Study Notes on Statistical Skewness and Distribution Analysis

Statistical Box Plot Analysis Exercises

Exercise 16 involves the analysis of a box plot representing interstate credit charges per hour for four-year degrees of graduate students at public universities. The y-axis of the chart ranges from $0$ to $1500$ with major increments every $300$ units ($0, 300, 600, 900, 1200, 1500$). To solve this, students are required to calculate the median, identify the first ($Q_1$) and third ($Q_3$) quartiles, and determine the interquartile range ($IQR$). Furthermore, the exercise asks for the calculation of the threshold beyond which a value is considered an outlier, the identification of specific outliers, and a determination of whether the distribution is symmetric, positively skewed, or negatively skewed.

Exercise 17 focuses on a study of fuel efficiency (miles per gallon) for 2011 model cars. The dataset is characterized by a mean of $27.5\,mpg$ and a median of $26.8\,mpg$. The extreme values in the data are a minimum of $12.70\,mpg$ and a maximum of $50.20\,mpg$. The first quartile ($Q_1$) is reported as $17.95\,mpg$, and the third quartile ($Q_3$) is $35.45\,mpg$. The task is to construct a box plot and comment on the distribution and its symmetry.

Exercise 18 presents a sample of 28 timeshare departments in Orlando, Florida, listing daily rates for a one-bedroom suite. The data, ordered from smallest to largest, is as follows: $\$116$, $\$121$, $\$157$, $\$192$, $\$207$, $\$209$, $\$209$, $229$, $232$, $236$, $236$, $239$, $243$, $246$, $260$, $264$, $276$, $281$, $283$, $289$, $296$, $307$, $309$, $312$, $317$, $324$, $341$, and $353$. The objective is to construct a box plot, identify the first quartile, third quartile, and median, and provide a commentary on the distribution shape.

Concepts and Definitions of Skewness

Skewness, or "sesgo," refers to the lack of symmetry in a set of observations. While Chapter 3 introduced measures of central location (mean, median, mode) and measures of dispersion (range, standard deviation), skewness addresses the "shape" of the data. There are four primary shapes: symmetric, positively skewed (skewed to the right), negatively skewed (skewed to the left), and bimodal.

In a symmetric distribution, the mean and the median are equal ($\bar{X} = \text{Median}$). The data observations are spread evenly around these central values, such that the data points below the mean form a mirror image of the data points above it. A symmetric frequency polygon has a single peak in the center.

A distribution is positively skewed, or skewed to the right, if it has a single peak but the observations extend much further to the right of the peak than to the left. In this configuration, the mean is typically larger than the median ($\bar{X} > \text{Median}$). This is a common pattern in economic data, such as salary distributions. For instance, in a company of 100 employees, most workers may earn a modest wage, but the president and a few high-level executives earn significantly more, pulling the mean to the right.

A distribution is negatively skewed, or skewed to the left, when it has a single peak but the observations extend further to the left, or the negative direction. In such cases, the mean is less than the median ($\bar{X} < \text{Median}$). Finally, a bimodal distribution is characterized by having two or more distinct peaks, which often happens when the data set contains values sampled from two or more different populations.

Pearson's Coefficient of Skewness

The most basic approach to quantifying skewness was developed by Professor Karl Pearson ($1857-1936$). This measure, known as Pearson's Coefficient of Skewness ($sk$), is based on the difference between the mean and the median of the dataset. The formula is defined as:

$sk = \frac{3(\bar{X} - \text{Median})}{s}$

Based on this expression, the value of the coefficient of skewness can range from $-3$ to $3$. A value close to $-3$, such as $-2.57$, indicates substantial negative skewness. A value of $0$ indicates that the distribution is symmetric, occurring when the mean and median are equal. A positive value, such as $1.63$, indicates moderate positive skewness.

Software-Based Skewness Calculations

Standard statistical software programs like Minitab and Excel use a more complex formula for skewness. This method is based on the cubes of the deviations from the mean. The formula for the software-calculated coefficient of skewness is:

$sk = \frac{n}{(n-1)(n-2)} \left[ \sum \left( \frac{X - \bar{X}}{s} \right)^3 \right]$

This formula utilizes the concept of standardization, where each value is converted into standard deviation units via the expression $\frac{X - \bar{X}}{s}$. If a value is larger than the mean, the standardized result is positive; if it is smaller, the result is negative. Unlike the standard deviation formula which squares differences to make all values non-negative, this formula cubes the differences, thereby preserving the direction (sign) of the variation. In a symmetric dataset, the sum of these cubed standardized values tends toward zero. If there are several large outliers on the high end, the sum will be a large positive value. If the outliers are on the low end, the sum will be a negative value.

