Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data.

8,9,10,11,12,13,14,15,16,17


To calculate the range, population variance, and population standard deviation for the given data set, we will follow these steps:

### 1. Calculate the Range
The range is the difference between the maximum and minimum values in the data set.

\[
\text{Range} = \text{Max} - \text{Min} = 17 - 8 = 9
\]

### 2. Calculate the Population Variance

The formula for population variance (\(\sigma^2\)) is:

\[
\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
\]

Where:
- \(N\) is the number of data points
- \(x_i\) is each individual data point
- \(\mu\) is the population mean

**Step 2.1: Calculate the Population Mean (\(\mu\))**

\[
\mu = \frac{8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17}{10} = \frac{125}{10} = 12.5
\]

**Step 2.2: Calculate Each Squared Deviation from the Mean and Sum Them**

\[
\begin{aligned}
(8 - 12.5)^2 &= (-4.5)^2 = 20.25 \\
(9 - 12.5)^2 &= (-3.5)^2 = 12.25 \\
(10 - 12.5)^2 &= (-2.5)^2 = 6.25 \\
(11 - 12.5)^2 &= (-1.5)^2 = 2.25 \\
(12 - 12.5)^2 &= (-0.5)^2 = 0.25 \\
(13 - 12.5)^2 &= (0.5)^2 = 0.25 \\
(14 - 12.5)^2 &= (1.5)^2 = 2.25 \\
(15 - 12.5)^2 &= (2.5)^2 = 6.25 \\
(16 - 12.5)^2 &= (3.5)^2 = 12.25 \\
(17 - 12.5)^2 &= (4.5)^2 = 20.25 \\
\end{aligned}
\]

The sum of these squared deviations is:

\[
\sum (x_i - \mu)^2 = 20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25 = 82.5
\]

**Step 2.3: Calculate the Population Variance**

\[
\sigma^2 = \frac{82.5}{10} = 8.25
\]

### 3. Calculate the Population Standard Deviation

The population standard deviation (\(\sigma\)) is the square root of the population variance:

\[
\sigma = \sqrt{8.25} \approx 2.87
\]

### Summary of Results:
- **Range**: 9
- **Population Variance (\(\sigma^2\))**: 8.25
- **Population Standard Deviation (\(\sigma\))**: 2.87

Would you like more details or have any questions?

Here are five related questions you might consider:
1. How would the calculations change if this was a sample instead of a population?
2. What is the coefficient of variation for this data set?
3. How would the variance change if an additional data point (e.g., 18) was added to the set?
4. How can these statistical measures be interpreted in real-world scenarios?
5. What is the relationship between variance and standard deviation?

**Tip:** Remember that population variance uses \(N\) in the denominator, while sample variance uses \(N-1\) to account for the degrees of freedom.

Learn how to calculate the range, population variance, and population standard deviation for the given data set [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] with detailed step-by-step solutions. Understand the statistical concepts of range, variance, and standard deviation, including formulas used in population statistics. Suitable for advanced high school students and beyond.

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