3. Describing Data Numerically

Using and Interpreting Concepts

Finding the Range of a Data Set In Exercises 9 and 10, find the range of the data set represented by the graph.

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.

In Exercises 13 and 14, find the range, mean, variance, and standard deviation of the population data set.

Drunk Driving The number of alcohol-impaired crash fatalities (in thousands) per year from 2010 through 2019 (Source: National Highway Traffic Safety Administration)

10.1 9.9 10.3 10.1 9.9 10.3 11.0 10.9 10.7 10.1

Finding Sample Statistics In Exercises 15 and 16, find the range, mean, variance, and standard deviation of the sample data set.

Pregnancy Durations The durations (in days) of pregnancies for a random sample of pregnant people

277 291 295 280 268 278 291

277 282 279 296 285 269 293

267 281 286 269 264 299 275

Estimating the Sample Mean and Standard Deviation for Grouped Data In Exercises 41–44, make a frequency distribution for the data. Then use the table to estimate the sample mean and the sample standard deviation of the data set.

Weekly Study Hours The distribution of the number of hours that a random sample of college students study per week is shown in the pie chart. Use 32 as the midpoint for “30+ hours.”

Finding the Sample Mean and Standard Deviation for Grouped Data In Exercises 39 and 40, make a frequency distribution for the data. Then use the table to find the sample mean and the sample standard deviation of the data set.

3 3 5 3 8 0 3 9 6 6 7 1 6 3 2 6 9 1 8 5 0 2 3 4 9

5 8 1 9 7 6 9 6 7 0 6 3 8 6 8 7 3 8 9 3 7 2 4 4 1

In Exercises 25 and 26, find the range, mean, variance, and standard deviation of the population data set.

The mileages (in thousands of miles) for a rental car company’s fleet.

4 2 9 12 15 3 6 8 1 4 14 12 3 3

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

n = 6

x̄ = 7

s ≈ 2

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

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.

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.

Use frequency distribution formulas to estimate the sample mean and the sample standard deviation of the data set in Exercise 2.

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

c. Each employee in the sample takes a pay cut of \$2000 from their original salary. Find the sample mean and the sample standard deviation for the revised data set.

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.

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.