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To solve this problem, we need to calculate the **mean**, **median**, **variance**, and **standard deviation** for the provided data set. I'll extract the data and work through these statistics step by step.

### Extracted Data (Monthly Entertainment Spending):
\[
\begin{align*}
50, 120, 10, \\
75, 130, 180, \\
100, 65, 140, \\
60, 55, 35, \\
90, 85, 175, \\
80, 160, 15, \\
45, 125, 25, \\
150, 20, 5, \\
200, 30, 70, \\
110, 40, 95
\end{align*}
\]

#### Step 1: **Calculate the Mean**
The mean is the average of all the data points. The formula for the mean ($\mu$) is:
\[
\mu = \frac{\sum{x}}{N}
\]
where $x$ represents each data point, and $N$ is the total number of data points.

#### Step 2: **Calculate the Median**
The median is the middle value when all data points are arranged in ascending order. If there is an even number of data points, the median is the average of the two middle values.

#### Step 3: **Calculate the Variance**
Variance measures the spread of the data points. The formula for variance ($\sigma^2$) is:
\[
\sigma^2 = \frac{\sum{(x - \mu)^2}}{N}
\]
where $x$ is each data point, and $\mu$ is the mean.

#### Step 4: **Calculate the Standard Deviation**
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{\sigma^2}
\]

Let me calculate these values for you.Here are the calculations for the provided data set:

1. **Mean**: $ 84.67 $ (rounded to two decimal places)
2. **Median**: $ 77.5 $
3. **Variance**: $ 2881.56 $ (rounded to two decimal places)
4. **Standard Deviation**: $ 53.68 $ (rounded to two decimal places)

Would you like more details on how these were calculated, or do you have any further questions?

### Related questions:
1. How does the variance give us insight into data spread compared to standard deviation?
2. What impact do extreme values (like 200 and 5) have on the mean?
3. Why is the median often used alongside the mean?
4. How would you calculate the range of this data set?
5. What is the interquartile range (IQR), and how is it calculated for this data?

#### Tip:
The **standard deviation** is useful for understanding how spread out the values are around the mean. A higher standard deviation indicates more variability in the data set.

Calculate the mean, median, variance, and standard deviation of a given data set of monthly entertainment spending.

Learn how to calculate the mean, median, variance, and standard deviation for a data set of monthly entertainment spending. The data points range from $5 to $200, making this a real-world application of descriptive statistics, suitable for students in Grades 10-12.

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