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.