance**:
   \[
   \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \dots + (x_n - \mu)^2}{n}
   \]
   
4. **Take the square root** of the variance to get the standard deviation \( \sigma \):
   \[
   \sigma = \sqrt{\sigma^2}
   \]

For **sample data**, we divide by \( n - 1 \) instead of \( n \) in the variance formula to get an **unbiased estimate** of the population standard deviation.

### Example:
If the data points are: \( 4, 8, 6, 5, 3 \),
1. Mean \( \mu = \frac{4 + 8 + 6 + 5 + 3}{5} = 5.2 \)
2. Squared deviations: \( (4-5.2)^2, (8-5.2)^2, (6-5.2)^2, (5-5.2)^2, (3-5.2)^2 \)
3. Variance \( \sigma^2 = \frac{1.44 + 7.84 + 0.64 + 0.04 + 4.84}{5} = 2.96 \)
4. Standard deviation \( \sigma = \sqrt{2.96} \approx 1.72 \)

Would you like further clarification or more detailed steps?

### Relative questions:
1. How does standard deviation differ from variance?
2. How do you interpret a high vs. low standard deviation in real-life data?
3. What are the uses of standard deviation in finance?
4. How does the standard deviation change if you add or subtract a constant to each value in the data?
5. Can you explain the significance of using \( n-1 \) for sample standard deviation?

**Tip**: Standard deviation is useful when comparing the variability of two different datasets, even if their means differ.

Explore the concept of standard deviation with a step-by-step explanation. Learn how to calculate the standard deviation, variance, and mean using real data points like 4, 8, 6, 5, and 3. Suitable for students in Grades 9-12.

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