For the following data, calculate the 5 number summary:

Write the 5 number summary in order.

Five number summary:  , , , ,

Data: 

5

5

5

5

5

6

7

7

8

6

7

5

3

5

8

7

9

1

2

3

4

5

3

5

8

7

9

6

3

5

2

4

5


To calculate the five-number summary for the given dataset, we first need to organize the data in ascending order, then determine the minimum, maximum, median, and the first (Q1) and third quartiles (Q3).

### **Given Data:**
5, 5, 5, 5, 5, 6, 7, 7, 8, 6, 7, 5, 3, 5, 8, 7, 9, 1, 2, 3, 4, 5, 3, 5, 8, 7, 9, 6, 3, 5, 2, 4, 5

### **Step 1: Sort the Data in Ascending Order**
Sorted Data:
1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9

### **Step 2: Calculate the Five-Number Summary**

1. **Minimum**: The smallest number in the dataset is **1**.
2. **Maximum**: The largest number in the dataset is **9**.

To find the median and quartiles, we need to identify the middle values.

3. **Median (Q2)**:
   - With 31 numbers, the median is the 16th value (since \( \frac{31 + 1}{2} = 16 \)). 
   - The 16th value in the sorted data is **5**.

4. **First Quartile (Q1)**: The median of the first half of the dataset (excluding the median if the number of data points is odd):
   - The first half of the dataset (15 values): 1, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5
   - The median of these 15 values (8th value) is **4**.

5. **Third Quartile (Q3)**: The median of the second half of the dataset:
   - The second half of the dataset (15 values): 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9
   - The median of these 15 values (8th value) is **7**.

### **The Five-Number Summary is:**
- Minimum: **1**
- Q1: **4**
- Median (Q2): **5**
- Q3: **7**
- Maximum: **9**

Would you like more details or have any questions?

Here are 5 related questions that expand on this topic:

1. How would the five-number summary change if we added a new value to the dataset?
2. Can a dataset have multiple modes, and how does that affect the five-number summary?
3. What are some other measures of spread, and how do they compare to the five-number summary?
4. How can outliers be identified using the five-number summary and the interquartile range?
5. How would you interpret a dataset with a very small interquartile range compared to a large one?

**Tip:** To quickly find the quartiles, it can help to divide the sorted data into four equal parts and find the median of each part for Q1 and Q3.

Learn how to calculate the five-number summary from a dataset with this step-by-step guide. Understand the concepts of minimum, maximum, median, and quartiles. Example dataset included.

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