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Chapter 3: Problem 21

Explain how to determine the shape of a distribution using the box plot and quartiles.

Short Answer

Expert verified
Examine the box plot's whiskers and quartiles (Q1, Q2, Q3) to determine symmetry or skewness of the distribution.

Step by step solution

01

- Understand the Components of a Box Plot

A box plot displays the minimum, first quartile (Q1), median (Q2), third quartile (Q3), and maximum of a data set. These components are essential in assessing the shape of the distribution.
02

- Identify the Quartiles

Locate the first quartile (Q1), median (Q2), and third quartile (Q3) on the box plot. Q1 is the left edge of the box, Q2 is the line inside the box, and Q3 is the right edge of the box.
03

- Analyze the Whiskers

Examine the length of the whiskers (lines that extend from the box to the minimum and maximum values). Observe if they are approximately the same length or if one is significantly longer than the other.
04

- Determine Skewness

If the right whisker (extending to the maximum value) is longer than the left whisker (extending to the minimum value), the distribution is skewed to the right (positively skewed). If the left whisker is longer, it is skewed to the left (negatively skewed).
05

- Assess Symmetry

If the box and whiskers are roughly symmetrical (equal lengths on both sides), the distribution is approximately symmetric. Also check if Q2 is centered within the box.

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Key Concepts

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

quartiles
Quartiles are key values that divide your data set into four equal parts. Understanding quartiles helps in determining the spread and center of your data.
There are three main quartiles in a data set:
  • The first quartile (Q1), also known as the lower quartile, marks the 25th percentile. In a box plot, it is the left edge of the box.
  • The second quartile (Q2), or the median, represents the 50th percentile. It is the middle value of the data set and is shown as the line inside the box.
  • The third quartile (Q3), or the upper quartile, indicates the 75th percentile and is represented by the right edge of the box.
To determine the shape of a distribution, locate these quartiles on the box plot. Evaluating how Q1, Q2, and Q3 position can reveal whether your data is skewed or symmetrical.
skewness
Skewness identifies whether your data leans more towards the lower or higher values. A skewed distribution means that the data has a longer tail on one side.
If the right whisker of the box plot (extending to the maximum value) is longer than the left whisker (extending to the minimum value), the distribution is positively skewed, or skewed to the right. Conversely, if the left whisker is longer, the distribution is negatively skewed, or skewed to the left.
This skewness helps in understanding the spread and possible outliers in the data. Such insights can be crucial for interpreting the data correctly and making informed decisions.
symmetry
Assessing the symmetry of a distribution using a box plot is straightforward. Symmetry means that the data is evenly distributed on both sides of the center.
To check for symmetry, observe the box plot:
  • If the box and whiskers are approximately equal in length on both sides of the median (Q2), the data distribution is symmetric.
  • Make sure Q2 is centered within the box, not skewed to one side, to ensure symmetry.
Symmetrical distributions imply that data points are spread consistently around the center. This is useful in many statistical analyses where normal distribution is assumed.
box plot components
Understanding the components of a box plot is crucial for interpreting data effectively. A box plot consists of:
  • A rectangular box, which spans from Q1 to Q3. This box represents the interquartile range (IQR), covering the middle 50% of your data.
  • A line inside the box indicating the median (Q2).
  • Whiskers extending from the box to the minimum and maximum values not considered outliers.
  • Potential outliers, which can be shown as individual points beyond the whiskers.
Each component provides different insights into your data set. For instance, the IQR highlights the data spread, while whiskers show the range. Evaluating these parts collectively allows you to determine the distribution shape and identify any potential anomalies.

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Most popular questions from this chapter

According to the U.S. Census Bureau, the mean of the commute time to work for a resident of Boston, Massachusetts, is 27.3 minutes. Assume that the standard deviation of the commute time is 8.1 minutes to answer the following: (a) What minimum percentage of commuters in Boston has a commute time within 2 standard deviations of the mean? (b) What minimum percentage of commuters in Boston has a commute time within 1.5 standard deviations of the mean? What are the commute times within 1.5 standard deviations of the mean? (c) What is the minimum percentage of commuters who have commute times between 3 minutes and 51.6 minutes?

A survey of 40 randomly selected full-time Joliet Junior College students was conducted in the Fall 2015 semester. In the survey, the students were asked to disclose their weekly spending on entertainment. The results of the survey are as follows: $$ \begin{array}{rrrrrrrr} \hline 21 & 54 & 64 & 33 & 65 & 32 & 21 & 16 \\ \hline 22 & 39 & 67 & 54 & 22 & 51 & 26 & 14 \\ \hline 115 & 7 & 80 & 59 & 20 & 33 & 13 & 36 \\ \hline 36 & 10 & 12 & 101 & 1000 & 26 & 38 & 8 \\ \hline 28 & 28 & 75 & 50 & 27 & 35 & 9 & 48 \\ \hline \end{array} $$ (a) Check the data set for outliers. (b) Draw a histogram of the data and label the outliers on the histogram. (c) Provide an explanation for the outliers.

The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds. $$ \begin{array}{lll} \hline 2.0 & 1.9 & 1.8 \\ \hline 3.2 & 2.4 & 3.4 \\ \hline 1.6 & 2.7 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 8.4 & 7.2 & 7.6 \\ \hline 7.4 & 6.9 & 9.4 \\ \hline 9.1 & 8.1 & \\ \hline \end{array} $$ (a) Determine the mean and standard deviation of each data set. (b) Based only on the standard deviation, which data set has more spread? (c) What proportion of the observations is within one standard deviation of the mean for each data set? (d) The coefficient of variation, \(C V\), is defined as the ratio of the standard deviation to the mean of a data set, so $$ C V=\frac{\text { standard deviation }}{\text { mean }} $$ The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more "spread"? (e) Let's take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters. $$ \begin{array}{lll} \hline 74 & 68 & 71 \\ \hline 66 & 72 & 69 \\ \hline 69 & 71 & \\ \hline \end{array} $$ $$ \begin{array}{lll} \hline 187.96 & 172.72 & 180.34 \\ \hline 167.64 & 182.88 & 175.26 \\ \hline 175.26 & 180.34 & \\ \hline \end{array} $$

One variable that is measured by online homework systems is the amount of time a student spends on homework for each section of the text. The following is a summary of the number of minutes a student spends for each section of the text for the fall 2014 semester in a College Algebra class at Joliet Junior College. $$ Q_{1}=42 \quad Q_{2}=51.5 \quad Q_{3}=72.5 $$ (a) Provide an interpretation of these results. (b) Determine and interpret the interquartile range. (c) Suppose a student spent 2 hours doing homework for a section. Is this an outlier? (d) Do you believe that the distribution of time spent doing homework is skewed or symmetric? Why?

A professor has recorded exam grades for 20 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 19 readable scores is \(84,\) what is the value of the unreadable score?
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