To find the standard deviation (SD) of the given data set, follow these steps:

Given values:

$18, 19, 15, 20, 25, 31, 17, 35, 27, 22, 34, 29, 40, 33, 21$

Step 1: Find the Mean ($\mu$)

The mean is calculated as:

$\mu = \frac{\sum X}{N}$

where $X$ represents the values and $N$ is the number of values.

$\mu = \frac{18 + 19 + 15 + 20 + 25 + 31 + 17 + 35 + 27 + 22 + 34 + 29 + 40 + 33 + 21}{15}$

$\mu = \frac{385}{15} = 25.67$

Step 2: Find Each Value’s Deviation from the Mean and Square It

$(X - \mu)^2$

We'll compute these in the next step.

Step 3: Compute the Variance ($\sigma^2$)

The variance formula is:

$\sigma^2 = \frac{\sum (X - \mu)^2}{N}$

Let's calculate this.

Step 4: Compute the Standard Deviation ($\sigma$)

The standard deviation is:

$\sigma = \sqrt{\sigma^2}$

Now, let’s calculate the exact standard deviation. numpy as np

Given data set

data = np.array([18, 19, 15, 20, 25, 31, 17, 35, 27, 22, 34, 29, 40, 33, 21])

Compute standard deviation (population standard deviation)

std_dev = np.std(data, ddof=0) # Using population formula

std_devThe standard deviation of the given data set is approximately 7.38.

Would you like a detailed breakdown of the calculations?

Related Questions:

1. What is the difference between population and sample standard deviation?
2. How does standard deviation help in data analysis?
3. Can standard deviation be negative? Why or why not?
4. How does standard deviation relate to variance?
5. What are real-life examples where standard deviation is useful?

Tip: A lower standard deviation means the data points are closer to the mean, while a higher standard deviation indicates more spread.