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Chapter 3: Problem 6

Consider the following ordered data: \(\begin{array}{llllllllll}2 & 5 & 5 & 6 & 7 & 8 & 8 & 9 & 10 & 12\end{array}\) (a) Find the low, \(Q_{1}\), median, \(Q_{3}\) high. (b) Find the interquartile range. (c) Make a box-and-whisker plot.

Short Answer

Expert verified
Low: 2, Q1: 5, Median: 7.5, Q3: 9, High: 12. IQR: 4.

Step by step solution

01

Identify the Low and High Values

The lowest value is the smallest number in the dataset, and the highest value is the largest. From the ordered data set: \(2, 5, 5, 6, 7, 8, 8, 9, 10, 12\), the low value is 2, and the high value is 12.
02

Find the Median

To determine the median, find the middle value of the dataset. Since we have ten numbers, the median is the average of the 5th and 6th values. These are 7 and 8. Thus, the median is \(\frac{7 + 8}{2} = 7.5\).
03

Calculate Q1

\(Q_1\) is the median of the first half of the dataset (excluding the overall median). The first half is \(2, 5, 5, 6, 7\). The median of this set is 5, which is the 3rd value, so \(Q_1 = 5\).
04

Calculate Q3

\(Q_3\) is the median of the second half of the dataset (excluding the overall median). The second half is \(8, 8, 9, 10, 12\). The median of this set is 9, which is the 3rd value, so \(Q_3 = 9\).
05

Compute the Interquartile Range

The interquartile range is the difference between \(Q_3\) and \(Q_1\). So, \(IQR = Q_3 - Q_1 = 9 - 5 = 4\).
06

Draft a Box-and-Whisker Plot

To draw a box-and-whisker plot, plot the five-number summary: low (2), \(Q_1 (5)\), median (7.5), \(Q_3 (9)\), and high (12). Draw a number line, then a box from \(Q_1\) to \(Q_3\) with a line at the median. Extend 'whiskers' from the box to the low and high values.

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Key Concepts

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

Box-and-Whisker Plot
A box-and-whisker plot, often just called a box plot, is a graphical representation of data that highlights its distribution. This type of plot shows the central tendency and dispersion of a dataset at a glance. It's particularly useful for spotting outliers and comparing different sets of data. The process to create this plot involves drawing a horizontal number line and stacking relevant elements on it.

Here's how it works:
  • A box is drawn from the first quartile ( Q_1 ) to the third quartile ( Q_3 ). This box shows where the central 50% of data lies.
  • A line within the box marks the median.
  • 'Whiskers' extend from the box to the smallest (Low) and largest (High) values in the dataset. They indicate the range of the data beyond the middle 50%.
This simple yet informative visualization helps understand the spread and skewness of the data. Creating a box-and-whisker plot involves plotting the five-number summary, which we will discuss next.
Interquartile Range
The interquartile range (IQR) is a measure of statistical dispersion and is one of the most widely used methods to understand the variability in a dataset. It represents the middle 50% of the data.
  • The formula for calculating IQR is straightforward: IQR = Q_3-Q_1 .
  • This range helps in identifying the spread of the middle half of the data, which can be critical for statistical analyses.
  • In our example dataset, Q_3 equals 9, and Q_1 equals 5, so the IQR = 9-5 = 4.
Using the IQR helps to identify outliers by measuring how far each data point is from the central range. Outliers often have values larger than Q_3 + 1.5 imes ext{IQR} or smaller than Q_1 - 1.5 imes ext{IQR} . These observations might require further investigation as they can have implications on the interpretation of data.
Five-Number Summary
The five-number summary is a way of summarizing a dataset using five key metrics:
  • **Low**: the smallest number in the dataset
  • **First Quartile ( Q_1 )**: the median of the first half of the data
  • **Median**: the middle value of the dataset
  • **Third Quartile ( Q_3 )**: the median of the second half of the data
  • **High**: the largest number in the dataset
This summary provides a comprehensive snapshot of the dataset's core characteristics, serving as the foundation for more in-depth data analysis. In our dataset example, the five-number summary includes Low = 2, Q_1 = 5 , Median = 7.5, Q_3 = 9 , and High = 12. With these numbers, you can quickly assess the dataset's distribution, detect any skewness, and prepare to draw the box-and-whisker plot. This tool is essential for anyone analyzing data, as it identifies not just extremes but positions of spread.

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Most popular questions from this chapter

Nurses At Center Hospital there is some concern about the high turnover of nurses. A survey was done to determine how long (in months) nurses had been in their current positions. The responses (in months) of 20 nurses were \(\begin{array}{rrrrrrrrrr}23 & 2 & 5 & 14 & 25 & 36 & 27 & 42 & 12 & 8 \\ 7 & 23 & 29 & 26 & 28 & 11 & 20 & 31 & 8 & 36\end{array}\) Make a box-and-whisker plot of the data. Find the interquartile range.

Find the mean, median, and mode of the data set \(\begin{array}{llllll}8 & 2 & 7 & 2 & 6 & 5\end{array}\)

Consider the numbers \(\begin{array}{lllll}2 & 3 & 4 & 5 & 5\end{array}\) (a) Compute the mode, median, and mean. (b) If the numbers represent codes for the colors of T-shirts ordered from a catalog, which average(s) would make sense? (c) If the numbers represent one-way mileages for trails to different lakes, which average(s) would make sense? (d) Suppose the numbers represent survey responses from 1 to 5, with \(1=\) disagree strongly, \(2=\) disagree, \(3=\) agree, \(4=\) agree strongly, and \(5=\) agree very strongly. Which averages make sense?

A job-performance evaluation form has these categories: \(1=\) excellent; \(2=\) good; \(3=\) statisfactory; \(4=\) poor; \(5=\) unacceptable Based on 15 client reviews, one employee had median rating of \(4 ;\) mode rating of 1 The employee was pleased that most clients had rated her as excellent. The supervisor said improvement was needed because at least half the clients had rated the employee at the poor or unacceptable level. Comment on the different perspectives.

Consider the data set \(\begin{array}{lllll}1 & 2 & 3 & 4 & 5\end{array}\) (a) Find the range. (b) Use the defining formula to compute the sample standard deviation \(s\). (c) Use the defining formula to compute the population standard deviation \(\sigma\).
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