3. Describing Data Numerically

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.

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)

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.

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

N = 6

μ = 5

σ ≈ 2

Use the frequency distribution in Exercise 4 to estimate the sample mean and sample standard deviation of the data. Do the formulas for grouped data give results that are as accurate as the individual entry formulas? Explain.

Graphical Analysis In Exercises 21–24, you are asked to compare three data sets.

(c) Estimate the sample standard deviations. Then determine how close each of your estimates is by finding the sample standard deviations.

i.

ii.

iii.

Constructing a Confidence Interval In Exercises 25–28, use the data set to (b) find the sample standard deviation. Assume the population is normally distributed.

SAT Scores The SAT scores of 12 randomly selected high school seniors