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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
All textbooksLarson 8th EditionCh. 2 - Descriptive StatisticsProblem 2.4.25
Chapter 2, Problem 2.4.25

Constructing Data Sets In Exercises 25–28, construct a data set that has the given statistics.


N = 6
μ = 5
σ ≈ 2

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Step 1: Understand the problem. You are tasked with constructing a data set of size N = 6 (6 data points) such that the mean (μ) is 5 and the standard deviation (σ) is approximately 2. Recall that the mean is the average of the data points, and the standard deviation measures the spread of the data around the mean.
Step 2: Start by using the formula for the mean: μ = (Σxᵢ) / N. Rearrange this formula to find the sum of the data points: Σxᵢ = μ × N. Substitute μ = 5 and N = 6 to calculate the total sum of the data points, which will guide the construction of the data set.
Step 3: Use the formula for the standard deviation: σ = √((Σ(xᵢ - μ)²) / N). Rearrange this formula to express the sum of squared deviations: Σ(xᵢ - μ)² = σ² × N. Substitute σ ≈ 2 and N = 6 to calculate the total sum of squared deviations from the mean.
Step 4: Construct a data set of 6 numbers that satisfies both the mean and standard deviation conditions. Start with a set of numbers that sum to the total calculated in Step 2. Adjust the numbers so that their squared deviations from the mean sum to the value calculated in Step 3. Ensure the spread of the numbers is consistent with the desired standard deviation.
Step 5: Verify your constructed data set. Check that the mean of the data set is 5 by calculating (Σxᵢ) / N. Then, calculate the standard deviation using the formula σ = √((Σ(xᵢ - μ)²) / N) to ensure it is approximately 2. Adjust the data points if necessary to meet both conditions.

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Key Concepts

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

Population Size (N)

Population size, denoted as N, refers to the total number of observations or data points in a data set. In this context, N = 6 indicates that the constructed data set must contain exactly six values. Understanding population size is crucial for calculating various statistics, as it directly influences measures like the mean and standard deviation.
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Mean (μ)

The mean, represented by the symbol μ, is the average of a data set and is calculated by summing all the values and dividing by the number of observations. In this case, μ = 5 means that the average of the six data points must equal 5. This concept is fundamental in statistics as it provides a central value around which the data points are distributed.
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Standard Deviation (σ)

Standard deviation, denoted as σ, measures the amount of variation or dispersion in a set of values. An approximate standard deviation of 2 indicates that the data points tend to deviate from the mean (5) by about 2 units. This concept is essential for understanding the spread of data and how individual data points relate to the mean.
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Related Practice
Textbook Question

Grades In Exercise 46, one of the student’s B grades gets changed to an A. What is the student’s new grade point average?

Textbook Question

Putting Graphs in Context In Exercises 5–8, match the plot with the description of the sample.

a. Times (in minutes) it takes a sample of employees to drive to work

b. Grade point averages of a sample of students with finance majors

c. Top speeds (in miles per hour) of a sample of high-performance sports cars

d. Ages (in years) of a sample of residents of a retirement home


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Textbook Question

Comparing Variation in Different Data Sets In Exercises 45–50, find the coefficient of variation for each of the two data sets. Then compare the results.

Heights and Weights The heights (in inches) and weights (in pounds) of every France national soccer team player that started the 2018 FIFA Men’s World Cup final are listed. (Source: ESPN)

Textbook Question

Construct a frequency distribution and a frequency histogram for the data set using the indicated number of classes. Describe any patterns.

Reaction Times

Number of classes: 8

Data set: Reaction times (in milliseconds) of 30 adult females to an auditory stimulus 507 389 305 291 336 310 514 442 373 428 387 454 323 441 388 426 411 382 320 450 309 416 359 388 307 337 469 351 422 413

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Textbook Question

Extending Concepts


A Misleading Graph? A misleading graph is not drawn appropriately, which can misrepresent data and lead to false conclusions. In Exercises 37–40, (a) explain why the graph is misleading, and (b) redraw the graph so that it is not misleading.


Textbook Question

Graphical Analysis In Exercises 59 and 60, the letters A, B, and C are marked on the horizontal axis. Describe the shape of the data. Then determine which is the mean, which is the median, and which is the mode. Justify your answers.


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