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Chapter 2: Problem 74

Box plot for exam The scores on an exam have mean \(=88\), standard deviation \(=10,\) minimum \(=65\), \(\mathrm{Q} 1=77,\) median \(=85, \mathrm{Q} 3=91,\) maximum \(=100 .\) Sketch a box plot, labeling which of these values are used in the plot.

Short Answer

Expert verified
Draw a box on a number line from 77 to 91, with a median line at 85, and whiskers extending from 65 to 100.

Step by step solution

01

Understand Box Plot Components

A box plot represents data using a five-number summary: minimum, first quartile (\(Q1\)), median, third quartile (\(Q3\)), and maximum. These values divide the data into sections and show variability outside the upper and lower quartiles.
02

Identify the Required Values

We are given the following values for the box plot: minimum = 65, \(Q1 = 77\), median = 85, \(Q3 = 91\), and maximum = 100. These are the values we will use in the box plot.
03

Draw the Box

On a number line, draw a box from \(Q1\) (77) to \(Q3\) (91). The length of the box shows the interquartile range (IQR), which is the range of the middle 50% of the data.
04

Draw the Median Line

Draw a line inside the box at the median (85). This line shows the middle value of the data set.
05

Draw the Whiskers

Extend lines (whiskers) from the ends of the box to the minimum value (65) and the maximum value (100). These lines indicate the range of the data outside the middle 50%.
06

Label the Plot

Label each part of the box plot: the minimum (65), first quartile \(Q1\) (77), median (85), third quartile \(Q3\) (91), and maximum (100).

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Key Concepts

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

Five-Number Summary
The five-number summary is a simple way to describe a dataset. It consists of five key metrics: the minimum, first quartile (\(Q1\)), median, third quartile (\(Q3\)), and maximum. These values break down the data into quartiles, providing a clear snapshot of its distribution.
  • **Minimum**: This is the smallest value in the dataset.
  • **First Quartile (\(Q1\))**: Represents the 25th percentile, marking the point below which 25% of the data falls.
  • **Median**: The middle value, splitting the dataset into two equal halves.
  • **Third Quartile (\(Q3\))**: Corresponds to the 75th percentile, the point below which 75% of the data lies.
  • **Maximum**: The largest value in the dataset.
The five-number summary helps identify the spread and center of the data, making it easier to visualize and analyze statistical information.
Interquartile Range (IQR)
The Interquartile Range (IQR) is a valuable measure of variability, capturing the spread of the central 50% of a dataset. It is the difference between the third quartile (\(Q3\)) and the first quartile (\(Q1\)).

The formula for IQR is: \[IQR = Q3 - Q1\]
By using the IQR, we can identify how spread out these middle values are, giving a more reliable measure of dispersion than the range, which can be influenced by extreme values.
  • **Resilience to Outliers**: Unlike the range, which involves the entire dataset, the IQR focuses solely on the middle half, making it robust to outliers.
  • **Box Plot Representation**: In a box plot, the length of the box represents the IQR, visually displaying the variability within the central portion of the data.
Understanding the IQR is crucial for comparing different datasets and assessing their consistency.
Data Visualization
Data visualization is a powerful tool in statistics that helps translate numerical data into visual context, like graphs and charts, for easier understanding. A box plot, for example, is an effective way to summarize and depict data distribution with just a few key figures.

Box plots provide insights into data through:
  • **Quarters and Spread**: Displaying the five-number summary, including the minimum, \(Q1\), median, \(Q3\), and maximum.
  • **Outliers**: Easily identifying if there are any outliers that fall outside the "whiskers" of the plot.
  • **Symmetry and Skewness**: Showing the symmetry or skewness in the data by examining the position of the median line within the box and the lengths of the whiskers.
Data visualization, such as box plots, allows for quick assessment and comparison of datasets, making complex data more accessible and understandable.
Statistics Education
Statistics education equips learners with the skills and knowledge to understand, analyze, and interpret data effectively. By mastering concepts like the five-number summary and IQR through tools like box plots, students gain critical insights into statistical data.

Teaching statistics involves:
  • **Concept Application**: Encouraging the practical use of the five-number summary to assess data distribution.
  • **Problem Solving**: Engaging students in exercises that require creating and interpreting box plots from real-world data.
  • **Critical Thinking**: Cultivating the ability to make data-driven decisions by recognizing patterns and anomalies.
By integrating hands-on activities and visual aids into the curriculum, statistics education fosters a deeper, practical understanding of data analysis, preparing learners for future challenges and opportunities in data-focused fields.

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Most popular questions from this chapter

Great pay (on the average) The six full-time employees of Linda's Tanning Salon near campus had annual incomes last year of \(\$ 8900, \$ 9200, \$ 9200, \$ 9300, \$ 9500,\) \(\$ 9800 .\) Linda herself made \(\$ 250,000\). a. For the seven annual incomes at Linda's Salon, report the mean and median. b. Why is it misleading for Linda to boast to her friends that the average salary at her business is more than \(\$ 40,000 ?\)

Hamburger sales The manager of a fast-food restaurant records each day for a year the amount of money received from sales of food that day. Using software, he finds a bellshaped histogram with a mean of \(\$ 1165\) and a standard deviation of \(\$ 220 .\) Today the sales equaled \(\$ 2000\). Is this an unusually good day? Answer by providing statistical justification.

What shape do you expect? For the following variables, indicate whether you would expect its histogram to be bell shaped, skewed to the right, or skewed to the left. Explain why. a. Number of times arrested in past year b. Time needed to complete difficult exam (maximum time is 1 hour \()\) c. Assessed value of home d. Age at death

Health insurance In \(2004,\) the five-number summary of positions for the distribution of statewide percentage of people without health insurance had a minimum of \(8.9 \%\) (Minnesota), \(\mathrm{Q} 1=11.6,\) Median \(=14.2,\) \(\mathrm{Q} 3=17.0,\) and maximum of \(25.0 \%\) (Texas) (Statistical Abstract of the United States, 2006 ). a. Do you think the distribution is symmetric, skewed right, or skewed left? Why? b. Which is most plausible for the standard deviation: \(-16,0,4,15,\) or \(25 ?\) Why? Explain what is unrealistic about the other values.

Variability of cigarette taxes Here's the five-number summary for the distribution of cigarette taxes (in cents) among the 50 states and Washington, D.C. in the United States. $$ \begin{array}{c} \text { Minimum }=2.5, \mathrm{Q} 1=36, \text { Median }=60 \\ \mathrm{Q} 3=100, \text { Maximum }=205 \end{array} $$ a. About what proportion of the states have cigarette taxes (i) greater than 36 cents and (ii) greater than \(\$ 1 ?\) b. Between which two values are the middle \(50 \%\) of the observations found? c. Find the interquartile range. Interpret it. d. Based on the summary, do you think that this distribution was bell shaped? If so, why? If not, why not, and what shape would you expect?
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