Descriptive Measures

In each of Exercises 3.42-3.50, use the technology of your choice to obtain the measures of center that are appropriate from among the mean, median, and mode. Discuss your results and decide which measure of center is most appropriate. Provide a reason for your answer. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

The Beatles. In the article, "Length of The Beatles' Songs" (Chance, Vol. 25, No. 1, pp. 30-33), T. Koyama discusses aspects and interpretations of the lengths of songs by The Beatles. Data on the length, in seconds, of 229 Beatles' songs are presented on the WeissStats site. 

In each of Exercises 3.42-3.50, use the technology of your choice to obtain the measures of center that are appropriate from among the mean, median, and mode. Discuss your results and decide which measure of center is most appropriate. Provide a reason for your answer. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Body Temperature. A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is 98.6° F. The results of the study by P. Mackowiak et al . appeared in the article "A Critical Appraisal of 98.6° F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Journal of the American Medical Association, Vol. 268, pp. 1578-1580). Among other data, the researchers obtained the body temperatures of 93 healthy humans, as provided on the WeissStats site.

In each of Exercises 3.42-3.50, use the technology of your choice to obtain the measures of center that are appropriate from among the mean, median, and mode. Discuss your results and decide which measure of center is most appropriate. Provide a reason for your answer. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Bachelor's Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who have completed a bachelor's degree is provided on the WeissStats site.

In each of Exercises 3.51-3.52,

(a) use the technology of your choice to determine the mean and median of each of the two data sets.

(b) compare the two data sets by using your results from part (a).

The Etruscans. Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly advanced Italic civilization formed around the eighth century B.C. in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from the East by land or sea? The maximum head breadth, in millimeters, of 70 modern Italian male skulls and that of 84 preserved Etruscan male skulls were analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats site. [SOURCE: N. Barnicot and D. Brothwell, "The Evaluation of Metrical Data in the Comparison of Ancient and Modern Bones." In Medical Biology and Etruscan Origins, G. Wolstenholme and C. O'Connor, eds., Little, Brown & Co., 1959}

In each of Exercises 3.51-3.52,

(a) use the technology of your choice to determine the mean and median of each of the two data sets.

(b) compare the two data sets by using your results from part (a).

Treating Psychotic Illness. L. Petersen et al . evaluated the effects of integrated treatment for patients with a first episode of psychotic illness in the paper "A Randomised Multicentre Trial of Integrated Versus Standard Treatment for Patients with a First Episode of Psychotic Illness" (British Medical Journal, Vol. 331, (7517): 602). Part of the study included a questionnaire that was designed to measure client satisfaction for both the integrated treatment and a standard treatment. The data on the WeissStats site are based on the results of the client questionnaire.

Outliers and trimmed means. Some data sets contain outliers, observation that fall well outside the overall pattern of the data(We discuss outliers in more detail in section 3.4) Suppose, for instance that you are interested in the ability o high school algebra  student to compute square roots. You decide to give a square root exam to 10 of these students Unfortunately, one of the student  had a fight with his girlfriend and cannot concentrate he gets a 0. The 10 score are displayed in increasing order in the following table. The score of 0 is an outlier.

Statisticians have a systematic method for avoiding extreme observation and outliers when they calculated means. They compute trimmed means, in which high and low observation are deleted or "trimmed off" before the mean is calculated . For instance, to compute the 10% trimmed mean of the test score data. we first delete both the bottom 10% and the top 10% of the ordered data. that is,0 and 80. Then we calculated the mean of the remaining data. Thus the10% trimmed mean of the test score data is

The following table displays a set of score for a 40 question algebra final

Part (a) Do any the score look like outliers?

Part (b) Compute the usual mean of the data.

Part (c) Compute the 5% trimmed mean of the data.

Part (d) Compute the 10% trimmed mean of the data.
Part (e) Compare the means you obtained in parts (b) (d) which of the three provides the best measure of center for the data?

Explain the difference between the quantities ${\left(\sum _{}{x}_i\right)}^2$and $\sum _{}{x^2}_i$ Construct an example to show that in general those two quantities are unequal.

Explain the difference between the quantities $\sum _{}{x}_i{y}_i$and $\left(\sum _{}{x}_i\right)\left(\sum _{}{y}_i\right)$.Provide an example to show that, in general, those two quantities are unequal.

Discuss the pros and cons of Chebyshev's rule.

If the condition for using the empirical rule is met,why should that rule be used insted of Chebyshev's rule.

In Exercises $3.65- 3.70$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation 

$4,0,5$.

