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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
For an odd number of observations, such as [1, 3, 5, 7, 9], the first quartile, median, and third quartile are 2, 5, and 8 respectively. For an even number of observations, as in [2, 4, 6, 8, 10, 12], the quartiles are calculated as 4, 7, and 10 respectively.

Step by step solution

01

Understanding Quartiles

Quartiles are statistical values that divide an ordered data set into four equal parts. The data set is ranked from smallest to largest values:- First quartile (Q1) is the middle number between the smallest number in the data set and the median of the data set. - Second quartile (Q2) is the median of the data set. - Third quartile (Q3) is the middle value between the median and the highest value in the data set.
02

Handling data sets with an odd number of observations

Let's take the following data set as example: \[1, 3, 5, 7, 9\]. This data set has an odd number of observations (5). To calculate the quartiles:- First, find the median (Q2). Since the number of observations is odd, the median is the middle number, which is 5. - For Q1, consider the lower half of the data (not including the median). The numbers are 1 and 3. The median of this half is Q1, which is 2. - For Q3, consider the upper half of the data (not including the median). The numbers are 7 and 9. The median of this half is Q3, which is 8.
03

Handling data sets with an even number of observations

Now, consider a data set with an even number of observations: \[2, 4, 6, 8, 10, 12\]. To calculate quartiles:- First, find the median (Q2). As there is an even number of observations, Q2 is the arithmetic average of the middle two numbers, (6+8)/2 = 7. - For Q1, consider the lower half of the data (this includes the lower of the 2 median values). The numbers are 2, 4, and 6. The median of this half is Q1, which is 4. - For Q3, consider the upper half of the data (this includes the higher of the 2 median values). The numbers are 8, 10, and 12. The median of this half is Q3, which is 10.

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Most popular questions from this chapter

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