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Chapter 3: Problem 103

The following data give the 2009 estimates of crude oil reserves (in billions of barrels) of Saudi Arabia, Iran, Iraq, Kuwait, Venezuela, the United Arab Emirates, Russia, Libya, Nigeria, Canada, the United States, China, Brazil, and Mexico (source: www.eia.gov). \(\begin{array}{rrrrrrr}266.7 & 136.2 & 115.0 & 107.0 & 99.4 & 97.8 & 60.0 \\ 43.7 & 36.2 & 27.7 & 21.3 & 16.0 & 12.6 & 10.5\end{array}\) Prepare a box-and-whisker plot. Is the distribution of these data symmetric or skewed? Are there any outliers? If so, classify them as mild or extreme.

Short Answer

Expert verified
The oil reserve data is skewed right. There is one extreme outlier at 266.7 billion barrels.

Step by step solution

01

Prepare Data

Reorganize the data into numerical order. The sorted data are: \[10.5, 12.6, 16.0, 21.3, 27.7, 36.2, 43.7, 60.0, 97.8, 99.4, 107.0, 115.0, 136.2, 266.7\]
02

Compute Five-Number Summary

Determine the five-number summary which includes minimum, first quartile (Q1), median, third quartile (Q3), and maximum. Here it goes: Minimum = 10.5, Q1 = 21.3, Median = 60.0, Q3 = 107.0, Maximum = 266.7.
03

Plotting a Box-and-Whisker Plot

Draw a number line that covers the data range. Draw a box with left edge at Q1 and right edge at Q3. Draw a line (whisker) from Q1 to the minimum and another whisker from Q3 to the maximum. Draw a line in the box for the median.
04

Evaluate Distribution Symmetry

Look at the boxplot to evaluate if the data is symmetric or skewed. The data is skewed to the right because upper whisker is longer than the lower whisker, and because Q3 to maximum is greater than minimum to Q1.
05

Detect and Classify Outliers

Calculate $1.5*Q_{IQR}$ (Q3 - Q1), which is $1.5*(107.0 - 21.3) = 128.55$. Any data point below $Q1 - 1.5*Q_{IQR}= -107.25$ or above $Q3 + 1.5*Q_{IQR}= 236.55$ is an outlier. Only 266.7 qualifies, so it is the only extreme outlier.

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Key Concepts

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

Crude Oil Reserves
Crude oil reserves are essentially the untapped oil that can be extracted. They are crucial for countries as they impact their economy and energy policies. In our data, various countries have reported their reserves which range from 10.5 to 266.7 billion barrels. Knowing the reserves helps in planning for future energy needs and economic development.
  • The term 'reserves' refers to oil that is currently recoverable.
  • Countries with larger reserves tend to have considerable influence in the global oil market.
  • The distribution of reserves across countries can highlight global energy distributions.
Understanding the distribution of these reserves gives insight into energy security and market dynamics, making it essential for strategizing in both national and global contexts.
Data Skewness
Data skewness refers to the asymmetry in the distribution of data points. In a box-and-whisker plot, this term helps us understand how data is spread relative to its median. Skewness affects the conclusions drawn about a dataset鈥檚 trend and central tendency.
  • A right-skewed or positively skewed distribution has a longer tail on the right.
  • In our example, the reserves' distribution is right-skewed due to heavier values pulling the mean higher than the median.
  • Typically, skewness can indicate data outliers or highlight underlying patterns.
Recognizing skewness in data is crucial because it helps investors, analysts, and policymakers interpret the real implications of data much more accurately.
Five-Number Summary
The five-number summary provides a snapshot of a dataset's distribution. It consists of the minimum, first quartile (Q1), median, third quartile (Q3), and maximum value.
  • Minimum: The smallest data value (10.5 in the exercise).
  • First Quartile (Q1): The median of the first half of data (21.3).
  • Median: The middle value (60.0).
  • Third Quartile (Q3): The median of the second half of data (107.0).
  • Maximum: The largest data value (266.7).
With this summary, we can understand how data values cluster around the center and how spread out the values are. It forms the backbone of a box-and-whisker plot and further insights like identifying skewness or outliers rely on understanding this foundational analysis.
Outliers Classification
Outliers are data points significantly different from others in a dataset. Identifying these points is vital because they can affect the mean and skew the distribution.
  • They are classified based on their position relative to the rest of the data, quantified using formulas like the IQR method.
  • Mild outliers appear within 1.5 to 3 times the interquartile range (IQR) beyond the quartiles.
  • Extreme outliers, such as the 266.7 value in our dataset, lie beyond 3 times the IQR.
Being aware of outliers is important as they might represent anomalies, errors, or significant insight, thereby affecting predictions and statistical conclusions. Their correct identification and classification guide better decision-making and accurate data representations.

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