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.