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From the images, it appears that you are asked to calculate the **variance** and **standard deviation** for a dataset representing the number of faculty in a college's departments.

### The dataset provided:
76, 42, 20, 35, 5, 15, 20, 2

Let's calculate both variance and standard deviation:

1. **Variance** is a measure of how much the values in the dataset vary from the mean.
   \[
   \text{Variance} = \frac{\sum (x_i - \bar{x})^2}{N}
   \]
   Where \( x_i \) represents each value, \( \bar{x} \) is the mean, and \( N \) is the number of values.

2. **Standard Deviation** is the square root of variance:
   \[
   \text{Standard Deviation} = \sqrt{\text{Variance}}
   \]

Let me calculate both.The calculations for the given dataset (76, 42, 20, 35, 5, 15, 20, 2) are as follows:

1. **Variance**: 505.109375 (no rounding).
2. **Standard Deviation**: 22.474638 (rounded to six decimal places).

Let me know if you need further clarification or additional details!

Here are five related questions:
1. What is the formula for population variance vs. sample variance?
2. How is variance affected by adding extreme values (outliers)?
3. How does standard deviation interpret the spread of data?
4. How would variance change if all data points increased by the same constant?
5. Why is standard deviation always non-negative?

**Tip:** Variance is useful when comparing datasets with similar means but different dispersions, while standard deviation offers an intuitive measure of spread in the original units.

Calculate the variance and standard deviation for the dataset representing the number of faculty in a private college's departments: 76, 42, 20, 35, 5, 15, 20, 2.

Learn how to calculate the variance and standard deviation for a dataset representing the number of faculty members in different departments of a private college. This problem involves using key statistical concepts and is suitable for college-level students.

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