Go to www.pearsonhighered.com/sullivanstats and download the...

Go to www.pearsonhighered.com/sullivanstats and download the data $3_{-} 5_{-} 17$ using the file format of your choice. The data contains the depth (in kilometers) and magnitude, measured using the Richter Scale, of all earthquakes worldwide that occurred between $12 / 2 / 2014$ and $12 / 9 / 2014 .$ Depth is the distance below the surface of Earth at which the earthquake originates. A one unit increase in magnitude represents ground shaking ten times as strong (an earthquake with magnitude 4 is ten times as strong as an earthquake with magnitude 3 ). (a) Find the mean, median, range, standard deviation, and quartiles for both the depth and magnitude of the earthquakes. Based on the values of the mean, median, and quartiles conjecture the shape of the distribution for depth and magnitude. (b) Draw a relative frequency histogram for both depth and magnitude. Label the mean and median on the histogram. Describe the shape of each histogram. (c) Draw a box plot for both depth and magnitude. Are there outliers? (d) Determine the lower and upper fences for identifying outliers for both depth and magnitude.

Go to www.pearsonhighered.com/sullivanstats and download the data $3_{-} 5_{-} 17$ using the file format of your choice. The data contains the depth (in kilometers) and magnitude, measured using the Richter Scale, of all earthquakes worldwide that occurred between $12 / 2 / 2014$ and $12 / 9 / 2014 .$ Depth is the distance below the surface of Earth at which the earthquake originates. A one unit increase in magnitude represents ground shaking ten times as strong (an earthquake with magnitude 4 is ten times as strong as an earthquake with magnitude 3 ).
(a) Find the mean, median, range, standard deviation, and quartiles for both the depth and magnitude of the earthquakes. Based on the values of the mean, median, and quartiles conjecture the shape of the distribution for depth and magnitude.
(b) Draw a relative frequency histogram for both depth and magnitude. Label the mean and median on the histogram. Describe the shape of each histogram.
(c) Draw a box plot for both depth and magnitude. Are there outliers?
(d) Determine the lower and upper fences for identifying outliers for both depth and magnitude.

Key Concepts

Outlier Detection and Fences

Outliers are data points that deviate markedly from the rest of the dataset. A common method for detecting them involves calculating 'fences' using the interquartile range (IQR). The lower fence is usually defined as Q1 minus 1.5 times the IQR, and the upper fence as Q3 plus 1.5 times the IQR. Data points that fall outside these fences are considered potential outliers, which could indicate variability in measurement or interesting anomalies worth further investigation.

Box Plot

A box plot is a visual summary of important descriptive statistics, including the median, quartiles, and potential outliers. It displays the central 50% of the data in a box, with 'whiskers' extending to the smallest and largest values within a calculated limit. Box plots are effective tools for comparing distributions between different datasets and highlighting symmetry, spread, and anomalies.

Relative Frequency Histogram

A relative frequency histogram is a graphical representation of the distribution of data where the height of each bar reflects the proportion of data within a given interval. Such histograms are useful to visualize the overall distribution, identify patterns, and observe how frequently certain data ranges occur. Annotating features like the mean and median on the histogram further helps to contextualize these central tendency measures within the spread of data.

Range and Standard Deviation

These are measures of dispersion that reveal how spread out the data is. The range indicates the difference between the maximum and minimum values, giving an overall sense of spread. The standard deviation provides a more nuanced understanding by measuring the average distance of data points from the mean, thus capturing variability within the distribution.

Quartiles and the Interquartile Range

Quartiles divide the data into four equal parts and are essential for understanding the data's distribution. The first quartile (Q1), the median (Q2), and the third quartile (Q3) provide benchmarks for the lower, middle, and upper parts of the dataset. The Interquartile Range (IQR), defined as Q3 minus Q1, is particularly valuable for identifying the spread of the central half of the data and is used in methods for detecting outliers.

Mean, Median, and Other Measures of Central Tendency

These summary statistics quantitatively describe the 'center' of a data set. The mean is the arithmetic average, while the median is the middle value when the data is ordered. Together, they provide insight into the central location of the data and help in understanding how representative the average is, especially in distributions with outliers or skewness.

Distribution Shape Analysis

By comparing central tendency measures like the mean and median, as well as analyzing the quartiles, one can infer the shape of the distribution. For example, a discrepancy between the mean and median may suggest skewness (asymmetry) in the data. Understanding the distribution shape is essential for selecting appropriate statistical models and methods.

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