To find the standard deviation of a data set, follow these steps:

Step-by-Step Process:

1. Find the mean (average):

   • Add up all the numbers in your data set.
   • Divide the sum by the number of data points (n). $\text{Mean} (\mu) = \frac{\sum x_i}{n}$ where $x_i$ represents each value in the data set.

2. Calculate the squared differences from the mean:

   • For each data point $x_i$, subtract the mean $\mu$ and then square the result. $(x_i - \mu)^2$

3. Find the average of the squared differences (this is the variance):

   • Add up all the squared differences and divide by the number of data points.
   • If you’re calculating for a sample, divide by $n - 1$ instead of $n$. $\text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{n} \quad \text{for population}$ or $\text{Variance} (s^2) = \frac{\sum (x_i - \mu)^2}{n-1} \quad \text{for sample}$

4. Take the square root of the variance:

   • This gives you the standard deviation. $\text{Standard Deviation} (\sigma) = \sqrt{\sigma^2}$ or $\text{Standard Deviation} (s) = \sqrt{s^2}$

Example:

Given the data set: 2, 4, 4, 4, 5, 5, 7, 9

1. Mean: $\mu = \frac{2 + 4 + 4 + 4 + 5 + 5 + 7 + 9}{8} = 5$

2. Squared differences: $(2 - 5)^2 = 9, \quad (4 - 5)^2 = 1, \quad (4 - 5)^2 = 1, \quad (4 - 5)^2 = 1$ $(5 - 5)^2 = 0, \quad (5 - 5)^2 = 0, \quad (7 - 5)^2 = 4, \quad (9 - 5)^2 = 16$

3. Variance: $\sigma^2 = \frac{9 + 1 + 1 + 1 + 0 + 0 + 4 + 16}{8} = \frac{32}{8} = 4$

4. Standard Deviation: $\sigma = \sqrt{4} = 2$

Would you like more details on any part of this process?

Related Questions:

1. What is the difference between population and sample standard deviation?
2. How do you interpret standard deviation in a data set?
3. Can standard deviation ever be negative?
4. How does standard deviation relate to variance?
5. What is the formula for standard deviation in grouped data?

Tip:

For large data sets, consider using a calculator or software (Excel, Python, etc.) to quickly compute the standard deviation.