Let's go through each calculation step-by-step:

a. Calculate the Mean

The mean (average) is calculated by summing all the values and dividing by the number of values.

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

$\text{Mean} = \frac{4 + 5 + 7 + 3 + 2 + 0 + 2 + 3 + 2 + 1 + 2 + 4}{12}$

$\text{Mean} = \frac{35}{12} = 2.92 \ (\text{rounded to 2 decimal places})$

b. Find the Range

The range is the difference between the highest and lowest values.

$\text{Range} = \text{Max value} - \text{Min value}$

$\text{Range} = 7 - 0 = 7$

c. Calculate the Variance

Variance is the average of the squared differences from the mean.

1. Subtract the mean from each value, and square the result.
2. Find the average of those squared differences.

Steps:

1. Subtract the mean from each value: $4 - 2.92 = 1.08, \quad 5 - 2.92 = 2.08, \quad 7 - 2.92 = 4.08, \quad 3 - 2.92 = 0.08, \quad 2 - 2.92 = -0.92, \quad 0 - 2.92 = -2.92$ Continue for all values.

2. Square the results: $1.08^2, \quad 2.08^2, \quad 4.08^2, \quad (-0.92)^2, \quad \dots$

3. Find the average of the squared differences. We will compute this in the next step.

d. Calculate the Standard Deviation

Standard deviation is the square root of the variance.

Let me calculate the variance and standard deviation for you.Here are the results:

a. Mean

$\text{Mean} = 2.92 \ (\text{rounded to 2 decimal places})$

b. Range

$\text{Range} = 7$

c. Variance

$\text{Variance} = 3.24 \ (\text{rounded to 2 decimal places})$

d. Standard Deviation

$\text{Standard Deviation} = 1.80 \ (\text{rounded to 2 decimal places})$

Would you like further details or have any questions?

Here are five related questions:

1. How does variance change if the dataset had fewer extreme values?
2. Can you calculate the median for this dataset?
3. What is the interpretation of standard deviation in this context?
4. How does the variance formula differ for samples vs. populations?
5. What would be the effect of adding a new data point to the dataset?

Tip: Always check whether your dataset is a sample or population before calculating variance or standard deviation, as the formulas differ slightly!