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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 2 - Descriptive StatisticsProblem 2.4.55e
Chapter 2, Problem 2.4.55e

Pearson’s Index of Skewness The English statistician Karl Pearson (1857–1936) introduced a formula for the skewness of a distribution.
P = 3 (x̄ - median) / s
Most distributions have an index of skewness between -3 and 3. When P > 0, the data are skewed right. When P < 0, the data are skewed left. When P = 0, the data are symmetric. Calculate the coefficient of skewness for each distribution. Describe the shape of each.


e. x̄ = 155, s = 20.0, median = 175

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1
Step 1: Understand the formula for Pearson's Index of Skewness, which is given as P = 3 * (x̄ - median) / s. Here, x̄ represents the mean, 'median' is the median of the data, and 's' is the standard deviation.
Step 2: Substitute the given values into the formula. From the problem, x̄ = 155, median = 175, and s = 20.0. The formula becomes P = 3 * (155 - 175) / 20.0.
Step 3: Simplify the numerator of the formula by calculating the difference between the mean and the median, which is (155 - 175).
Step 4: Divide the result of the numerator by the standard deviation (20.0) to compute the fraction.
Step 5: Multiply the result of the fraction by 3 to find the value of P. Based on the sign of P, interpret the skewness: if P > 0, the data are skewed right; if P < 0, the data are skewed left; if P = 0, the data are symmetric.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Skewness

Skewness is a statistical measure that describes the asymmetry of a distribution. A positive skewness indicates that the tail on the right side of the distribution is longer or fatter than the left side, while a negative skewness indicates the opposite. A skewness of zero suggests a symmetric distribution. Understanding skewness helps in interpreting the shape and behavior of data distributions.
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Pearson’s Index of Skewness

Pearson’s Index of Skewness is a specific formula used to quantify the skewness of a distribution, defined as P = 3(x̄ - median) / s, where x̄ is the mean, median is the median value, and s is the standard deviation. This index provides insight into the direction and degree of skewness, allowing statisticians to assess the distribution's shape and make informed decisions based on the data.
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Mean, Median, and Standard Deviation

The mean is the average of a data set, calculated by summing all values and dividing by the number of observations. The median is the middle value when the data is ordered, providing a measure of central tendency that is less affected by outliers. The standard deviation measures the dispersion of data points around the mean, indicating how spread out the values are. Together, these measures are essential for calculating skewness and understanding data distributions.
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Related Practice
Textbook Question

Extending Concepts


Golf The distances (in yards) for nine holes of a golf course are listed.

336 393 408 522 147 504 177 375 360


d. Use your results from part (c) to explain how to quickly find the mean and the median of the original data set when the distances are converted to inches.

Textbook Question

Song Lengths Side-by-side box-and-whisker plots can be used to compare two or more different data sets. Each box-and-whisker plot is drawn on the same number line to compare the data sets more easily. The lengths (in seconds) of songs played at two different concerts are shown.

d. Can you determine which concert lasted longer? Explain.

Textbook Question

Use the frequency histogram

d. describe any patterns with the data..