/*! 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 86 Briefly describe how the three q... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 86

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
Quartiles are values that split the data set into four equal parts. For the first example with an odd number of observations \{1, 3, 5, 7, 9\}, the quartiles are Q1=3, Q2=5, Q3=7. For the second example with an even number of observations \{2, 4, 6, 8, 10, 12\}, the quartiles are Q1=3, Q2=7, Q3=11.

Step by step solution

01

Understanding Quartiles

The three quartiles in a data set are values that divide the data into four equal parts. The first quartile (Q1) represents the 25th percentile of the data, the second quartile (Q2) represents the 50th percentile or the median, and the third quartile (Q3) represents the 75th percentile.
02

Calculation with Odd Number of Observations

Imagine a data set with an odd number of observations, e.g. \{1, 3, 5, 7, 9\}. Here, the second quartile (Q2) is the median of the entire data set, which can be obtained by {(n+1) / 2}th observation = 3rd observation, which is 5. The first quartile (Q1) is the median of the lower half not including Q2, which is the 2nd observation, which is 3. The third quartile (Q3) is the median of the upper half not including Q2; thus, it is the 4th observation, or 7.
03

Calculation with Even Number of Observations

Now consider a data set with an even number of observations, e.g. \{2, 4, 6, 8, 10, 12\}. The second quartile (Q2) here is the average of the two middle values (6 + 8) / 2 = 7. The first quartile (Q1) is the median of the lower half, which here is (4 + 2) / 2 = 3. The third quartile (Q3) is the median of the upper half, thus it is (10 + 12) / 2 = 11.

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!

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

The following data represent the total points scored in each of the NFL championship games played from 2000 through 2009 in that order. \(\begin{array}{lllllllllll}39 & 41 & 37 & 69 & 61 & 45 & 31 & 46 & 31 & 50\end{array}\)

The prices of all college textbooks follow a bell-shaped distribution with a mean of \(\$ 105\) and a standard deviation of \(\$ 20\). a. Using the empirical rule, find the percentage of all college textbooks with their prices between i. \(\$ 85\) and \(\$ 125\) ii. \(\$ 65\) and \(\$ 145\) \({ }^{*} \mathrm{~b}\). Using the empirical rule, find the interval that contains the prices of \(99.7 \%\) of college textbooks.

The mean monthly mortgage paid by all home owners in a town is \(\$ 2365\) with a standard deviation of \(\$ 340\) a. Using Chebyshev's theorem, find at least what percentage of all home owners in this town pay a monthly mortgage of i. \(\$ 1685\) to \(\$ 3045\) ii. \(\$ 1345\) to \(\$ 3385\) \({ }^{*} \mathbf{b}\). Using Chebyshev's theorem, find the interval that contains the monthly mortgage payments of at least \(84 \%\) of all home owners.

The following data give the numbers of new cars sold at a dealership during a 20-day period. \(\begin{array}{llrlrlrlrll}8 & 5 & 1 & 2 & 3 & 9 & 1 & 06 & 1 & 28 & 8 & \\ 4 & 1 & 61 & 01 & 17 & 7 & 3 & 5 & 9 & 1 & 1\end{array}\) Make a box-and-whisker plot. Comment on the skewness of these data.

When studying phenomena such as inflation or population changes that involve periodic increases or decreases, the geometric mean is used to find the average change over the entire period under study. To calculate the geometric mean of a sequence of \(n\) values \(x_{1}, x_{2}, \ldots, x_{n}\), we multiply them together and then find the \(n\) th root of this product. Thus $$ \text { Geometric mean }=\sqrt[n]{x_{1} \cdot x_{2} \cdot x_{3} \cdot \ldots \cdot x_{n}} $$ Suppose that the inflation rates for the last five years are \(4 \%, 3 \%, 5 \%, 6 \%\), and \(8 \%\), respectively. Thus at the end of the first year, the price index will be \(1.04\) times the price index at the beginning of the year, and so on. Find the mean rate of inflation over the 5 -year period by finding the geometric mean of the data set \(1.04,1.03,1.05,1.06\), and \(1.08 .\) (Hint: Here, \(n=5, x_{1}=1.04, x_{2}=1.03\), and so on. Use the \(x^{1 / n}\) key on your calculator to find the fifth root. Note that the mean inflation rate will be obtained by subtracting 1 from the geometric mean.)
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

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.