To find the standard deviation of the given numbers $4.7, 6.7, 4.9, 6.3, 5.5$, we follow these steps:

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

The mean is given by:

$\mu = \frac{\sum x_i}{n}$

where $x_i$ are the data points and $n$ is the number of data points.

$\mu = \frac{4.7 + 6.7 + 4.9 + 6.3 + 5.5}{5}$

Step 2: Compute Each Value’s Squared Difference from the Mean

For each $x_i$, compute $(x_i - \mu)^2$.

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

The variance for a population is:

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{n}$

For a sample, use $n-1$ instead of $n$:

$s^2 = \frac{\sum (x_i - \mu)^2}{n-1}$

Step 4: Take the Square Root to Get the Standard Deviation

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

Let's calculate it now. Given data points data = [4.7, 6.7, 4.9, 6.3, 5.5]

Number of data points

n = len(data)

Step 1: Calculate the mean

mean = sum(data) / n

Step 2: Compute squared differences from the mean

squared_diffs = [(x - mean) ** 2 for x in data]

Step 3: Compute the sample variance (since it's a small dataset)

variance = sum(squared_diffs) / (n - 1)

Step 4: Compute the standard deviation

std_dev = variance ** 0.5

std_devThe sample standard deviation of the given numbers is approximately 0.87.

Would you like a detailed breakdown of the calculations?

Here are some related questions for deeper understanding:

1. How does standard deviation differ from variance?
2. When should we use population vs. sample standard deviation?
3. How does standard deviation help in data analysis?
4. Can standard deviation ever be negative?
5. What is the effect of adding an outlier to this dataset?

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