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Chapter 15: Problem 63

(a) Give an example of 10 numbers with an average less than the median. (b) Give an example of 10 numbers with a median less than the average. (c) Give an example of 10 numbers with an average less than the first quartile. (d) Give an example of 10 numbers with an average more than the third quartile.

Short Answer

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(a) Example could be {1, 2, 3, 4, 5, 6, 7, 8, 9, 20} with average 6.5 and median 7. \n(b) Example could be {1, 2, 3, 4, 5, 6, 7, 8, 100, 200} with average 33.6 and median 6.5. \n(c) Example could be {1, 2, 2, 3, 4, 10, 11, 12, 13, 14} with average 7.2 and first quartile 2.75. \n(d) Example could be {5, 6, 7, 8, 9, 10, 11, 99, 100, 101} with average 35.6 and third quartile 10.75.

Step by step solution

01

Part (a) - An example of 10 numbers with an average less than the median

Choose 10 numbers such that the average is less than the median. For instance, {1, 2, 3, 4, 5, 6, 7, 8, 9, 20}. The average of these is 6.5 while the median is 7.
02

Part (b) - An example of 10 numbers with a median less than the average

Choose 10 numbers such that the median is less than the average. For instance, numbers {1, 2, 3, 4, 5, 6, 7, 8, 100, 200}. Here the median of this set is 6.5, while the average is 33.6.
03

Part (c) - An example of 10 numbers with an average less than the first quartile

Choose 10 numbers such that the average is less than the first quartile. For instance, numbers {1, 2, 2, 3, 4, 10, 11, 12, 13, 14}. The average of these numbers is 7.2, while the first quartile (25% percentile) is 2.75.
04

Part (d) - An example of 10 numbers with an average more than the third quartile

Choose 10 numbers such that the average is more than the third quartile. For instance, numbers {5, 6, 7, 8, 9, 10, 11, 99, 100, 101}. The average of these numbers is 35.6, while the third quartile (75% percentile) is 10.75.

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Key Concepts

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

Statistics
Statistics is a branch of mathematics that helps us make sense of complex data. It allows us to describe, analyze, and draw conclusions from sets of data. Among the most important features we analyze in statistics are measures of central tendency and dispersion.

Central tendency helps us understand where the center of a data set lies by using measures like the mean (average), median, and mode. Dispersion, on the other hand, describes how spread out the data is, using concepts like variance and standard deviation. Understanding these measures helps in identifying patterns and making informed decisions based on data.
  • Mean (Average): Sum of all data points divided by the number of data points.
  • Median: The middle value when all data points are ordered.
  • Mode: The most frequently occurring value in a data set.
Selecting the right measure of central tendency depends on the data and the situation you are analyzing. For example, the median is often more informative than the mean in skewed distributions.
Median and Average
The median and the average (mean) are crucial concepts in analyzing data sets. Each offers different insights and is useful in different scenarios.

The average is found by adding all the numbers in a data set and then dividing by the number of values. It's an excellent tool for understanding the general tendency of most data points. However, it can be heavily influenced by outliers or extreme values.
To calculate the average for the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 20}, you would sum these values to get 65 and then divide by 10, resulting in an average of 6.5.

The median, however, is the middle value of an ordered data set. If you have an even number of data points, you average the two middle numbers. It provides a better reflection of the 'central' value in a skewed distribution since it is not affected by outliers.
For the same set {1, 2, 3, 4, 5, 6, 7, 8, 9, 20}, the median is 7, the middle of the ordered sequence. In some cases, the median can be greater than the average (as in the example above) which suggests that the data is skewed towards lower numbers.
Quartiles
Quartiles are values that divide your data set into four equal parts or quarters. Each quartile represents 25% of the data set.

The first quartile (Q1) marks the cutoff for the lowest 25% of the data. It acts as a boundary within which the smallest values reside. Calculating the first quartile involves finding the median of the lower half of the data set.
For example, in the set {1, 2, 2, 3, 4, 10, 11, 12, 13, 14}, the first quartile is 2.75. This means 25% of the numbers are 2.75 or less.
  • Second Quartile (Median or Q2): Divides the data set in half.
  • Third Quartile (Q3): Marks the cutoff for the highest 25% of the data.
The third quartile indicates the upper 25% boundary of your data set. In {5, 6, 7, 8, 9, 10, 11, 99, 100, 101}, the third quartile is 10.75, showing that 75% of the numbers are below this value.
Quartiles offer a robust way to understand the spread of data across its full range. They are particularly useful when comparing different data sets or identifying outliers and trends.

