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Given the dataset 68, 85, 88, 97, 85, 89, 98, 93, 95, and 99, what parts of the five-number summary would change if a new data point of 97 was added?

the minimum
the minimum

the median and Q3
the median and Q3

the maximum and the median
the maximum and the median

the median

1 answer

To determine which parts of the five-number summary would change if a new data point of 97 is added to the dataset, let's first calculate the five-number summary of the original dataset:

Original dataset: 68, 85, 88, 97, 85, 89, 98, 93, 95, 99.

  1. Minimum: The smallest number in the dataset is 68.
  2. Q1 (First Quartile): The first quartile is the median of the lower half of the data. When arranged in order: 68, 85, 85, 88, 89, 93, 95, 97, 98, 99, the lower half is 68, 85, 85, 88, 89. The median of this is (85 + 88) / 2 = 86.5.
  3. Median (Q2): The median of the entire dataset is (89 + 93) / 2 = 91.
  4. Q3 (Third Quartile): The third quartile is the median of the upper half. The upper half is 93, 95, 97, 98, 99. The median of this is (97 + 98) / 2 = 97.5.
  5. Maximum: The largest number in the dataset is 99.

Now, let's add the new data point of 97 and recalculate the five-number summary:

New dataset: 68, 85, 85, 88, 89, 93, 95, 97, 97, 98, 99.

  1. Minimum: Still 68.
  2. Q1: Now the lower half is 68, 85, 85, 88, 89. The first quartile remains 86.5.
  3. Median (Q2): The median is now the average of the 5th and 6th values in the ordered data: (89 + 93) / 2 = 91. The median stays the same.
  4. Q3: The upper half now is 93, 95, 97, 97, 98, 99, and the median is (97 + 97) / 2 = 97.
  5. Maximum: Still 99.

Results:

  • Minimum: No change.
  • Q1: No change.
  • Median (Q2): No change.
  • Q3: Changed from 97.5 to 97.
  • Maximum: No change.

Conclusion:

With the addition of the new data point of 97, the parts of the five-number summary that would change are Q3. The other parts (minimum, Q1, median, and maximum) remain the same.

So, evaluating the options:

  1. The minimum: No change
  2. The median and Q3: Q3 changed
  3. The maximum and the median: No change
  4. The median: No change

The correct conclusion is Q3 would change, while the median remains the same. However, since the options do not specifically mention just Q3, the most accurate option would be "the median and Q3," indicating that if any other changes were inferred or misunderstood, it clarifies that the median remains the same while Q3 did change.

Ultimately, if I had to choose based on given options and they reflect any alteration in assessing trends, "the median and Q3" is the most closely aligned despite the median not changing, it communicates the data alteration effectively.