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What will be the 25th percentile of the dataset? What will be the 75th percentile of the dataset? Find IQR of the dataset

To find the 25th percentile, 75th percentile, and the interquartile range (IQR) of the dataset provided in the stem-and-leaf diagram, we need to follow these steps:

1. **List the data in ascending order:**
   The data provided in the stem-and-leaf diagram can be listed as:
   - 14, 15, 18
   - 21, 22, 24
   - 31, 31, 32, 33

   Combining and sorting these values, we get:
   \[
   \{14, 15, 18, 21, 22, 24, 31, 31, 32, 33\}
   \]

2. **Find the position of the 25th and 75th percentiles:**
   For a dataset with \(n\) observations, the \(k\)th percentile is located at position \(P_k = \frac{k}{100} \times (n + 1)\).

   Here, \(n = 10\).

   - **25th percentile (P_25):**
     \[
     P_{25} = \frac{25}{100} \times (10 + 1) = 2.75
     \]
     The 25th percentile lies between the 2nd and 3rd values.

   - **75th percentile (P_75):**
     \[
     P_{75} = \frac{75}{100} \times (10 + 1) = 8.25
     \]
     The 75th percentile lies between the 8th and 9th values.

3. **Calculate the 25th and 75th percentiles:**
   - For the 25th percentile, \(P_{25}\):
     \[
     \text{Value at } 2.75 = \text{Value at } 2 + 0.75 \times (\text{Value at } 3 - \text{Value at } 2)
     \]
     \[
     = 15 + 0.75 \times (18 - 15) = 15 + 0.75 \times 3 = 15 + 2.25 = 17.25
     \]

   - For the 75th percentile, \(P_{75}\):
     \[
     \text{Value at } 8.25 = \text{Value at } 8 + 0.25 \times (\text{Value at } 9 - \text{Value at } 8)
     \]
     \[
     = 31 + 0.25 \times (32 - 31) = 31 + 0.25 \times 1 = 31 + 0.25 = 31.25
     \]

4. **Calculate the Interquartile Range (IQR):**
   \[
   \text{IQR} = P_{75} - P_{25} = 31.25 - 17.25 = 14
   \]

**Summary:**
- 25th percentile (P_25): \(17.25\)
- 75th percentile (P_75): \(31.25\)
- Interquartile Range (IQR): \(14\)

Would you like more details or have any questions?

Here are some questions you might want to ask next:
1. How do I interpret the interquartile range in a dataset?
2. What other measures of spread can be used besides IQR?
3. How do I find the median of this dataset?
4. What is the range of this dataset?
5. How can I identify outliers using the IQR?

**Tip:** The interquartile range (IQR) is a robust measure of variability that is not influenced by outliers, making it useful for summarizing the spread of a dataset.

Learn how to calculate the 25th percentile, 75th percentile, and interquartile range (IQR) from a given dataset. Follow step-by-step instructions with detailed formulas and examples. Ideal for undergraduate students studying statistics or data analysis.

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