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Chapter 10: Problem 35

What proportion of the observations from a normal sample would you expect to be marked by an asterisk on a boxplot?

Short Answer

Expert verified
Approximately 5% of observations.

Step by step solution

01

Understanding Boxplots and Normal Distribution

In a boxplot, outliers are marked by asterisks or dots. Observations may be considered outliers if they lie below the lower whisker or above the upper whisker. In a normal distribution, data is symmetrically distributed around the mean.
02

Calculating Whiskers in a Boxplot

In a boxplot, the lower whisker is typically set to the first quartile minus 1.5 times the interquartile range (IQR), and the upper whisker is the third quartile plus 1.5 times the IQR. Observations beyond these whiskers are deemed outliers.
03

Identifying Outliers for a Normal Distribution

In a normal distribution, roughly 95% of the data will lie within 1.96 standard deviations from the mean. This implies about 5% of the data may fall beyond these whiskers.
04

Estimating Proportion of Asterisks

Since the whiskers aim to encapsulate central 95% of the data, and outliers lie outside this, we can expect approximately 5% of the observations to be marked by an asterisk in a boxplot.

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Key Concepts

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

Normal Distribution
In statistics, a normal distribution is one of the most important concepts due to its frequent occurrence and useful properties. This distribution is often depicted as a bell-shaped curve, symmetric around the mean. The normal distribution describes how the values of a variable are distributed.

Some key characteristics of a normal distribution include:
  • The mean, median, and mode are all equal and located at the center of the distribution.
  • The curve is symmetric, which means that if you fold it down the middle, both sides would match perfectly.
  • The total area under the curve equals 1, representing the entirety of data points.
  • The spread of the distribution is determined by the standard deviation, which measures the average distance of each data point from the mean.
Understanding the normal distribution helps in estimating probabilities related to different statistical methodologies and tests.
Outliers
Outliers are observations that deviate significantly from other observations in a dataset. They can arise due to variability in the data, experimental measurement errors, or sometimes indicate a novel discovery that deviates from the expected norm.

In a boxplot, observations are considered outliers if they lie outside the "whiskers." More specifically:
  • Data points below the lower whisker (first quartile minus 1.5 times the interquartile range) are deemed lower outliers.
  • Data points above the upper whisker (third quartile plus 1.5 times the interquartile range) are identified as upper outliers.
Outliers are often marked by symbols such as asterisks or circles on a boxplot. While they can provide valuable insights, such as identifying variability or errors, they may also skew the analysis if not handled properly.
Interquartile Range
The interquartile range (IQR) is a statistical measure that represents the middle 50% of a dataset. It is calculated by subtracting the first quartile (Q1) from the third quartile (Q3).

This can be represented mathematically as:\[\text{IQR} = Q3 - Q1\]The IQR is an important aspect of data analysis because:
  • It provides a measure of variability, showing how spread out the central portion of the data is.
  • It is resistant to outliers, making it a robust way to understand the spread in your data.
  • It is used in boxplots to draw the whiskers, which help identify potential outliers.
Understanding the interquartile range helps in visualizing and summarizing the data distribution efficiently, making it easier to identify any unusual data points.

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Most popular questions from this chapter

Give an example of a probability distribution with increasing failure rate.

Let \(X_{1}, \ldots, X_{n}\) be a sample (i.i.d.) from a distribution function, \(F,\) and let \(F_{n}\) denote the ecdf. Show that $$ \operatorname{Cov}\left[F_{n}(u), F_{n}(v)\right]=\frac{1}{n}[F(m)-F(u) F(v)] $$ where \(m=\min (u, v) .\) Conclude that \(F_{n}(u)\) and \(F_{n}(v)\) are positively correlated: If \(F_{n}(u)\) overshoots \(F(u),\) then \(F_{n}(v)\) will tend to overshoot \(F(v)\)

