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Let's solve the problems from the image step by step.

### Problem 1: Finding Mean, Median, and Mode

Data Set: 5, 3, 6, 4, 7, 3, 5, 8, 4, 5

1. **Mean** (average) number of books read:
   - Formula: 
   \[
   \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
   \]
   Sum of values: \( 5 + 3 + 6 + 4 + 7 + 3 + 5 + 8 + 4 + 5 = 50 \)
   Number of values: \( 10 \)

   \[
   \text{Mean} = \frac{50}{10} = 5
   \]

2. **Median** (middle value) of the dataLet's solve the problems from the image step by step.

### Problem 1: Finding Mean, Median, and Mode

Data Set: 5, 3, 6, 4, 7, 3, 5, 8, 4, 5

1. **Mean** (average) number of books read:
   - Formula: 
   \[
   \text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}
   \]
   Sum of values: \( 5 + 3 + 6 + 4 + 7 + 3 + 5 + 8 + 4 + 5 = 50 \)
   Number of values: \( 10 \)

   \[
   \text{Mean} = \frac{50}{10} = 5
   \]

2. **Median** (middle value) of the 
   - First, arrange the data in ascending order: \( 3, 3, 4, 4, 5, 5, 5, 6, 7, 8 \)
   - Since there are 10 values (even number), the median is the average of the 5th and 6th values:
   \[
   \text{Median} = \frac{5 + 5}{2} = 5
   \]

3. **Mode** (most frequent value):
   - The mode is the value that appears most frequently.
   - In this data set, the number \( 5 \) appears 3 times.
   
   Thus, the mode is \( 5 \).

### Problem 2: Variance and Standard Deviation

Data Set: 160, 165, 170, 175, 180, 185

1. **Variance**:
   - Formula for variance:
   \[
   \text{Variance} = \frac{\sum{(x_i - \mu)^2}}{N}
   \]
   where \( x_i \) is each value, \( \mu \) is the mean, and \( N \) is the number of values.
   
   First, calculate the mean:
   \[
   \mu = \frac{160 + 165 + 170 + 175 + 180 + 185}{6} = \frac{1035}{6} = 172.5
   \]

   Now, calculate \( (x_i - \mu)^2 \) for each value:
   \[
   (160 - 172.5)^2 = (-12.5)^2 = 156.25
   \]
   \[
   (165 - 172.5)^2 = (-7.5)^2 = 56.25
   \]
   \[
   (170 - 172.5)^2 = (-2.5)^2 = 6.25
   \]
   \[
   (175 - 172.5)^2 = (2.5)^2 = 6.25
   \]
   \[
   (180 - 172.5)^2 = (7.5)^2 = 56.25
   \]
   \[
   (185 - 172.5)^2 = (12.5)^2 = 156.25
   \]

   Sum of squared differences:
   \[
   156.25 + 56.25 + 6.25 + 6.25 + 56.25 + 156.25 = 437.5
   \]

   Variance:
   \[
   \text{Variance} = \frac{437.5}{6} = 72.92 \, (\text{rounded})
   \]

2. **Standard Deviation**:
   - Formula for standard deviation is the square root of the variance:
   \[
   \text{Standard Deviation} = \sqrt{72.92} \approx 8.54
   \]

Let me know if you need any further details!

---

Here are 5 related questions for further exploration:
1. How do you calculate the range of a data set?
2. What is the difference between population and sample variance?
3. How does the shape of a data set affect the mean, median, and mode?
4. Can a data set have more than one mode?
5. How would you interpret a high standard deviation in a data set?

**Tip:** Always arrange the data in ascending order when calculating the median to avoid mistakes.

Solve problems involving mean, median, mode, variance, and standard deviation based on provided data sets.

Solve problems involving finding the mean, median, mode, variance, and standard deviation using real-world data on books read and heights. Learn detailed step-by-step calculations to understand basic statistics. Suitable for Grades 7-10.

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