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The dataset 3, 6, 2, 8, 6, 6, 9, 10, 3 represents the scores on the most recent quiz. Describe the change in the distribution of data if an additional number is added. If you add the score of 4, what part of the five-number summary will change?(1 point)

Responses

Q1
Q1

minimum
minimum

Q3
Q3

median

1 answer

To analyze the effect of adding the score of 4 to the dataset, let's first calculate the existing five-number summary for the dataset \(3, 6, 2, 8, 6, 6, 9, 10, 3\).

  1. Organize the data in ascending order:

    • Sorted dataset: \(2, 3, 3, 6, 6, 6, 8, 9, 10\)

  2. Minimum: The smallest value is \(2\).

  3. Maximum: The largest value is \(10\).

  4. Median (Q2): The median is the middle value. For an odd number of observations (9), the median is the fifth value in the sorted list:

    • Median = \(6\).

  5. First Quartile (Q1): This is the median of the first half of the data (values less than the median):

    • First half: \(2, 3, 3, 6\) (the median here is \((3 + 3)/2 = 3\)).
    • Q1 = \(3\).

  6. Third Quartile (Q3): This is the median of the second half of the data (values greater than the median):

    • Second half: \(6, 6, 8, 9, 10\) (the median here is \(8\)).
    • Q3 = \(8\).

So the five-number summary before adding 4 is:

  • Minimum = \(2\)
  • Q1 = \(3\)
  • Median = \(6\)
  • Q3 = \(8\)
  • Maximum = \(10\)

Next, let's add the score of \(4\) and recalculate the five-number summary:

  1. New dataset: \(2, 3, 3, 4, 6, 6, 6, 8, 9, 10\)

  2. Organize the data in ascending order:

    • Sorted new dataset: \(2, 3, 3, 4, 6, 6, 6, 8, 9, 10\) (now 10 values).

  3. Minimum: The smallest value remains \(2\).

  4. Maximum: The largest value remains \(10\).

  5. Median (Q2): With an even number of observations (10), the median is the average of the fifth and sixth values:

    • Median = \((6 + 6)/2 = 6\).

  6. First Quartile (Q1): For the first half \(2, 3, 3, 4, 6\) (the median of this is \(3\)).

    • Q1 = \(3\).

  7. Third Quartile (Q3): For the second half \(6, 6, 8, 9, 10\) (the median of this is \(8\)).

    • Q3 = \(8\).

After adding the score of \(4\), the five-number summary is:

  • Minimum = \(2\)
  • Q1 = \(3\)
  • Median = \(6\)
  • Q3 = \(8\)
  • Maximum = \(10\)

Conclusion:

The five-number summary that changes after adding the score of \(4\) is:

  • None of the values have changed; Minimum, Q1, Median, Q3, and Maximum all remain the same. Therefore, if we specifically refer to which part of the five-number summary changes, none of them do, and thus no specific part changes. In this case, it seems that the answer options provided (Q1, minimum, Q3, median) do not apply since they remain invariant.