Explain the purpose of a measure of  variation.

Why is the standard deviation preferable to the range as a measure of variation.

Why you use the standard deviation as a measure of variation. What is the reference point.

The following dashboard represents dart thrown by two players. Tracey and Joan.

For the variable distance from the center" which player's board represents data with a smaller sample standard deviation? Explain your answer. 

Consider the data set 1,2,3,4,5,6,7,8,9.

Part (a)  Use the defining formula to obtain the sample standard deviation.

Part (b) Replace the 9 in the data set by 99, and again use the defining formula to compute the sample standard deviation.

Part (c) Compare your answer in parts (a) and (b). The lack of what property of the standard deviation accounts for its extreme sensitivity to the change of 9 to 99? 

Age of U.S Residents. The U.S census Bureau publishes information about ages of people in the United States in Current Population Reports. A sample of five U.S residents has the following ages in year.

Part (a) Determine the range of the ages.

Part (b) Find the sample standard deviation of these ages by using the defining formula. Definition 3.6 on page 108.

Part (c) Find the sample standard deviation of these ages by using the computing formula. Formula 3.1 on page 110.

Part (d)Compare your work in part (b) and (c).

Consider the data set 3,3,3,3,3,3.

Part (a) Guess the value of the sample standard deviation without calculating it. Explain your reasoning

Part (b) Use the defining formula to calculate the sample standard deviation.

Part (c) Complete the following statement and explain your reasoning. If all observation in a data set are equal, the sample standard deviation is

Part (d) Complete the following statement and explain your reasoning. If the sample standard deviation of a data set is 0 then... 

 Consider the following four data sets. 

a. Compute the mean of each data set.

b. Although the four data sets have the same means, in what respect are they quite different?

c. Which data set appears to have the least variation? the greatest variation?

d. Compute the range of each data set.

e. Use the defining formula to compute the sample standard deviation of each data set.

f. From your answers to parts (d) and (e), which measure of variation better distinguishes the spread in the four data sets: the range or the standard deviation? Explain your answer.

g. Are your answers from parts (c) and (e) consistent?

In Exercises $3.65-3.70$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range.

b) sample standard deviation.

$4,0,5$

In Exercises $3.65-3,67$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation 

$1,2,4,4$

In Exercises , we $3.65-3.70$have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation 

2,5,0,-1

In Exercises $3.65-3.70$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation 

$1,9,8,4,3$

In Exercises $3.65-3.70$, we have provided simple data sets for you to practice the basics of finding measures of variation . For each data set determine.

a) Range

b) sample standard deviation

4,2,0,2,2

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation.  

Hurricanes. An article by D. Schaefer et al. (Journal of Tropical Ecology, Vol16, pp. 189-207) reported on a long-term study of the effects of hurricanes on tropical streams of the Luquillo Experimental Forest in Puerto Rico. The study showed that Hurricane Hugo had a significant impact on stream water chemistry. The following table shows a sample of 10 ammonia fluxes in the first year after Hugo. Data are in kilograms per hectare per year .

$96, 66, 147, 147, 175, 116, 57, 154, 88, 154$

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation. 

Tornado Touchdowns. Each year, tornadoes that touch down are recorded by the Storm Prediction Center and published in Monthly Tornado Statistics. The following table gives the number of tornadoes that touched down in the United States during each month of one year.

 $3, 2, 47, 118, 204,97, 68, 86,62,57, 98, 99$

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation. 

 Amphibian Embroyes. In a study of the effects of radiation on amphibian embryos titled "Shedding Light on Ultraviolet Radiation and Amphibian Embryos" ( Bio Science, Vol. 53. No. 6. pp. 551561) L. Licht recorded the time it took for a sample of seven different species of frogs and toads' eggs to hatch. The following table shows the times to hatch, in days 

$6, 7, 11, 6, 5, 5, 11$

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation 

Technical Merit. In one Winter Olympics, Michelle Kwan competed in the Short Program ladies singles event. From nine judges, she received scores ranging from poor 6 perfect. The following cable provides the scores that the judges gave her on trackiest merit found in an article by S. Berry.

$5.8, 5.7, 5.9, 5.7, 5.5, 5.7, 5.7, 5.7, 5.6$

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation.

 Billionaires' Club, Each year, Forbes magazine compiles a list of the $400$ richest Americans. As of September 19, . the top 10 on the list areas in the following table.

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation. 

