To calculate the five-number summary, we need to determine the minimum, first quartile (Q1), median, third quartile (Q3), and maximum of the data set.
Original Data Set: 68, 85, 85, 88, 89, 93, 95, 97, 98, 99
Step 1: Order the Original Data Set The original data set is already ordered.
Five-Number Summary of Original Data Set:
- Minimum: 68
- Maximum: 99
- Median: The median is the average of the 5th and 6th values (89 and 93): \[ \text{Median} = \frac{89 + 93}{2} = 91 \]
- First Quartile (Q1): The median of the lower half (68, 85, 85, 88, 89): \[ Q1 = 85 \]
- Third Quartile (Q3): The median of the upper half (93, 95, 97, 98, 99): \[ Q3 = 97 \]
Summary:
- Minimum: 68
- Q1: 85
- Median: 91
- Q3: 97
- Maximum: 99
Step 2: Add the New Data Point (97) The new dataset becomes: 68, 85, 85, 88, 89, 93, 95, 97, 97, 98, 99
Step 3: Order the New Data Set 68, 85, 85, 88, 89, 93, 95, 97, 97, 98, 99
Five-Number Summary of New Data Set:
- Minimum: 68
- Maximum: 99
- Median: Now, the median is the 6th value in this ordered set (93).
- First Quartile (Q1): The median of the lower half (68, 85, 85, 88, 89): \[ Q1 = 85 \]
- Third Quartile (Q3): The median of the upper half (93, 95, 97, 97, 98, 99): \[ Q3 = \frac{97 + 97}{2} = 97 \]
Summary of Changes:
- Minimum: 68 (no change)
- Q1: 85 (no change)
- Median: From 91 to 93 (this changed)
- Q3: 97 (no change)
- Maximum: 99 (no change)
In conclusion, the only part of the five-number summary that changes with the addition of the new data point of 97 is the median, which increases from 91 to 93.