/*! 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 1 Explain how the value of the med... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 1

Explain how the value of the median is determined for a data set that contains an odd number of observations and for a data set that contains an even number of observations.

Short Answer

Expert verified
For an odd number of data points, the median is the value at position \((n+1)/2\), where n is the number of observations, in the ordered dataset. For an even number of data points, the median is the average of the two middle numbers, at positions \(n/2\) and \(n/2 + 1\) in the ordered dataset.

Step by step solution

01

Determine the Median for Odd Number of Observations

For a dataset with an odd number of observations, first, arrange the data in ascending or descending order. The median is then the value exactly in the middle of this arrangement. If the number of observations is \(n\), the position of the median can be determined using the formula \(\frac{n+1}{2}\). This will tell you the position of the median in the ordered dataset.
02

Determine the Median for Even Number of Observations

If the number of data points is even, after arranging the data in ascending or descending order, there will be two values in the middle. The median is calculated as the average of these two. If the total number of observations is \(n\), the two central positions are \(\frac{n}{2}\) and \(\frac{n}{2} + 1\). To find the median, take these two values from the ordered dataset and calculate their average: \(\frac{value1 + value2}{2}\).

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Most popular questions from this chapter

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.)

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