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Chapter 15: Problem 78

Show that if \(A\) is the mean and \(M\) is the median of the data set \(\\{1,2,3, \ldots, N\\},\) then for all values of \(N, A=M\).

Short Answer

Expert verified
After calculating both the mean and median for the arithmetic sequence of numbers from 1 to N, we find A = M = (N + 1) / 2 for all N.

Step by step solution

01

Determine the Mean

To find the mean (A), sum up all the values in the data set and divide by the number of values. The data set is an arithmetic sequence with the first term a_1 = 1, the last term a_N = N, and N terms in total. The mean A is given by the formula A = (a_1 + a_N) / 2, which simplifies to A = (1 + N) / 2.
02

Determine the Median

The median (M) is the middle value of the data set when the values are arranged in ascending order. For odd N, M is the middle value, (N+1)/2. For even N, M is the average of the two middle values which gives (N/2 + (N/2 + 1)) / 2, which simplifies to (N + 1) / 2.
03

Compare Mean and Median

For both odd and even N, the mean A and median M have been found to be (N + 1) / 2. This means that A = M for all values of N.

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Key Concepts

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

Mathematical Statistics
Mathematical statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It provides us with methods to make sense of data by allowing us to describe, summarize, and infer information.

In the given exercise, mathematical statistics helps us understand the relationship between two measures of central tendency: the mean and the median. By using a formulaic approach from this field, we are able to show that, within an arithmetic sequence, these two measures coincide, regardless of the number of terms in the sequence. This result emphasizes the power of mathematical statistics in explaining patterns and relationships within data sets.
Arithmetic Sequence
An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. It's a simplistically structured progression that can be visualized easily and has predictable characteristics which can be used to analyze data sets.

In our exercise, the data set \(\{1,2,3, \ldots, N\}\) is a perfect example of such a sequence, with a common difference of 1. This regularity allows for straightforward computation of mean and median, demonstrating the simplicity and efficiency of arithmetic sequences in data analysis. When we apply formulas for the arithmetic sequence to find the mean and median, we are able to show concisely that they equal each other irrespective of the number of terms.
Data Set Analysis
Data set analysis involves inspecting, cleansing, transforming, and modeling data with the goal of uncovering useful information, concluding, and supporting decision-making. Measures of central tendency, such as the mean and median, are crucial in data set analysis as they provide a single value that is representative of a collection of data.

The exercise we are discussing is a fundamental example of data set analysis, where we observe a known integer sequence to draw conclusions about its central tendencies. This type of analysis not only supports academic pursuits but also aids in real-world applications where understanding the typical value of a data set is key.
Central Tendency
Central tendency is a statistical measure that identifies a single value as representative of an entire distribution. It aims to provide an accurate picture of the 'center' of a data set. The mean and median are two of the most commonly used measures of central tendency.

In relation to our exercise, the mean is calculated as the average of all numbers in the sequence, while the median is the middle value when all numbers are listed in order. The fact that the mean and median are equivalent for an arithmetic sequence is a distinctive property that highlights the symmetry of the sequence. This concept is fundamental in understanding the homogeneity of certain types of data sets and the mean-median relationship is pivotal in the study of the shape and distribution of data.

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Most popular questions from this chapter

A data set is called constant if every value in the data set is the same. Explain why any data set with standard deviation 0 must be a constant data set.

The purpose of Exercises 55 through 58 is to practice computing standard deviations by using the definition. 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) \\{5,5,5,5\\} (b) \\{0,5,5,10] (c) \\{0,10,10,20\\}

Table 16 shows the percentage of U.S. working married couples in which the wife's income is higher than the husband's \((1999-2009)\) (a) Draw a pictogram for the data in Table 16 . Assume you are trying to convince your audience that things are looking great for women in the workplace and that women's salaries are catching up to men's very quickly. (b) Draw a different pictogram for the data in Table 16 , where you are trying to convince your audience that women's salaries are catching up with men's very slowly. $$ \begin{array}{|l|c|c|c|c|c|c|}= \hline \text { Year } & 1999 & 2000 & 2001 & 2002 & 2003 & 2004 \\ \hline \text { Percent } & 28.9 & 29.9 & 30.7 & 31.9 & 32.4 & 32.6 \\ \hline \text { Year } & 2005 & 2006 & 2007 & 2008 & 2009 & \\\ \hline \text { Percent } & 33.0 & 33.4 & 33.5 & 34.5 & 37.7 & \end{array} $$

Chebyshev's theorem. The Russian mathematician P. L. Chebyshev \((1821-1894)\) showed that for any data set and any constant \(k\) greater than \(1,\) at least \(1-\left(1 / k^{2}\right)\) of the data must lie within \(k\) standard deviations on either side of the mean \(A\). For example, when \(k=2\), this says that \(1-\frac{1}{4}=\frac{3}{4}(i, e ., 75 \%)\) of the data must lie within two standard deviations of \(A\) (i.e., somewhere between \(A-2 o\) and \(A+2 \sigma)\) (a) Using Chebyshev's theorem, what percentage of a data set must lie within three standard deviations of the mean? (b) How many standard deviations on each side of the mean must we take to be assured of including \(99 \%\) of the data? (c) Suppose that the average of a data set is \(A\). Explain why there is no number \(k\) of standard deviations for which we can be certain that \(100 \%\) of the data lies within \(k\) standard deviations on either side of the \(\operatorname{mean} A\)

Suppose that the standard deviation of the data set \(\left\\{x_{1}, x_{2}, x_{3} \ldots . x_{N}\right\\}\) is \(d\). Explain why the standard deviation of the data set \(\left\\{a \cdot x_{1}, a \cdot x_{2}, a \cdot x_{3}, \ldots, a \cdot x_{N}\right\\}\) (where \(a\) is a positive number) is \(a \cdot \sigma\).
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