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Most popular questions from this chapter

Mike's average on the first five exams in Econ \(1 \mathrm{~A}\) is 88 What must he earn on the next exam to raise his overall average to \(90 ?\)

For each data set, find the 10 th and the 25 th percentiles. (a) \(\\{1,2,3, \ldots, 49,50,50,49, \ldots, 3,2,1\\}\) (b) \(\\{1,2,3, \ldots, 49,50,49, \ldots, 3,2,1\\}\) (c) \(\\{1,2,3, \ldots, 49,49, \ldots, 3,2,1\\}\)

Refer to Table \(15-13,\) which gives the home-to-school distance \(d\) (rounded to the nearest half-mile) for each of the 27 kindergarten students at Cleansburg Elementary School. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \boldsymbol{d} \\ \hline 1362 & 1.5 & 3921 & 5.0 \\ \hline 1486 & 2.0 & 4355 & 1.0 \\ \hline 1587 & 1.0 & 4454 & 1.5 \\ \hline 1877 & 0.0 & 4561 & 1.5 \\ \hline 1932 & 1.5 & 5482 & 2.5 \\ \hline 1946 & 0.0 & 5533 & 1.5 \\ \hline 2103 & 2.5 & 5717 & 8.5 \\ \hline 2877 & 1.0 & 6307 & 1.5 \\ \hline 2964 & 0.5 & 6573 & 0.5 \\ \hline 3491 & 0.0 & 8436 & 3.0 \\ \hline 3588 & 0.5 & 8592 & 0.0 \\ \hline 3711 & 1.5 & 8964 & 2.0 \\ \hline 3780 & 2.0 & 9205 & 0.5 \\ \hline & & 9658 & 6.0 \\ \hline \end{array} $$ (a) Make a frequency table for the distances in Table \(15-13 .\) (b) Draw a line graph for the data in Table \(15-13\).

Refer to the data set shown in Table \(15-12 .\) The table shows the scores on a Chem 103 test consisting of 10 questions worth 10 points each. $$ \begin{array}{c|c|c|c} \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } & \begin{array}{c} \text { Student } \\ \text { ID } \end{array} & \text { Score } \\ \hline 1362 & 50 & 4315 & 70 \\ \hline 1486 & 70 & 4719 & 70 \\ \hline 1721 & 80 & 4951 & 60 \\ \hline 1932 & 60 & 5321 & 60 \\ \hline 2489 & 70 & 5872 & 100 \\ \hline 2766 & 10 & 6433 & 50 \\ \hline 2877 & 80 & 6921 & 50 \\ \hline 2964 & 60 & 8317 & 70 \\ \hline 3217 & 70 & 8854 & 100 \\ \hline 3588 & 80 & 8964 & 80 \\ \hline 3780 & 80 & 9158 & 60 \\ \hline 3921 & 60 & 9347 & 60 \\ \hline 4107 & 40 & & \end{array} $$ Suppose that the grading scale for the test is \(\mathrm{A}: 80-100 ; \mathrm{B}:\) \(70-79 ; \mathrm{C}: 60-69 ; \mathrm{D}: 50-59 ;\) and \(\mathrm{F}: 0-49\) (a) Make a frequency table for the distribution of the test grades. (b) Draw a relative frequency bar graph for the test grades.

The purpose is to practice computing standard deviations the old fashioned way (by hand). Granted, computing standard deviations this way is not the way it is generally done in practice; a good calculator (or a computer package) will do it much faster and more accurately. The point is that computing a few standard deviations the old-fashioned way should help you understand the concept a little better. If you use a calculator or a computer to answer these exercises, you are defeating their purpose. Find the standard deviation of each of the following data sets. (a) \\{3,3,3,3\\} (b) \\{0,6,6,8\\} (c) \\{-6,0,0,18\\} (d) \\{6,7,8,9,10\\}
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