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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

The following stem-and-leaf diagram gives the distances (in thousands of miles) driven during the past year by a sample of drivers in a city. $$ \begin{array}{l|llllll} 0 & 3 & 6 & 9 & & & \\ 1 & 2 & 8 & 5 & 1 & 0 & 5 \\ 2 & 5 & 1 & 6 & & & \\ 3 & 8 & & & & & \\ 4 & 1 & & & & & \\ 5 & & & & & & \\ 6 & 2 & & & & \end{array} $$ a. Compute the sample mean, median, and mode for the data on distances driven. b. Compute the range, variance, and standard deviation for these data. c. Compute the first and third quartiles. d. Compute the interquartile range. Describe what properties the interquartile range has. When would the IQR be preferable to using the standard deviation when measuring variation?

The distribution of the lengths of fish in a certain lake is not known, but it is definitely not bell shaped. It is estimated that the mean length is 6 inches with a standard deviation of 2 inches. a. At least what proportion of fish in the lake are between 3 inches and 9 inches long? b. What is the smallest interval that will contain the lengths of at least \(84 \%\) of the fish? c. Find an interval so that fewer than \(36 \%\) of the fish have lengths outside this interval.

When is the value of the standard deviation for a data set zero? Give one example. Calculate the standard deviation for the example and show that its value is zero.

The following table gives the standard deductions and personal exemptions for persons filing with "single" status on their 2011 state income taxes in a random sample of 9 states. Calculate the mean and median for the data on standard deductions for these states. $$ \begin{array}{lcc} \hline \text { State } & \begin{array}{c} \text { Standard Deduction } \\ \text { (in dollars) } \end{array} & \begin{array}{c} \text { Personal Exemption } \\ \text { (in dollars) } \end{array} \\ \hline \text { Delaware } & 3250 & 110 \\ \text { Hawaii } & 2000 & 1040 \\ \text { Kentucky } & 2190 & 20 \\ \text { Minnesota } & 5450 & 3500 \\ \text { North Dakota } & 5700 & 3650 \\ \text { Oregon } & 1945 & 176 \\ \text { Rhode Island } & 5700 & 3650 \\ \text { Vermont } & 5700 & 3650 \\ \text { Virginia } & 3000 & 930 \\ \hline \end{array} $$

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