To calculate the variance and standard deviation of the sample data, we follow these steps:

1. Mean of the sample:

The formula for the sample mean is:

$\bar{x} = \frac{\sum{x_i}}{n}$

where $x_i$ represents each data point and $n$ is the number of data points.

2. Variance of the sample:

The formula for the sample variance is:

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

where $\bar{x}$ is the sample mean.

3. Standard deviation of the sample:

The standard deviation is simply the square root of the variance:

$s = \sqrt{s^2}$

I'll now calculate both the variance and standard deviation for the given sample:

$x = \{ 29.6, 22.7, 3.7, 22.8, 7.3, 20, 9 \}$

Let me calculate these values.The calculated values for the sample data are:

• Variance: 94.45
• Standard deviation: 9.72

Would you like more details on the calculations, or do you have any questions?

Here are five related questions you might find useful:

1. How does sample variance differ from population variance?
2. How is standard deviation used to measure data spread?
3. Can you explain the significance of degrees of freedom in variance calculation?
4. How would the variance change if all data points were doubled?
5. What is the relationship between variance and standard deviation?

Tip: Standard deviation is a key tool in identifying how far individual data points are from the mean, helping in understanding data variability.