3. Describing Data Numerically

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

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.

Threaded Problem: Tornado The data set “Tornadoes_2017” located at www.pearsonhighered.com/sullivanstats contains a variety of variables that were measured for all tornadoes in the United States in 2017.

b. Determine the population mean and standard deviation length of all tornadoes in 2017.

Putting It Together: Bike Sharing Bicycle sharing exists in a variety of cities around the country. Los Angeles has the Metro Bike Share system. Users pick up a bike from one station, go for a ride, and return the bike to any station. Go to www.pearsonhighered.com/sullivanstats and download the file 8_1_34. The data represent the duration (in minutes) of all rides in the fourth quarter (October through December) of 2018.

b. Determine the population mean and standard deviation duration.

[DATA] Homeruns Go to www.pearsonhighered.com/sullivanstats to obtain the data file 12_2_24 using the file format of your choice for the version of the text you are using. The variable “TrueDist” represents the distance, in feet, that the homerun traveled for all homeruns hit in the 2014 season.

b. Find the population mean and population standard deviation distance.

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated.

Sample: 83, 65, 91, 87, 84

In Problems 5–10, by hand, find the population variance and standard deviation or the sample variance and standard deviation as indicated.

Population: 1, 19, 25, 15, 12, 16, 28, 13, 6

Exam Time The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in Sullivan’s Statistics course. Compute the range, sample variance, and sample standard deviation time.

60.5, 128.0, 84.6, 122.3, 78.9, 94.7, 85.9, 89.9

Which Professor? Suppose Professor Alpha and Professor Omega each teach Introductory Biology. You need to decide which professor to take the class from and have just completed your Introductory Statistics course. Records obtained from past students indicate that students in Professor Alpha’s class have a mean score of 80% with a standard deviation of 5%, while past students in Professor Omega’s class have a mean score of 80% with a standard deviation of 10%. Decide which instructor to take for Introductory Biology using a statistical argument.

Comparing Standard Deviations The standard deviation of batting averages of all teams in the American League is 0.008. The standard deviation of all players in the American League is 0.02154. Why is there less variability in team batting averages?

Mean Absolute Deviation Another measure of variation is the mean absolute deviation. It is computed using the formula:

MAD = (Σ |xi − x̄|) / n

Compute the mean absolute deviation of the data in Problem 11 and compare the results with the sample standard deviation.