Consider a sample with data values of 27, 28, 20, 15, 30, 34, 25, and 28.
Compute the range.
 
Compute the interquartile range.
 
Compute the sample variance. (Round your answer to two decimal places.)
 
Compute the sample standard deviation. (Round your answer to two decimal places.)
 

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.

Consider a sample with data values of 27, 28, 20, 15, 30, 34, 25, and 28. Compute the range. Compute the interquartile range. Compute the sample variance. (Round your answer to two decimal places.) Compute the sample standard deviation. (Round your answer to two decimal places.)

This tutorial walks through the calculations of range, interquartile range (IQR), sample variance, and sample standard deviation for a dataset: 27, 28, 20, 15, 30, 34, 25, 28. Learn step-by-step how to compute these measures of spread with detailed explanations and examples. Perfect for statistics students at the college level.

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