To calculate the 5-number summary of the dataset \(5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16\), we will find the following values:
- Minimum value: The smallest number in the dataset.
- Maximum value: The largest number in the dataset.
- Median: The middle value of the ordered dataset.
- Quartile 1 (Q1): The median of the lower half of the dataset.
- Quartile 3 (Q3): The median of the upper half of the dataset.
- IQR (Interquartile Range): The difference between Q3 and Q1.
Let's go through the dataset step by step.
Step 1: Find the minimum and maximum values
- Minimum value: 5
- Maximum value: 16
Step 2: Find the median
To find the median, we need the number of elements in the dataset. In this case, there are 12 elements (an even number), so the median will be the average of the 6th and 7th values in the ordered dataset.
Ordered dataset: \[5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16\]
The 6th value is \(8\) and the 7th value is \(8\). Therefore: \[ \text{Median} = \frac{8 + 8}{2} = 8 \]
Step 3: Find Quartile 1 (Q1)
Q1 is the median of the lower half of the dataset (the first 6 elements): \[5, 6, 6, 7, 7, 8\]
To find Q1, we take the average of the 3rd and 4th elements:
- 3rd value: \(6\)
- 4th value: \(7\)
\[ Q1 = \frac{6 + 7}{2} = \frac{13}{2} = 6.5 \]
Step 4: Find Quartile 3 (Q3)
Q3 is the median of the upper half of the dataset (the last 6 elements): \[8, 8, 9, 9, 14, 16\]
To find Q3, we take the average of the 3rd and 4th elements:
- 3rd value: \(9\)
- 4th value: \(9\)
\[ Q3 = \frac{9 + 9}{2} = 9 \]
Step 5: Calculate the IQR
The IQR is the difference between Q3 and Q1: \[ \text{IQR} = Q3 - Q1 = 9 - 6.5 = 2.5 \]
Summary of Values
- Minimum value: 5
- Maximum value: 16
- Median: 8
- Quartile 1 (Q1): 6.5
- Quartile 3 (Q3): 9
- IQR: 2.5
Final Answer
- Minimum value: 5
- Maximum value: 16
- Median: 8
- Quartile 1: 6.5
- Quartile 3: 9
- IQR: 2.5