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Chapter 3: Problem 9

Each of the following data sets has a mean of \(\bar{x}=10 .\) \(\begin{array}{llllll}\text { (i) } 8 & 9 & 10 & 11 & 12\end{array}\) \(\begin{array}{llllll}\text { (ii) } 7 & 9 & 10 & 11 & 13\end{array}\) \(\begin{array}{llllll}\text { (iii) } 7 & 8 & 10 & 12 & 13\end{array}\) (a) Without doing any computations, order the data sets according to increasing value of standard deviations. (b) Why do you expect the difference in standard deviations between data sets (i) and (ii) to be greater than the difference in standard deviations between data sets (ii) and (iii)? Hint: Consider how much the data in the respective sets differ from the mean.

Short Answer

Expert verified
Order is (i), (ii), (iii); (i) to (ii) has greater difference due to larger range introduction.

Step by step solution

01

Understanding the Concept of the Mean

We are given three data sets, each with a mean value of 10. The mean, or average, is calculated by summing all the values in a set and dividing by the number of values. This exercise requires us to examine the spread of the data, not calculate the mean.
02

Defining Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation means the values tend to be close to the mean, while a high standard deviation means they are spread out over a wider range.
03

Ordering Data Sets by Standard Deviation Intuition

Look at the data values in each set. In set (i), the values are 8, 9, 10, 11, and 12, which are evenly spaced around the mean, indicating a smaller spread. In set (ii), values shift slightly further at the ends as 7 and 13 appear, which suggests a larger spread than set (i). In set (iii), numbers like 7, 8, 12, and 13 create even greater distances from the mean compared to (ii). Thus, the order by increasing standard deviation should be: (i), (ii), (iii).
04

Comparing Differences in Standard Deviation Between Data Sets

To understand why the difference in standard deviation between (i) and (ii) is greater than between (ii) and (iii), consider how the data differs from the mean. Set (i) has numbers evenly distributed around the mean, while set (ii) has one additional extreme value (7) compared to set (i), creating greater deviation from the mean. Set (iii) further spreads the numbers without drastically increasing the range compared to set (ii), so the increase in standard deviation is less drastic.

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Key Concepts

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

Understanding the Mean
The mean is a key statistical measure that represents the central point of a data set. It is the sum of all the data points divided by the number of points. For example, if you have the numbers 8, 9, 10, 11, and 12, you first add them up, which gives you 50, and then divide by 5, resulting in a mean of 10. In this exercise, each of the given data sets has a mean of 10, providing a straightforward comparison.
  • The mean is a way to find the 'average' but does not always reflect the variety within a data set.
  • Having the same mean does not imply that data sets will have the same level of variation or dispersion.
  • It's important to consider other statistics, like standard deviation, to fully understand the distribution of data.
Exploring Data Sets
A data set comprises a collection of numbers or values. In statistics, examining the features of data sets enables us to understand more about the underlying phenomenon the data represents.
When comparing multiple data sets with the same mean, as in this exercise, analyze how values in each set deviate from that average.
  • Linear data sets, like 8, 9, 10, 11, 12, have values streamlined around the mean, suggesting less variation.
  • Data sets such as 7, 9, 10, 11, 13 introduce more variation around the average, adding complexity to their evaluation.
  • Different numbers introduce different degrees of skewness, affecting statistical concepts like standard deviation.
Reflecting on these aspects helps understand why statistics beyond mean are vital in assessing data's behavior.
Variation or Dispersion in Data Sets
Dispersion or variation in a data set refers to how spread out the numbers are. The standard deviation is the most common measure used to quantify this dispersion.
A lower standard deviation means the values cluster closely to the mean and there's less variation. Conversely, a higher standard deviation indicates that values are more spread out and vary more.
This is evident in the given data sets:
  • Data set (i) with values closer to each other and the mean shows minimal dispersion, hence a lower standard deviation.
  • Data set (ii) has values like 7 and 13 that introduce more variation than set (i), presenting a moderate dispersion level.
  • Finally, data set (iii) expands the distance between numbers even further, resulting in a higher standard deviation.
The difference in standard deviation between sets (i) and (ii) is noticeable due to the introduction of these outliers; however, the movement in set (iii) causes a lesser addition of dispersion compared to set (ii), making the differences smaller.

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Most popular questions from this chapter

Consider two data sets with equal sample standard deviations. The first data set has 20 data values that are not all equal, and the second has 50 data values that are not all equal. For which data set is the difference between \(s\) and \(\sigma\) greater? Explain. Hint: Consider the relationship \(\sigma=s \sqrt{(n-1) / n}\).

The Grand Canyon and the Colorado River are beautiful, rugged, and sometimes dangerous. Thomas Myers is a physician at the park clinic in Grand Canyon Village. Dr. Myers has recorded (for a 5 -year period) the number of visitor injuries at different landing points for commercial boat trips down the Colorado River in both the Upper and Lower Grand Canyon (Source: Fateful Journey by Myers, Becker, and Stevens). Upper Canyon: Number of Injuries per Landing Point Between North Canyon and Phantom Ranch \(\begin{array}{lllllllllll}2 & 3 & 1 & 1 & 3 & 4 & 6 & 9 & 3 & 1 & 3\end{array}\) Lower Canyon: Number of Injuries per Landing Point Between Bright Angel and Lava Falls \(\begin{array}{llllllllllllll}8 & 1 & 1 & 0 & 6 & 7 & 2 & 14 & 3 & 0 & 1 & 13 & 2 & 1\end{array}\) (a) Compute the mean, median, and mode for injuries per landing point in the Upper Canyon. (b) Compute the mean, median, and mode for injuries per landing point in the Lower Canyon. (c) Compare the results of parts (a) and (b). (d) The Lower Canyon stretch had some extreme data values. Compute a \(5 \%\) trimmed mean for this region, and compare this result to the mean for the Upper Canyon computed in part (a).

Consider the following types of data that were obtained from a random sample of 49 credit card accounts. Identify all the averages (mean, median, or mode) that can be used to summarize the data. (a) Outstanding balance on each account (b) Name of credit card (e.g., MasterCard, Visa, American Express, etc.) (c) Dollar amount due on next payment

Consider two data sets. Set \(\mathrm{A}: n=5 ; \bar{x}=10 \quad\) Set B: \(n=50 ; \bar{x}=10\) (a) Suppose the number 20 is included as an additional data value in Set \(\mathrm{A}\). Compute \(\bar{x}\) for the new data set. Hint: \(\Sigma x=n \bar{x} .\) To compute \(\bar{x}\) for the new data set, add 20 to \(\Sigma x\) of the original data set and divide by 6 . (b) Suppose the number 20 is included as an additional data value in Set \(\mathrm{B}\). Compute \(\bar{x}\) for the new data set. (c) Why does the addition of the number 20 to each data set change the mean for Set A more than it does for Set B?

If a data set has an even number of data, is it true or false that the median is never equal to a value in the data set? Explain.
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