Using the following stem \& leaf plot, find the five number summary for the data by hand. 1123 $$ 2 \mid 234 $$ $$ 3 \mid 23 $$ $$ 4 \mid 4588 $$ 5112344 $$ 6 \mid 01 $$ Min $$ =12 $$ $$ 0^{6} $$ Q1 $$ =23 $$ Med $$ =45 $$ Q3 $$ =53 $$ $$ 0^{6} $$ Max $$ =61 $$ $$ O^{6} $$

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To determine the **five-number summary** from the given **stem-and-leaf plot**, we'll follow these steps:

1. **Understand the Stem-and-Leaf Plot:**

The provided stem-and-leaf plot appears as follows:

```
   1 | 1 2 3
   2 | 2 3 4
   3 | 2 3
   4 | 4 5 8 8
   5 | 1 1 2 3 4 4
   6 | 0 1
   ```

Here, the **stem** represents the **tens place**, and the **leaves** represent the **units place**. For example, the stem `2 | 2 3 4` corresponds to the numbers **22, 23, 24**.

2. **List All Data Points in Order:**

Extracting all the data points from the stem-and-leaf plot:

```
   11, 12, 13, 22, 23, 24, 32, 33, 44, 45, 48, 48, 51, 51, 52, 53, 54, 54, 60, 61
   ```

**Total Data Points (N):** 20

3. **Identify the Five-Number Summary Components:**

The five-number summary consists of:
   
   - **Minimum (Min):** The smallest data point.
   - **First Quartile (Q1):** The median of the lower half of the data.
   - **Median:** The middle value of the data.
   - **Third Quartile (Q3):** The median of the upper half of the data.
   - **Maximum (Max):** The largest data point.

4. **Calculate Each Component:**

- **Minimum (Min):**  
     The smallest number in the dataset is **11**.

- **Maximum (Max):**  
     The largest number in the dataset is **61**.

- **Median:**
     - Since there are **20** data points (an even number), the median is the average of the 10th and 11th values.
     - 10th value: **45**
     - 11th value: **48**
     - **Median = (45 + 48) / 2 = 46.5**

- **First Quartile (Q1):**
     - Q1 is the median of the first half of the data (first 10 numbers).
     - First 10 numbers: **11, 12, 13, 22, 23, 24, 32, 33, 44, 45**
     - The median of these is the average of the 5th and 6th values: **23** and **24**.
     - **Q1 = (23 + 24) / 2 = 23.5**

- **Third Quartile (Q3):**
     - Q3 is the median of the second half of the data (last 10 numbers).
     - Last 10 numbers: **48, 48, 51, 51, 52, 53, 54, 54, 60, 61**
     - The median of these is the average of the 5th and 6th values: **52** and **53**.
     - **Q3 = (52 + 53) / 2 = 52.5**

5. **Final Five-Number Summary:**

Based on the calculations:

- **Minimum (Min):** 11
   - **First Quartile (Q1):** 23.5
   - **Median:** 46.5
   - **Third Quartile (Q3):** 52.5
   - **Maximum (Max):** 61

```
   Five-Number Summary:
   Min = 11
   Q1  = 23.5
   Median = 46.5
   Q3  = 52.5
   Max = 61
   ```

**Note:** The values provided by you (Min = 12, Q1 = 23, Median = 45, Q3 = 53, Max = 61) are slightly different from the calculated ones. This discrepancy might be due to different interpretations of the stem-and-leaf plot or rounding conventions. Ensure that all data points are correctly extracted and that the quartiles are calculated using the appropriate method based on the dataset size.
The five-number summary is: - Minimum (Min): 11 - First Quartile (Q1): 23.5 - Median: 46.5 - Third Quartile (Q3): 52.5 - Maximum (Max): 61

Using the following stem \& leaf plot, find the five number summary for the data by hand. 1123 \( 2 \mid 234 \) \( 3 \mid 23 \) \( 4 \mid 4588 \) 5112344 \( 6 \mid 01 \) Min \( =12 \) \( 0^{6} \) Q1 \( =23 \) Med \( =45 \) Q3 \( =53 \) \( 0^{6} \) Max \( =61 \) \( O^{6} \)

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