6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:45 minutesProblem 2.4.55a
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.
a. x̄ = 17, s = 2.3, median = 19
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 = 17, median = 19, and s = 2.3. The formula becomes P = 3 * (17 - 19) / 2.3.
Step 3: Simplify the numerator of the formula by calculating the difference between the mean and the median. This gives (17 - 19), which simplifies to -2.
Step 4: Multiply the result of the numerator (-2) by 3, and then divide by the standard deviation (2.3). This will give the value of P.
Step 5: Interpret the result of P. If P > 0, the data are skewed right; if P < 0, the data are skewed left; and if P = 0, the data are symmetric. Based on the sign of P, describe the shape of the distribution.
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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. It is calculated 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, aiding in the analysis of data distributions.
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Mean, Median, and Standard Deviation
The mean is the average of a data set, the median is the middle value when the data is ordered, and the standard deviation measures the dispersion of data points around the mean. These three statistics are fundamental in understanding the characteristics of a distribution. They are essential for calculating ÃÛÌÒapp’s Index of Skewness and interpreting the shape of the distribution.
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