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Chapter 3: Problem 86

Briefly describe how the three quartiles are calculated for a data set. Illustrate by calculating the three quartiles for two examples, the first with an odd number of observations and the second with an even number of observations.

Short Answer

Expert verified
Quartiles are values that split the data set into four equal parts. For the first example with an odd number of observations \{1, 3, 5, 7, 9\}, the quartiles are Q1=3, Q2=5, Q3=7. For the second example with an even number of observations \{2, 4, 6, 8, 10, 12\}, the quartiles are Q1=3, Q2=7, Q3=11.

Step by step solution

01

Understanding Quartiles

The three quartiles in a data set are values that divide the data into four equal parts. The first quartile (Q1) represents the 25th percentile of the data, the second quartile (Q2) represents the 50th percentile or the median, and the third quartile (Q3) represents the 75th percentile.
02

Calculation with Odd Number of Observations

Imagine a data set with an odd number of observations, e.g. \{1, 3, 5, 7, 9\}. Here, the second quartile (Q2) is the median of the entire data set, which can be obtained by {(n+1) / 2}th observation = 3rd observation, which is 5. The first quartile (Q1) is the median of the lower half not including Q2, which is the 2nd observation, which is 3. The third quartile (Q3) is the median of the upper half not including Q2; thus, it is the 4th observation, or 7.
03

Calculation with Even Number of Observations

Now consider a data set with an even number of observations, e.g. \{2, 4, 6, 8, 10, 12\}. The second quartile (Q2) here is the average of the two middle values (6 + 8) / 2 = 7. The first quartile (Q1) is the median of the lower half, which here is (4 + 2) / 2 = 3. The third quartile (Q3) is the median of the upper half, thus it is (10 + 12) / 2 = 11.

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