Various chemical tests were conducted on beeswax by White, Riethof, and Kushnir (1960). In particular, the percentage of hydrocarbons in each sample of wax was determined. a. Plot the ecdf, a histogram, and a normal probability plot of the percentages of hydrocarbons given in the following table. Find the \(.90, .75, .50, .25,\) and .10 quantiles. Does the distribution appear Gaussian? a. Plot the ecdf, a histogram, and a normal probability plot of the percentages of hydrocarbons given in the following table. Find the \(.90, .75, .50, .25,\) and .10 quantiles. Does the distribution appear Gaussian? $$\begin{array}{llllllll} 14.27 & 14.80 & 12.28 & 17.09 & 15.10 & 12.92 & 15.56 & 15.38 \\ 15.15 & 13.98 & 14.90 & 15.91 & 14.52 & 15.63 & 13.83 & 13.66 \\ 13.98 & 14.47 & 14.65 & 14.73 & 15.18 & 14.49 & 14.56 & 15.03 \\ 15.40 & 14.68 & 13.33 & 14.41 & 14.19 & 15.21 & 14.75 & 14.41 \\ 14.04 & 13.68 & 15.31 & 14.32 & 13.64 & 14.77 & 14.30 & 14.62 \\ 14.10 & 15.47 & 13.73 & 13.65 & 15.02 & 14.01 & 14.92 & 15.47 \\ 13.75 & 14.87 & 15.28 & 14.43 & 13.96 & 14.57 & 15.49 & 15.13 \\ 14.23 & 14.44 & 14.57 & & & \end{array}$$ b. The average percentage of hydrocarbons in microcrystalline wax (a synthetic commercial wax is \(85 \%\). Suppose that beeswax was diluted with \(1 \%\) micro- crystalline wax. Could this be detected? What about a \(3 \%\) or a \(5 \%\) dilution? (Such questions were one of the main concerns of the beeswax study.)

Consider a sample of size 100 from an exponential distribution with parameter \(\lambda=1\) a. Sketch the approximate standard deviation of the empirical log survival function, \(\log S_{n}(t),\) as a function of \(t\) b. Generate several such samples of size 100 on a computer and for each sample plot the empirical log survival function. Relate the plots to your answer to (a).

Olive oil from Spain, Tunisia, and other countrics is imported into Italy and is then repackaged and exported with the label "Imported from Italy." Olive oils from different places have distinctive tastes. Can the oils from different regions and areas in Italy be distinguished based on their combinations of fatty acids? This question was considered by Forina et al. (1983). The data consists of the percentage composition of 8 fatty acids (palmitic, palmitoleic, stearic, oleic, linoleic, linolenic, arachidic, eicosenoic) found in the lipid fraction of 572 Italian olive oils. There are 9 collection areas, 4 from southern Italy (North and South Apulia, Calabria, Sicily), two from Sardinia (Inland and Coastal), and 3 from northern Italy (Umbria, East and West Liguria). The file olive contains the following variables for cach of the 572 samples: Region: South, North, or Sardinia Area (subregions within the larger regions): North and South Apulia, Calabria, Sicily, Inland and Coastal Sardinia, Umbria, East and West Liguria Palmitic Acid Percentage Palmitoleic Acid Percentage Stearic Acid Percentage Oleic Acid Percentage Linoleic Acid Percentage Linolenic Acid Percentage Arachidic Acid Percentage Eicoscnoic Acid Perccntage Examine this data with the aim of distinguishing between regions and areas by using fatty acid composition. a. Make a table of the mean and median values of percentages for each area, grouping the areas within regions. b. Complement the analysis by making parallel boxplots. Which variables look promising for separating the regions? c. It is possible that the regions can be more clearly separated by considering pairs of variables. Use the variables that appear to be informative from the analysis up to this point to make scatterplots. How well can the regions be scparated based on the scatterplots? d. How well can the areas within regions be distinguished? e. By interactively rotating point clouds, one can examine relationships among more than two variables at a time. Try this with the software ggobi available at ht tp: / /www. ggobi. oxg/.
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