To calculate the variance for a sample, we need to follow these steps:

Step 1: Calculate the Mean (Average)

The formula for the mean is:

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

Where:

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

Let's compute the mean:

Data points: $2, 0, 2, 0, 1, 2, 4, 2, 5, 1, 3, 0, 0, 1, 2, 4, 3, 2, 1, 7, 5, 2, 4, 0$

The sum of the data points is:

$2 + 0 + 2 + 0 + 1 + 2 + 4 + 2 + 5 + 1 + 3 + 0 + 0 + 1 + 2 + 4 + 3 + 2 + 1 + 7 + 5 + 2 + 4 + 0 = 57$

The number of data points $n = 24$.

So the mean is:

$\bar{x} = \frac{57}{24} = 2.375$

Step 2: Calculate the Squared Differences

For each data point, subtract the mean and then square the result.

For example, for the first data point $x_1 = 2$:

$(x_1 - \bar{x})^2 = (2 - 2.375)^2 = (-0.375)^2 = 0.140625$

Do this for all data points:

$(2 - 2.375)^2, (0 - 2.375)^2, \ldots, (0 - 2.375)^2$

Step 3: Compute the Variance

The sample variance formula is:

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

Now, I’ll calculate the squared differences, sum them up, and divide by $n - 1 = 24 - 1 = 23$. Let me perform this calculation.

The sample variance of the given data is approximately $3.476$.

Would you like further details or have any other questions?

Here are 5 related questions to consider:

1. How does sample variance differ from population variance?
2. Can you explain the concept of standard deviation based on the variance?
3. How does outlier data impact variance?
4. How would the variance change if one data point were significantly larger?
5. How can you use variance to compare the spread of two different datasets?

Tip: Variance measures the spread of data points around the mean, and larger values indicate greater dispersion.