To calculate the 5-number summary of the dataset \(5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16\), we need to determine the following:
- Minimum Value: The smallest number in the dataset.
- Maximum Value: The largest number in the dataset.
- Median: The middle value when the dataset is ordered.
- Quartile 1 (Q1): The median of the first half of the dataset.
- Quartile 3 (Q3): The median of the second half of the dataset.
Let's go through these calculations step by step.
Step 1: Minimum Value
The minimum value is: \[ \text{Minimum} = 5 \]
Step 2: Maximum Value
The maximum value is: \[ \text{Maximum} = 16 \]
Step 3: Median
To find the median, first determine the number of values. The dataset has 12 values, so the median will be the average of the 6th and 7th values when ordered.
Ordered dataset: \[ 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16 \]
The 6th value is 8 and the 7th value is also 8. Therefore: \[ \text{Median} = \frac{8 + 8}{2} = 8 \]
Step 4: Quartile 1 (Q1)
To find Q1, we consider the first half of the dataset, which is: \[ 5, 6, 6, 7, 7, 8 \] This half has 6 values. The median of these values (Q1) is the average of the 3rd and 4th values.
The 3rd value is 6 and the 4th value is 7: \[ Q1 = \frac{6 + 7}{2} = 6.5 \]
Step 5: Quartile 3 (Q3)
To find Q3, we consider the second half of the dataset, which is: \[ 8, 8, 9, 9, 14, 16 \] This half also has 6 values. The median of these values (Q3) is the average of the 3rd and 4th values.
The 3rd value is 9 and the 4th value is also 9: \[ Q3 = \frac{9 + 9}{2} = 9 \]
Summary of Values
Now we can summarize the 5-number summary:
- Minimum: \(5\)
- Maximum: \(16\)
- Median: \(8\)
- Quartile 1 (Q1): \(6.5\)
- Quartile 3 (Q3): \(9\)
Thus, the final results are:
- The minimum value = 5
- The maximum value = 16
- The median = 8
- Quartile 1 (Q1) = 6.5
- Quartile 3 (Q3) = 9