To calculate the five number summary of the given data set, we need to find the following:
- Minimum
- First Quartile (Q1)
- Median (Q2)
- Third Quartile (Q3)
- Maximum
The data set provided is: 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 14, 16
Step 1: Arrange the data in ascending order. The data is already arranged in ascending order.
Step 2: Find the minimum and maximum values.
- Minimum = 5
- Maximum = 16
Step 3: Find the median (Q2). To find the median, we calculate the middle value of the data set.
- The number of data points is 12 (even). The median is the average of the 6th and 7th values.
- Values in position 6 and 7 are: 8 and 8.
\[ \text{Median} (Q2) = \frac{8 + 8}{2} = 8 \]
Step 4: Find Q1 (First Quartile). Q1 is the median of the first half of the data (the lower half). The lower half of the data includes: 5, 6, 6, 7, 7, 8 (which are the first 6 values).
- The median of this data set is the average of the 3rd and 4th values (6 and 7).
\[ Q1 = \frac{6 + 7}{2} = 6.5 \]
Step 5: Find Q3 (Third Quartile). Q3 is the median of the second half of the data (the upper half). The upper half of the data includes: 8, 8, 9, 9, 14, 16 (which are the last 6 values).
- The median of this data set is the average of the 3rd and 4th values (9 and 9).
\[ Q3 = \frac{9 + 9}{2} = 9 \]
Final Summary: The five number summary is:
- Minimum: 5
- Q1: 6.5
- Median (Q2): 8
- Q3: 9
- Maximum: 16
Five Number Summary: (5, 6.5, 8, 9, 16)