To calculate the variance and standard deviation of the given data sample $-1, 7, 7, 5, 12$, we can follow these steps:

Step 1: Calculate the Mean

The mean ($\bar{x}$) is the average of all the numbers in the data set.

$\bar{x} = \frac{-1 + 7 + 7 + 5 + 12}{5}$

Step 2: Calculate Each Data Point’s Deviation from the Mean

Subtract the mean from each data point to find their deviations.

Step 3: Square Each Deviation

Square each of these deviations to eliminate negative values.

Step 4: Find the Sum of Squared Deviations

Add all the squared deviations together.

Step 5: Compute the Variance

Divide the sum of squared deviations by $n - 1$ (since this is a sample variance). Here, $n = 5$.

Step 6: Calculate the Standard Deviation

The standard deviation is the square root of the variance.

I'll calculate these values step by step.The variance of the data sample is $22.00$, and the standard deviation is $4.69$.

Do you want more details or have any questions?

Here are 5 related questions for further exploration:

1. How is the sample variance different from population variance?
2. What happens to the variance if every value in the dataset is increased by a constant?
3. Why do we use $n - 1$ instead of $n$ when calculating sample variance?
4. How does standard deviation help in understanding data spread?
5. Can variance ever be negative? Why or why not?

Tip: The standard deviation is a useful metric to assess the spread of your data; smaller values indicate that the data points are close to the mean.