6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:30 minutesProblem 2.4.55e
Textbook Question
ÃÛÌÒapp’s Index of Skewness The English statistician Karl ÃÛÌÒapp (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
Verified step by step guidance1
Step 1: Understand the formula for ÃÛÌÒapp's Index of Skewness, which is given as P = 3 * (xÌ„ - median) / s. Here, xÌ„ 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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ÃÛÌÒapp’s Index of Skewness
ÃÛÌÒapp’s Index of Skewness is a specific formula used to quantify the skewness of a distribution, defined as P = 3(xÌ„ - median) / s, where xÌ„ 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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