The daily number of hours worked by a student in recent few days are displayed below:

4, 4-2, 4+3, 4 + 1, 4 + 2,  4 -3,  4 -1, 4 + 4.

Simplify these eight numbers and then find the variance of these numbers.

First, let's simplify the given numbers:

1. \( 4 \)
2. \( 4 - 2 = 2 \)
3. \( 4 + 3 = 7 \)
4. \( 4 + 1 = 5 \)
5. \( 4 + 2 = 6 \)
6. \( 4 - 3 = 1 \)
7. \( 4 - 1 = 3 \)
8. \( 4 + 4 = 8 \)

So, the simplified numbers are:

\[ 4, 2, 7, 5, 6, 1, 3, 8 \]

Next, we calculate the variance of these numbers. 

1. **Calculate the mean** (\(\mu\)):

\[
\mu = \frac{4 + 2 + 7 + 5 + 6 + 1 + 3 + 8}{8} = \frac{36}{8} = 4.5
\]

2. **Calculate each squared deviation from the mean**:

\[
(4 - 4.5)^2 = (-0.5)^2 = 0.25
\]
\[
(2 - 4.5)^2 = (-2.5)^2 = 6.25
\]
\[
(7 - 4.5)^2 = (2.5)^2 = 6.25
\]
\[
(5 - 4.5)^2 = (0.5)^2 = 0.25
\]
\[
(6 - 4.5)^2 = (1.5)^2 = 2.25
\]
\[
(1 - 4.5)^2 = (-3.5)^2 = 12.25
\]
\[
(3 - 4.5)^2 = (-1.5)^2 = 2.25
\]
\[
(8 - 4.5)^2 = (3.5)^2 = 12.25
\]

3. **Calculate the variance** (\(\sigma^2\)):

\[
\sigma^2 = \frac{0.25 + 6.25 + 6.25 + 0.25 + 2.25 + 12.25 + 2.25 + 12.25}{8} = \frac{42.0}{8} = 5.25
\]

So, the variance of these numbers is \(5.25\).

Would you like further details or have any questions?

Here are 8 relative questions that expand on this topic:
1. What is the significance of variance in a data set?
2. How do you calculate the standard deviation from the variance?
3. Can the variance ever be negative?
4. How would the variance change if all the numbers were doubled?
5. What are some real-life applications of variance?
6. How does variance relate to the concept of dispersion in statistics?
7. What is the difference between sample variance and population variance?
8. How can you interpret a high variance versus a low variance in data?

**Tip:** Variance is a measure of how spread out the values in a data set are. Understanding variance helps in analyzing the consistency and reliability of the data.

Learn how to simplify and calculate the variance of a set of numbers. Step-by-step explanation with examples covering statistical concepts and formulas. Suitable for advanced high school students and learners interested in statistics.

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