Empirical Example: Software Company Earnings

To illustrate these concepts, consider the earnings per share for a sample of 15 software companies in 2010. The values, ordered from smallest to largest, are: $\$0.09$, $\$0.13$, $\$0.41$, $\$0.51$, $\$1.12$, $\$1.20$, $\$1.49$, $\$3.18$, $\$3.50$, $\$6.36$, $\$7.83$, $\$8.92$, $\$10.13$, $\$12.99$, and $\$16.40$.

To find the mean ($\bar{X}$), the sum of all values ($\$74.26$) is divided by the sample size ($n = 15$):

$\bar{X} = \frac{\sum X}{n} = \frac{74.26}{15} = 4.95$

The median is the middle value in the ordered list, which in this case is $\$3.18$. The sample standard deviation ($s$) is calculated by taking the square root of the sum of squared deviations from the mean divided by $n-1$:

$s = \sqrt{\frac{\sum(X - \bar{X})^2}{n-1}} = 5.22$

Using Pearson's method, the coefficient of skewness is calculated as:

$sk = \frac{3(4.95 - 3.18)}{5.22} = 1.017$

This indicates a moderate positive skew. When using the software method, the process involves standardizing each value by subtracting $\$4.95$ and dividing by $5.22$, then cubing that result. For the first company ($\$0.09$), the calculation is $\left( \frac{0.09 - 4.95}{5.22} \right)^3 = (-0.9310)^3 = -0.8070$. Summing all 15 cubed results yields a total of $11.8274$. Applying the software formula results in:

$sk = \frac{15}{(15 - 1)(15 - 2)} \times (11.8274) = 0.975$

The summary statistics from a software output (Minitab) confirm these findings, noting a mean of $4.9507$, a standard deviation of $5.228$, a minimum of $0.0900$, a median of $3.1800$, and a maximum of $16.4000$.

Statistics in Action: The Median and Survival Rates

Stephen Jay Gould ($1941-2002$), a professor of zoology and geology at Harvard University, used statistical skewness to interpret his own medical prognosis. Diagnosed with cancer in 1982 and given a life expectancy of eight months, Gould researched the survival distribution for his condition. He discovered that the distribution of survival times was drastically skewed to the right. While the median was eight months (meaning $50\%$ of patients lived longer than that), the right-side tail extended for years rather than months. This realization led him to write the famous essay "The Median Is Not the Message," emphasizing that individual survival can vary greatly from the central tendency in a skewed distribution.

Practice Datasets and Exercises

Exercise 19 provides starting salaries (in thousands of dollars) for five accounting graduates: $36.0$, $26.0$, $33.0$, $28.0$, and $31.0$. Exercise 20 lists salaries of 15 finance directors in the electronics industry (in thousands): $\$516.0$, $\$548.0$, $546.0$, $523.0$, $\$566.0$, $538.0$, $\$534.0$, $523.0$, $\$586.0$, $551.0$, $\$529.0$, $552.0$, $486.0$, $558.0$, and $574.0$. Exercise 21 covers commissions (in thousands) for Furniture Patch, Inc. sales representatives: $\$3.9$, $17.4$, $\$5.7$, $17.6$, $\$7.3$, $10.6$, $22.3$, $38.6$, $87.7$, $\$13.0$, $\$13.6$, $\$15.1$, $\$15.8$, $\$17.1$, and $43.2$.

Exercise 22 presents the 2010 New York Yankees roster and salaries (in thousands of dollars). The data includes: Aceves, Alfredo (Pitcher, $435.7$); Burnett, A.J. (Pitcher, $16500.0$); Cano, Robinson (Second Base, $9000.0$); Cervelli, Francisco (Catcher, $410.8$); Chamberlain, Joba (Pitcher, $488.0$); Gardner, Brett (Outfielder, $452.5$); Granderson, Curtis (Outfielder, $5500.0$); Hughes, Phil (Pitcher, $447.0$); Jeter, Derek (Shortstop, $22600.0$); Johnson, Nick (First Base, $5500.0$); Marte, Damaso (Pitcher, $4000.0$); Mitre, Sergio (Pitcher, $850.0$); Park, Chan Ho (Pitcher, $1200.0$); Pena, Ramiro (Defense, $412.1$); Pettitte, Andy (Pitcher, $11750.0$); Posada, Jorge (Catcher, $13100.0$); Rivera, Mariano (Pitcher, $15000.0$); Robertson, David (Pitcher, $426.7$); Rodriguez, Alex (Third Base, $33000.0$); Sabathia, CC (Pitcher, $24285.7$); Swisher, Nick (Pitcher, $6850.0$); Teixeira, Mark (First Base, $20625.0$); Thames, Marcus (Outfielder, $900.0$); Vazquez, Javier (Pitcher, $11500.0$); and Winn, Randy (Outfielder, $1100.0$). Students are to calculate the mean, median, standard deviation, and skewness for each set.