Textbook Question
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.
Ch. 2 - Descriptive Statistics
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 2, Problem 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
Verified step by step guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
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.
Recommended video:
04:41
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.
Recommended video:
04:41Creating Frequency Polygons
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.
Recommended video:
Guided course
08:45Calculating Standard Deviation
Related Practice
Textbook Question
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.
Textbook Question
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.
Textbook Question
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.
Textbook Question
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
Textbook Question
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.