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Chapter 3: Problem 65

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
The quartiles of the odd observation set {1, 2, 3, 4, 5, 6, 7, 8, 9} are Q1 = 2.5, Q2 = 5, Q3 = 7.5. The quartiles of the even observation set {1, 2, 3, 4, 5, 6, 7, 8} are Q1 = 2.5, Q2 = 4.5, Q3 = 6.5.

Step by step solution

01

Understanding Quartiles

Quartiles are statistical values that divide a set of data into four equal parts. The first quartile (Q1) marks the 25th percentile of the data set; the second quartile (Q2) marks the 50th percentile, also called the median of the data set; and the third quartile (Q3) marks the 75th percentile of the data set.
02

Calculating Quartiles for Odd Number of Observations

Suppose there are 9 observations: 1, 2, 3, 4, 5, 6, 7, 8, 9. The median (Q2) is the middle value, which is 5. Q1 is the median of the lower half non-inclusive of the median: 1, 2, 3, 4 which is 2.5. Q3 is the median of the upper half non-inclusive of the median: 6, 7, 8, 9 which is 7.5.
03

Calculating Quartiles for Even Number of Observations

Suppose there are 8 observations: 1, 2, 3, 4, 5, 6, 7, 8. The median (Q2) is the average of the two middle values, which is (4+5)/2 = 4.5. Q1 is the median of the lower half (inclusive of the median): 1, 2, 3, 4 which is 2.5. Q3 is the median of the upper half (inclusive of the median): 5, 6, 7, 8 which is 6.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Percentiles
Percentiles are a way of breaking down a large dataset into 100 equal parts. They provide understanding on how individual values within a dataset compare to the larger group. Each percentile represents a specific position in this ordered dataset. For example, if a student scores in the 80th percentile, it means they scored better than 80% of students. In statistical terms:
  • The 25th percentile is the first quartile (Q1), which means 25% of data falls below this point.
  • The 50th percentile is the second quartile (Q2), also known as the median, equating to the midpoint of the dataset.
  • The 75th percentile is the third quartile (Q3), where 75% of data points fall below this value.
Knowing percentiles helps with understanding data dispersion and can be a helpful tool to assess any data distribution more deeply. They are often used in educational assessments, income analysis, and other fields requiring statistical evaluation.
Median Calculation
The median is a key concept in statistics, representing the middle value in a dataset that has been arranged in numerical order. Calculating the median involves identifying the central value or average of two central values depending on whether the dataset has an odd or even number of observations. For an odd number of observations:
  • Data: 1, 2, 3, 4, 5, 6, 7, 8, 9
  • The median is the middle value, which in this case is 5.
For an even number of observations:
  • Data: 1, 2, 3, 4, 5, 6, 7, 8
  • The median is the average of the two middle values, (4+5)/2 = 4.5.
The median is integral in quartile calculations and provides a clear measure of the central location in a dataset without being influenced by outliers, unlike the mean. Understanding where the median lies gives a solid footing in both quartile and percentile calculations.
Data Distribution
Data distribution illustrates how data points are spread across a range. It鈥檚 an important statistical concept that allows one to visualize and understand how often values occur within a dataset, thereby uncovering patterns. Types of data distribution include:
  • Normal distribution, which is symmetric and bell-shaped, where most of the data points cluster around the mean.
  • Skewed distribution, which can lean left or right, affecting the position of the median and the mean in relation to each other.
When examining quartiles and the broader context of the dataset:
  • Quartiles divide the data into four equal parts, helping with the immediate understanding of the spread and variability.
  • Analyzing quartiles allows for insights into the presence of any skewness or outliers within the dataset.
Understanding data distribution is crucial for interpreting statistical results effectively, as it can highlight whether calculated statistics, like the mean or quartiles, are representative and useful for making real-world decisions.

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Most popular questions from this chapter

The following data give the number of patients who visited a walk-in clinic on each of 20 randomly selected days. \(\begin{array}{llllllllll}23 & 37 & 26 & 19 & 33 & 22 & 30 & 42 & 24 & 26 \\\ 28 & 32 & 37 & 29 & 38 & 24 & 35 & 20 & 34 & 38\end{array}\) a. Calculate the range, variance, and standard deviation for these data. b. Calculate the coefficient of variation.

A student washes her clothes at a laundromat once a week. The data below give the time (in minutes) she spent in the laundromat for each of 15 randomly selected weeks. Here, time spent in the laundromat includes the time spent waiting for a machine to become available. \(\begin{array}{rrrrrrrr}75 & 62 & 84 & 73 & 107 & 81 & 93 & 72 \\ 135 & 77 & 85 & 67 & 90 & 83 & 112 & \end{array}\) Prepare a box-and-whisker plot. Is the data set skewed in any direction? If yes, is it skewed to the right or to the left? Does this data set contain any outliers?

Explain how the interquartile range is calculated. Give one example.

The following data give the annual salaries (in thousand dollars) of 20 randomly selected health care workers. \(\begin{array}{llllllllll}50 & 71 & 57 & 39 & 45 & 64 & 38 & 53 & 35 & 62 \\\ 74 & 40 & 67 & 44 & 77 & 61 & 58 & 55 & 64 & 59\end{array}\) a. Compute the range, variance, and standard deviation for these data. b. Calculate the coefficient of variation. c. Are the values of these summary measures population parameters or sample statistics? Explain.

Jeffrey is serving on a six-person jury for a personal-injury lawsuit. All six jurors want to award damages to the plaintiff but cannot agree on the amount of the award. The jurors have decided that each of them will suggest an amount that he or she thinks should be awarded; then they will use the mean of these six numbers as the recommended amount to be awarded to the plaintiff. a. Jeffrey thinks the plaintiff should receive \(\$ 20,000\), but he thinks the mean of the other five jurors' recommendations will be about \(\$ 12,000 .\) He decides to suggest an inflated amount so that the mean for all six jurors is \(\$ 20,000\). What amount would Jeffrey have to suggest? b. How might this jury revise its procedure to prevent a juror like Jeffrey from having an undue influence on the amount of damages to be awarded to the plaintiff?
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