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
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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 2.3.57a
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.
Verified step by step guidance1
Step 1: To find the mean, calculate the sum of all the recorded contents. Add the values: 1525 + 1526 + 1502 + 1516 + 1529 + 1511.
Step 2: Divide the total sum obtained in Step 1 by the number of containers (6) to compute the mean. Use the formula: Mean = (Sum of all values) / (Number of values).
Step 3: To find the median, first arrange the recorded contents in ascending order: 1502, 1511, 1516, 1525, 1526, 1529.
Step 4: Since there is an even number of data points (6), the median is the average of the two middle values in the ordered list. Identify the two middle values (1516 and 1525).
Step 5: Compute the median by taking the average of the two middle values: Median = (Middle value 1 + Middle value 2) / 2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Mean
The mean, or average, is calculated by summing all the values in a dataset and then dividing by the number of values. In this case, to find the mean of the protein powder contents, you would add the weights of all six containers and divide by six. The mean provides a central value that represents the overall dataset.
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Calculating the Mean
Median
The median is the middle value of a dataset when it is ordered from least to greatest. If there is an even number of observations, the median is the average of the two middle numbers. For the protein powder contents, you would first sort the values and then identify the median to understand the central tendency without the influence of outliers.
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Descriptive Statistics
Descriptive statistics summarize and describe the main features of a dataset. This includes measures like mean, median, mode, and standard deviation. In the context of the protein powder question, calculating the mean and median helps to provide insights into the typical content of the protein powder containers, aiding in quality assurance.
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Related Practice
Textbook Question
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
Textbook Question
Extending Concepts
Data Analysis Students in an experimental psychology class did research on depression as a sign of stress. A test was administered to a sample of 30 students. The scores are shown in the table at the left.
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a. Find the mean and the median of the data.
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
Drawing a Box-and-Whisker Plot In Exercises 15–18,
(a) find the five-number summary
4 7 7 5 2 9 7 6 8 5 8 4 1 5 2 8 7 6 6 9
1
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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.