Let's break this down step-by-step for the given data set:

Data Set: 7, 8, 9, 10, 11, 12, 13, 14, 15, 16

1. Range

The range is the difference between the largest and smallest numbers in the data set.

$\text{Range} = \text{Largest Value} - \text{Smallest Value}$

$\text{Range} = 16 - 7 = 9$

2. Population Variance

Population variance is calculated using the formula:

$\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$

Where:

• $x_i$ is each data point
• $\mu$ is the population mean
• $N$ is the total number of data points

First, we calculate the mean ($\mu$):

$\mu = \frac{\sum x_i}{N}$

$\mu = \frac{7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16}{10} = \frac{105}{10} = 10.5$

Now, we calculate the squared differences from the mean for each data point:

$(7 - 10.5)^2 = (-3.5)^2 = 12.25$ $(8 - 10.5)^2 = (-2.5)^2 = 6.25$ $(9 - 10.5)^2 = (-1.5)^2 = 2.25$ $(10 - 10.5)^2 = (-0.5)^2 = 0.25$ $(11 - 10.5)^2 = (0.5)^2 = 0.25$ $(12 - 10.5)^2 = (1.5)^2 = 2.25$ $(13 - 10.5)^2 = (2.5)^2 = 6.25$ $(14 - 10.5)^2 = (3.5)^2 = 12.25$ $(15 - 10.5)^2 = (4.5)^2 = 20.25$ $(16 - 10.5)^2 = (5.5)^2 = 30.25$

Sum of squared differences:

$\sum (x_i - \mu)^2 = 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 + 30.25 = 92.5$

Now, divide by the total number of data points $N = 10$ to get the population variance:

$\sigma^2 = \frac{92.5}{10} = 9.25$

3. Population Standard Deviation

The population standard deviation is simply the square root of the variance:

$\sigma = \sqrt{\sigma^2} = \sqrt{9.25} \approx 3.04$

Final Answers:

• Range = 9
• Population Variance = 9.25
• Population Standard Deviation ≈ 3.0 (rounded to one decimal place)

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Would you like further details or have any questions on this? Here are some additional questions you might find helpful:

1. How do you calculate sample variance and standard deviation?
2. What is the difference between population and sample variance?
3. Why do we square the differences from the mean when calculating variance?
4. How does the range help in understanding the spread of data?
5. How would you interpret the standard deviation value in this context?

Tip: The standard deviation provides a way to measure the "average" distance each data point is from the mean, helping you understand the spread of the data more intuitively.