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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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Most popular questions from this chapter

When studying phenomena such as inflation or population changes that involve periodic increases or decreases, the geometric mean is used to find the average change over the entire period under study. To calculate the geometric mean of a sequence of \(n\) values \(x_{1}, x_{2}, \ldots, x_{n}\), we multiply them together and then find the \(n\) th root of this product. Thus $$ \text { Geometric mean }=\sqrt[n]{x_{1} \cdot x_{2} \cdot x_{3} \cdot \ldots \cdot x_{n}} $$ Suppose that the inflation rates for the last five years are \(4 \%, 3 \%, 5 \%, 6 \%\), and \(8 \%\), respectively. Thus at the end of the first year, the price index will be \(1.04\) times the price index at the beginning of the year, and so on. Find the mean rate of inflation over the 5 -year period by finding the geometric mean of the data set \(1.04,1.03,1.05,1.06\), and \(1.08 .\) (Hint: Here, \(n=5, x_{1}=1.04, x_{2}=1.03\), and so on. Use the \(x^{1 / n}\) key on your calculator to find the fifth root. Note that the mean inflation rate will be obtained by subtracting 1 from the geometric mean.)

The following data set belongs to a sample: \(\begin{array}{llllll}14 & 18 & -1 & 08 & 8 & -16\end{array}\) Calculate the range, variance, and standard deviation.

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