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To calculate the quartiles \( Q_1 \), \( Q_2 \), and \( Q_3 \), we first need to organize the data in ascending order and then use the following formulas:

1. **\( Q_1 \) (First Quartile)**: This is the median of the lower half of the dataset (excluding the overall median if the total number of values is odd).
2. **\( Q_2 \) (Second Quartile or Median)**: This is the median of the entire dataset.
3. **\( Q_3 \) (Third Quartile)**: This is the median of the upper half of the dataset (excluding the overall median if the total number of values is odd).

Let's go step by step:

### 1. Organize the data in ascending order:

\[
2, 2, 3, 4, 5, 5, 5, 6, 10, 10, 15, 16, 19, 20, 20, 22, 22, 22, 23, 25, 55
\]

### 2. Find \( Q_1 \), \( Q_2 \), and \( Q_3 \):

- **\( Q_2 \) (Median):** 
  Since there are 21 values, the median is the 11th value:
  \[
  Q_2 = 15
  \]

- **\( Q_1 \) (First Quartile):**
  The lower half of the data (first 10 values) is:
  \[
  2, 2, 3, 4, 5, 5, 5, 6, 10, 10
  \]
  The median of these 10 values (average of the 5th and 6th values) is:
  \[
  Q_1 = \frac{5 + 5}{2} = 5
  \]

- **\( Q_3 \) (Third Quartile):**
  The upper half of the data (last 10 values) is:
  \[
  16, 19, 20, 20, 22, 22, 22, 23, 25, 55
  \]
  The median of these 10 values (average of the 5th and 6th values) is:
  \[
  Q_3 = \frac{22 + 22}{2} = 22
  \]

### Final Answers:
- \( Q_1 = 5 \)
- \( Q_2 = 15 \)
- \( Q_3 = 22 \)

Do you need further details or have any questions?

Here are some related questions:
1. How do you calculate the interquartile range (IQR)?
2. Why is the second quartile also called the median?
3. How can quartiles help in understanding data distribution?
4. What is the significance of outliers in this context?
5. Can the quartiles help detect skewness in the data?

**Tip:** Quartiles divide data into four equal parts and are useful for summarizing the spread of the data.

A statistic center compiles data on the length of stay by patients in short-term hospitals. A random sample of 21 patients yielded the data on length of stay, in days. The question asks to calculate and interpret the quartiles.

Learn how to calculate and interpret the quartiles for hospital stay data. This problem involves the use of quartile formulas to understand the distribution of patient stays in short-term hospitals. Suitable for students in Grades 10-12.

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