Ch. 2 - Descriptive Statistics

Larson - Elementary Statistics: Picturing the World 8th Edition

Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook

Larson 8th Edition

Ch. 2 - Descriptive Statistics

Problem 2.R.19

Chapter 2, Problem 2.R.19

Describe the shape of the distribution for the histogram you made in Exercise 3 as symmetric, uniform, skewed left, skewed right, or none of these.

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Review the histogram created in Exercise 3 and observe the overall shape of the data distribution.

Check if the histogram is symmetric by determining if the left and right sides of the distribution are approximately mirror images of each other.

Examine if the histogram is uniform by checking if all the bars have roughly the same height, indicating equal frequencies across intervals.

Determine if the histogram is skewed left by observing if the tail of the distribution extends more to the left (towards smaller values).

Determine if the histogram is skewed right by observing if the tail of the distribution extends more to the right (towards larger values). If none of these apply, classify the shape as 'none of these.'

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Distribution Shape

The shape of a distribution refers to the visual representation of data points in a histogram. Common shapes include symmetric, where both sides mirror each other; uniform, where all values have similar frequencies; and skewed, where one tail is longer than the other. Understanding these shapes helps in interpreting the underlying data characteristics.

Skewness

Skewness measures the asymmetry of a distribution. A distribution is skewed left (negatively skewed) if it has a longer left tail, indicating that most data points are concentrated on the right. Conversely, a right skew (positively skewed) has a longer right tail, suggesting that most data points are on the left. Recognizing skewness is crucial for understanding data behavior.

Symmetry in Distributions

A symmetric distribution has equal frequencies on both sides of its center, resulting in a balanced shape. The mean, median, and mode of a symmetric distribution are all located at the center. Identifying symmetry is important for statistical analysis, as it influences the choice of statistical tests and the interpretation of results.

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Related Practice

Textbook Question

A student’s test grade of 75 represents the 65th percentile of the grades. What percent of students scored higher than 75?

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From a random sample of airplanes, the number of defects found in their fuselages are listed. Find the sample mean and the sample standard deviation of the data.

Textbook Question

The heights (in feet) and the number of stories of the ten tallest buildings in New York City are listed. Use a scatter plot to display the data. Describe any patterns. (Source: Emporis)

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Textbook Question

In Exercises 21 and 22, determine whether the approximate shape of the distribution in the histogram is symmetric, uniform, skewed left, skewed right, or none of these.

Textbook Question

In Exercises 27 and 28, find the range, mean, variance, and standard deviation of the sample data set.

Salaries (in dollars) of a random sample of teachers

62,222 56,719 50,259 45,120 47,692 45,985 53,489 71,534

Textbook Question

The mean sale per customer for 40 customers at a gas station is \$32.00, with a standard deviation of \$4.00. Using Chebychev’s Theorem, determine at least how many of the customers spent between \$24.00 and \$40.00.