/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 157 Examine issues of location and s... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 157

Examine issues of location and spread for boxplots. In each case, draw sideby- side boxplots of the datasets on the same scale. There are many possible answers. One dataset has median 50, interquartile range 20 , and range 40 . A second dataset has median 50, interquartile range 50 , and range 100 . A third dataset has median 50 , interquartile range 50 , and range 60 .

Short Answer

Expert verified
All three datasets are centred around the same value (Median = 50), but they differ in their spread. The interquartile ranges for the second and third datasets are larger than for the first dataset, indicating a greater spread of the middle 50% of data. The range is the largest for the second dataset, suggesting overall greater data dispersion. The third dataset, despite having an IQR equivalent to dataset 2, has less overall spread due to a smaller range.

Step by step solution

01

Understand the Data

Firstly, identify the median which is the middle value for each data set. Also, acknowledge the interquartile range (IQR) which is the range of the middle 50% of the data, and is calculated as Q3 - Q1. The 'range' is the difference between the maximum and minimum values. Dataset 1: median 50, IQR 20, range 40; Dataset 2: median 50, IQR 50, range 100; Dataset 3: median 50, IQR 50, range 60.
02

Create the Boxplots

Plot three boxplots side-by-side. For boxplot of each data set, draw a box from the Q1 to Q3 value. To find Q1 and Q3, add or subtract half of the IQR to the median. The line inside the box represents the median. The whiskers extend from Q1 and Q3 to the minimum and maximum value respectively. To find maximum and minimum, add or subtract half of the range to the median.
03

Examine Location and Spread

Examine issues of location and spread for the boxplots. All three datasets have the same median (50), so they have the same location. However, the spread (IQR and range) differ for each dataset. Dataset 2 and 3 have larger IQR showing greater middle 50% data dispersion than Dataset 1. Dataset 2 has the largest spread (range) indicating that its data is the most dispersed overall.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Interquartile Range (IQR)
One of the fundamental components of a boxplot is the interquartile range (IQR), which measures the spread of the middle 50% of a dataset. Essentially, the IQR reflects the range between the first quartile (Q1) and the third quartile (Q3) values of a dataset.

Calculating the IQR is straightforward: you subtract Q1 from Q3, and the result shows how much variability there is in the central portion of your data. In our exercise, for instance, Dataset 1 has an IQR of 20, which means there is less variance in the middle 50% compared to Dataset 2 and 3, which both have an IQR of 50. Understanding IQR is essential as it helps identify the compactness of the data and is less affected by outliers and extreme values than the full range.
Dataset Comparison
Comparing datasets is an integral part of statistical analysis, often to assess differences and similarities in their central tendency and variability. Boxplots are particularly useful for this purpose as they summarize the data through five-number summaries (minimum, Q1, median, Q3, and maximum) and showcase the data鈥檚 spread.

When comparing datasets with boxplots, you should check for the IQR, which indicates the concentration of the middle 50% of values. In our example, although all datasets share the same median of 50, indicating a similar central location, their IQRs differ significantly. Moreover, comparing the ranges (total spread) alongside the IQR provides insights into how spread out the entire set of values is, not just the middle 50%.
Data Spread
Data spread, or variability, is a key concept in statistics, exhibiting how much the data points differ from each other. A higher data spread means the values are more spread out from the center, and a lower spread means the values are closer to the center. The range, IQR, and boxplot whiskers are all indicators of data spread.

The boxplot visually communicates the spread of data: the wider the box (representing the IQR), the greater the variability within the central portion of the data. Whiskers on boxplots extend to the minimum and maximum values, demonstrating the total spread. For example, in the exercise, Dataset 2 has a much larger range than the others, signaling that its values are more widely dispersed.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A somewhat surprising fact about coffee is that the longer it is roasted, the less caffeine it has. Thus an "extra bold" dark roast coffee actually has less caffeine than a light roast coffee. What is the explanatory variable and what is the response variable? Do the two variables have a negative association or a positive association?

2Does Sexual Frustration Increase the Desire for Alcohol? Apparently, sexual frustration increases the desire for alcohol, at least in fruit flies. Scientists \(^{35}\) randomly put 24 fruit flies into one of two situations. The 12 fruit flies in the "mating" group were allowed to mate freely with many available females eager to mate. The 12 in the "rejected" group were put with females that had already mated and thus rejected any courtship advances. After four days of either freely mating or constant rejection, the fruit flies spent three days with unlimited access to both normal fruit fly food and the same food soaked in alcohol. The percent of time each fly chose the alcoholic food was measured. The fruit flies that had freely mated chose the two types of food about equally often, choosing the alcohol variety on average \(47 \%\) of the time. The rejected males, however, showed a strong preference for the food soaked in alcohol, selecting it on average \(73 \%\) of the time. (The study was designed to study a chemical in the brain called neuropeptide that might play a role in addiction.) (a) Is this an experiment or an observational study? (b) What are the cases in this study? What are the variables? Which is the explanatory variable and which is the response variable? (c) We are interested in the difference in means, where the means measure the average percent preference for alcohol \((0.47\) and 0.73 in this case). Find the difference in means and give the correct notation for your answer, using the correct notation for a mean, subscripts to identify groups, and a minus sign. (d) Can we conclude that rejection increases a male fruit fly's desire for alcohol? Explain.

The Honeybee dataset contains data collected from the USDA on the estimated number of honeybee colonies (in thousands) for the years 1995 through 2012.77 We use technology to find that a regression line to predict number of (thousand) colonies from year (in calendar year) is $$\text { Colonies }=19,291,511-8.358(\text { Year })$$ (a) Interpret the slope of the line in context. (b) Often researchers will adjust a year explanatory variable such that it represents years since the first year data were colleected. Why might they do this? (Hint: Consider interpreting the yintercept in this regression line.) (c) Predict the bee population in \(2100 .\) Is this prediction appropriate (why or why not)?

Online Cat Videos In Exercise 1.59 on page 28 , we introduced a study looking at the effect of watching cat videos on mood and energy. The authors asked participants how many cats they currently own and report 2.39 as the measure of center for this right-skewed distribution. (a) Is this measure of center the mean number of cats or the median number of cats? (Hint: Think about how the two numbers are calculated.) (b) Would we expect the mean number of cats to be greater than or less than the median?

Find and interpret the z-score for the data value given. The value 8.1 in a dataset with mean 5 and standard deviation 2
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.