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Chapter 2: Problem 1

Here are the data from Exercise 2.3 .10 on the number of virus-resistant bacteria in each of 10 aliquots: $$ \begin{array}{lllll} 14 & 15 & 13 & 21 & 15 \\ 14 & 26 & 16 & 20 & 13 \end{array} $$ (a) Determine the median and the quartiles. (b) Determine the interquartile range. (c) How large would an observation in this data set have to be in order to be an outlier?

Short Answer

Expert verified
Median is 15, Q1 is 14, Q3 is 20.5, IQR is 6.5. Any observation less than 4.25 or greater than 30.25 would be considered an outlier.

Step by step solution

01

Arrange the Data in Ascending Order

List the data values in order from smallest to largest. The arranged data will be used to find the median and quartiles.
02

Determine the Median

The median is the middle value of the arranged data set. If the number of observations is odd, the median is the middle number. If even, it is the average of the two middle numbers. In this data set with 10 values, the median will be the average of the 5th and 6th values.
03

Determine the First Quartile (Q1)

The first quartile (Q1) is the median of the lower half of the data (excluding the median if the number of values is odd).
04

Determine the Third Quartile (Q3)

The third quartile (Q3) is the median of the upper half of the data (excluding the median if the number of values is odd).
05

Calculate the Interquartile Range (IQR)

The interquartile range is the difference between the third and first quartiles (Q3 - Q1).
06

Determine Outliers

An outlier is defined as a value that is more than 1.5 times the IQR above the third quartile or below the first quartile. Calculate the lower and upper bounds for potential outliers.

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Key Concepts

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

Median
The median is a form of average that represents the middle point of a data set. It's where half the values are above and half are below. Unlike the mean, the median isn't affected by extremely high or low values, and therefore, gives a better idea of a 'typical' value when dealing with skewed distributions.

To find the median, first, list all numbers in the set from the smallest to the largest. For data with an odd number of entries, the median is the number right in the middle. If there's an even number, as in our exercise, the median is the average of the two middle numbers. In our case, after sorting, the 5th and 6th values are identified, their average is calculated and that's the median of our bacterial counts.
Quartiles
Quartiles divide a rank-ordered data set into four equal parts. These are Q1 (first quartile), Q2 (second quartile or median), and Q3 (third quartile). Think of them as 'milestones' along the data's range.

First Quartile (Q1)

Declares that 25% of the data falls below it. It's found by taking the median of the lower half of the dataset (not including the overall median if the data count is odd).

Third Quartile (Q3)

Indicates the point below which 75% of the data falls. It's the median of the upper half of your data set (again, excluding the median for odd-numbered sets). Knowing these allows one to understand how data spreads around the median.
Interquartile Range
The interquartile range (IQR) reflects the range within the middle 50% of the data. It is the difference between the third quartile (Q3) and first quartile (Q1). In simpler terms, IQR helps us understand the spread or variability of the middle part of our dataset, disregarding the extremes.

Calculating the IQR requires subtracting Q1 from Q3. This metric is crucial in detecting outliers since it defines the 'normal' spread around the median, giving us benchmarks for indicating whether certain data points are unusually distant from the rest of the sample.
Outliers
Outliers are data points that differ significantly from most of the data. They can arise due to variability in measurement or possibly experimental error; sometimes they just indicate a very unusual (but possible) event.

To identify potential outliers, we use the calculated IQR. Any value more than 1.5 times the IQR above Q3 or below Q1 is considered an outlier. Doing this for our dataset, we find the cutoff points by calculating the bounds. The values that fall outside these bounds would be potential outliers, warranting further scrutiny or analysis.

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Most popular questions from this chapter

The following boxplot shows the five-number summary for a data set. For these data the minimum is \(35,\) \(Q_{1}\) is \(42,\) the median is \(49, Q_{3}\) is \(56,\) and the maximum is 65 . Is it possible that no observation in the data set equals 42? Explain your answer.

The two claws of the lobster (Homarus americanus) are identical in the juvenile stages. By adulthood, however, the two claws normally have differentiated into a stout claw called a "crusher" and a slender claw called a "cutter." In a study of the differentiation process, 26 juvenile animals were reared in smooth plastic trays and 18 were reared in trays containing oyster chips (which they could use to exercise their claws). Another 23 animals were reared in trays containing only one oyster chip. The claw configurations of all the animals as adults are summarized in the table. \({ }^{31}\) $$ \begin{array}{|lccc|} \hline&& {\text { Claw Configuration }} \\ \text { Treatment } & \begin{array}{c} \text { Right } \\ \text { crusher, } \\ \text { left cutter } \end{array} & \begin{array}{c} \text { Right } \\ \text { cutter, } \\ \text { left crusher } \end{array} & \begin{array}{c} \text { Right and } \\ \text { left cutter } \\ \text { (no crusher) } \end{array} \\ \hline \text { Oyster chips } & 8 & 9 & 1 \\ \text { Smooth plastic } & 2 & 4 & 20 \\ \text { One oyster chip } & 7 & 9 & 7 \\ \hline \end{array} $$ (a) Create a stacked frequency bar chart to display these data. (b) Create a stacked relative frequency bar chart to display these data. (c) Of the two charts you created in parts (a) and (b), which is more useful for comparing the claw configurations across the three treatments? Why?

(i) identify the variable(s) in the study, (ii) for each variable tell the type of variable (e.g., categorical and ordinal, discrete, etc.), (iii) identify the observational unit (the thing sampled), and (iv) determine the sample size. (a) A biologist measured the body mass \((\mathrm{g})\) and sex of each of 123 blue jays. (b) A biologist measured the lifespan (in days), the thorax length (in \(\mathrm{mm}\) ), and the percent of time spent sleeping for each of 125 fruit flies.

Agronomists measured the yield of a variety of hybrid corn in 16 locations in Illinois. The data, in bushels per acre, were \(^{17}\) \(\begin{array}{llllll}241 & 230 & 207 & 219 & 266 & 167 \\ 204 & 144 & 178 & 158 & 153 & \\ 187 & 181 & 196 & 149 & 183 & \end{array}\) (a) Construct a dotplot of the data. (b) Describe the shape of the distribution.

The rowan (Sorbus aucuparia) is a tree that grows in a wide range of altitudes. To study how the tree adapts to its varying habitats, researchers collected twigs with $$ \begin{array}{|ccc|} \hline & \text { Altitude of origin } & \text { Respiration rate } \\ \text { Tree } & X(\mathrm{~m}) & Y(\mu \mathrm{l} / \mathrm{hr} \cdot \mathrm{mg}) \\ \hline 1 & 90 & 0.11 \\ 2 & 230 & 0.20 \\ 3 & 240 & 0.13 \\ 4 & 260 & 0.15 \\ 5 & 330 & 0.18 \\ 6 & 400 & 0.16 \\ 7 & 410 & 0.23 \\ 8 & 550 & 0.18 \\ 9 & 590 & 0.23 \\ 10 & 610 & 0.26 \\ 11 & 700 & 0.32 \\ 12 & 790 & 0.37 \\ \hline \end{array} $$ attached buds from 12 trees growing at various altitudes in North Angus, Scotland. The buds were brought back to the laboratory and measurements were made of the dark respiration rate. The accompanying table shows the altitude of origin (in meters) of each batch of buds and the dark respiration rate (expressed as \(\mu\) l of oxygen per hour per mg dry weight of tissue). \(^{33}\) (a) Create a scatterplot of the data. (b) If your software allows, add a regression line to summarize the trend. (c) If your software allows, create a scatterplot with a lowess smooth to summarize the trend.
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