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QuestionStatistics
What is the five-number summary for the data 0, 2, 2, 4, 5, 5, 5, 5, 7, 11?
Minimum
Q^1
Median
Q^3
Maximum
When the minimum, 0, is removed from the data set, what is the five-number summary?
Minimum
Q^1
Median
Q^3
Maximum
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Answer
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Step 1:: Find the minimum, maximum, and ordered data set.
The minimum is the smallest value in the data set, which is 0. The maximum is the largest value in the data set, which is 11. The ordered data set is: 0, 2, 2, 4, 5, 5, 5, 5, 7, 11.
Step 2:: Calculate Q^1 (the first quartile).
\frac{2 + 2}{2} = 2
The first quartile is the median of the lower half of the data set. Here, the lower half of the data set is: 0, 2, 2, 4. The median of this set is the average of the two middle numbers: So, Q^1 is 2.
Step 3:: Calculate the median (the second quartile).
The median is the middle value in the data set. Since there is an odd number of data points (10), the median is the fifth value in the ordered data set, which is 5.
Step 4:: Calculate Q^3 (the third quartile).
\frac{5 + 7}{2} = 6
The third quartile is the median of the upper half of the data set. Here, the upper half of the data set is: 5, 5, 7, 11. The median of this set is the average of the two middle numbers: So, Q^3 is 6.
Step 5:: Summarize the five-number summary.
Minimum: 0 Q^1: 2 Median: 5 Q^3: 6 Maximum: 11
Step 6:: Find the five-number summary without the minimum.
Now, we will exclude the minimum (0) from the data set and recalculate the five-number summary. The ordered data set without the minimum is: 2, 2, 4, 5, 5, 5, 7, 11. The new minimum is 2, the new Q^1 is 3.5 (the average of 2 and 4), the median remains 5, the new Q^3 is 6.5 (the average of 5 and 7), and the maximum remains 11.
Step 7:: Summarize the five-number summary without the minimum.
Minimum: 2 Q^1: 3.5 Median: 5 Q^3: 6.5 Maximum: 11
Final Answer
Minimum: 2 Q^1: 3.5 Median: 5 Q^3: 6.5 Maximum: 11
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