Let's work through the requested computations step by step:

1. Range:

The range is calculated as the difference between the maximum and minimum values in the dataset.

• Maximum value: 34
• Minimum value: 15

$\text{Range} = 34 - 15 = 19$

2. Interquartile Range (IQR):

The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1).

To compute the IQR, we first need to order the data values and then find the quartiles:

• Ordered dataLet's work through the requested computations step by step:

1. Range:

The range is calculated as the difference between the maximum and minimum values in the dataset.

• Maximum value: 34
• Minimum value: 15

$\text{Range} = 34 - 15 = 19$

2. Interquartile Range (IQR):

The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1).

To compute the IQR, we first need to order the data values and then find the quartiles:

• Ordered 15, 20, 25, 27, 28, 28, 30, 34

• Median (Q2): The median is the average of the 4th and 5th data points in the ordered list:
  $Q2 = \frac{27 + 28}{2} = 27.5$

• First quartile (Q1): The median of the first half of the data (15, 20, 25, 27) is 22.5.

• Third quartile (Q3): The median of the second half of the data (28, 28, 30, 34) is 29.

$\text{IQR} = Q3 - Q1 = 29 - 22.5 = 6.5$

3. Sample Variance:

The formula for the sample variance is:

$s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2$

Where:

• $n$ is the number of data points (in this case, 8),

• $x_i$ represents each data point,

• $\bar{x}$ is the sample mean.

• Mean ($\bar{x}$): $\bar{x} = \frac{27 + 28 + 20 + 15 + 30 + 34 + 25 + 28}{8} = \frac{207}{8} = 25.875$

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

$(27 - 25.875)^2 = 1.27, \quad (28 - 25.875)^2 = 4.52, \quad (20 - 25.875)^2 = 34.53$ $(15 - 25.875)^2 = 118.89, \quad (30 - 25.875)^2 = 17.02, \quad (34 - 25.875)^2 = 65.27$ $(25 - 25.875)^2 = 0.77, \quad (28 - 25.875)^2 = 4.52$

Summing these squared differences: $\sum (x_i - \bar{x})^2 = 1.27 + 4.52 + 34.53 + 118.89 + 17.02 + 65.27 + 0.77 + 4.52 = 246.79$

Finally, the sample variance: $s^2 = \frac{246.79}{8 - 1} = \frac{246.79}{7} = 35.26$

4. Sample Standard Deviation:

The standard deviation is the square root of the variance:

$s = \sqrt{35.26} = 5.94$

Summary of Results:

• Range: 19
• Interquartile Range (IQR): 6.5
• Sample Variance: 35.26
• Sample Standard Deviation: 5.94

Would you like further details on any of these steps? Here are some related questions for practice:

1. What is the difference between population and sample variance?
2. How would the calculations change if the dataset had one additional data point?
3. Why do we divide by $n-1$ for sample variance instead of $n$?
4. What are the implications of a large interquartile range in a dataset?
5. How does outlier data affect the standard deviation?

Tip: The interquartile range (IQR) is resistant to outliers, making it a useful measure of spread when data contains extreme values.