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Ch. 2 - Descriptive Statistics
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLarson 8th EditionCh. 2 - Descriptive StatisticsProblema 2.4.55a
Capitolo 2, Problema 2.4.55a

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.


a. x̄ = 17, s = 2.3, median = 19

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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 = 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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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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Creating Frequency Polygons

Pearson’s Index of Skewness

Pearson’s Index of Skewness is a specific formula used to quantify the skewness of a distribution. It is calculated 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, 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 Pearson’s Index of Skewness and interpreting the shape of the distribution.
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Calculating Standard Deviation
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Domanda del libro di testo

Archaeology The depths (in inches) at which 10 artifacts are found are listed. 

20.7 24.8 30.5 26.2 36.0 34.3 30.3 29.5 27.0 38.5


a. Find the range of the data set.

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Protein Powder During a quality assurance check, the actual contents (in grams) of six containers of protein powder were recorded as 1525, 1526, 1502, 1516, 1529, and 1511.


a. Find the mean and the median of the contents.

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Shifting Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

40   35   49   53   38   39   40

37   49   34   38   43   47   35


a. Find the sample mean and the sample standard deviation.

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Scaling Data Sample annual salaries (in thousands of dollars) for employees at a company are listed.

42   36   48   51   39   39   42

36   48   33   39   42   45   50

b. Each employee in the sample receives a 5% raise. Find the sample mean and the sample standard deviation for the revised data set.

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Drawing a Box-and-Whisker Plot In Exercises 15–18,

(b) draw a box-and-whisker plot that represents the data set.


2 7 1 3 1 2 8 9 9 2 5 4 7 3 7 5 4

2 3 5 9 5 6 3 9 3 4 9 8 8 2 3 9 5

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Extending Concepts


Alternative Formula You used SSₓ = Σ(x − x̄)² when calculating variance and standard deviation. An alternative formula that is sometimes more convenient for hand calculations is

SSₓ = Σ x² − (Σ x)² / n.

You can find the sample variance by dividing the sum of squares by n − 1 and the sample standard deviation by finding the square root of the sample variance.


b. Use the alternative formula to calculate the sample standard deviation for the data set in Exercise 15.

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