Tablet Computers. Tablet personal computers such as iPad and Kindle Fire are one piece portable computers that typically offer a touchscreen. Tablets are available in different sizes but generally offer a screen that is greater than $7$ inches diagonally. Consumer Reports reviews tablets of different sizes and Internet capabilities. The battery lives, in hours, for a sample of tablets with screen sizes between $9$ and $12$ inches are provided in the following table.

$8.4, 5.6, 10.9, 10.8, 11.5, 8, 8.5, 12.9$ 

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation.  

The Beatles. The English rock band, The Beatles, was formed in Liverpool in 1960. There are 12 studio albums that are considered part of their core catalogue In the article. "Length of The Beatles' Songs" (Chance, Vol. 25. No. 1. pp. 30-33), T. Koyama lists the album title, date of release, and number of songs on each album. Here are the album names and numbers of songs. 

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation. 

 Router Horsepower. In the article "Router Roundup" T. Klenck reported on tests of seven fixed-base routers for perform features, and handling. The following table gives the horsepower for each of the seven routers tested .

$1.75, 2.25, 2.25, 2.25, 1.75, 2.00, 1.50$

Determine the age and sample standard deviation for each of the due sets. For the sample standard deviation round each answer to one more decimal place than that used for the observation

 Amphibian Embryos. In a study of the effects of radiation on amphibian embryos titled "Shedding Light on Ultraviolet Radiation and Amphibian Embryos" (BioScience, Vol. 53, No. 6, pp. 551561). L. Licht recorded the time it took for a sample of seven different species of frogs and toads' eggs to hatch. The following table shows the times to hatch, in days.

 .  

 Medieval Cremation Burials. In the article "Material Culture as Memory: Combs and Cremations in Early Medieval Britain" Early  Medieval, Vol. 12. Issue 2, pp. 89-128, H. Williams discussed the frequency of cremation burials found in 17 archaeological sites in eastern England. Here are the data.

 $$\begin{gathered} 23, 64, 46, 48, 523, 35, 34, 265, 2484, \\ 46, 385, 21, 86, 429, 51, 258, 119 \end{gathered}$$

a) Obtain the sample standard deviation of these data.

b) Do you think that in this case, the sample standard deviation provides a good measure of variation? Explain your answer. 

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Japanese Vehicle Exports. The Japan Automobile Manufacturer's Association provides data on exported vehicles in Motor Vehicle Statistics of Japan. In 2010, cars, trucks, and buses constituted 88.3%, 9.3%, and 2.4% of vehicle exports, respectively. A random sample of last year's exports yielded the vehicle-type data on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

U.S. Hospitals. The American Hospital Association conducts annual surveys of hospitals in the United States and publishes its findings in AHA Hospital Statistics. Data on hospital type for U.S. registered hospitals can be found on the WeissStats site. For convenience, we use the following abbreviations: 

• NPC: Nongovernment not-for-profit community hospitals 
• IOC: Investor-owned (for-profit) community hospitals 
• SLC: State and local government community hospitals 
• FGH: Federal government hospitals 
• NFP: Nonfederal psychiatric hospitals
• NLT: Nonfederal long-term-care hospitals 
• HUI: Hospital units of institutions

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Marital Status and Drinking. Research by W. Clark and L. Midanik (Alcohol Consumption and Related Problems: Alcohol and Health Monograph 1. DHHS Pub. No. (ADM) 82-1190) examined, among other issues, alcohol consumption patterns of U.S. adults by marital status. Data for marital status and number of drinks per month, based on the researcher's survey results, are provided on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Ballot Preferences. In Issue 338 of the Amstat News, then president of the American Statistical Association, F. Scheuren reported the results of a survey on how members would prefer to receive ballots in annual elections. On the WeissStats site, you will find data for preference and highest degree obtained for the 566 respondents. 

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

The Great White Shark. In an article titled " Great White, Deep Trouble " ( National Geographic, Vol. 197 (4). pp. 2-29), Peter Benchley- the author of JAWS-discussed various aspects of the Great White Shark (Carcharodon carcharias). Data on the number of pups borne in a lifetime by each of 80 Great White Shark females are provided on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

The Beatles. In the article. "Length of The Beatles' Songs" (Chance, Vol. 25, No. 1. pp . 30-33), T. Koyama discusses aspects and interpretations of the lengths of songs by The Beatles. Data on the length, in seconds, of 229 Beatles' songs are presented on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

High School Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who have completed high school is provided on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Body Temperature. A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is $98.6° F$. The results of the study by P. Mackowiak et al. appeared in the article "A Critical Appraisal of 98.6 F. the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Journal of the American Medical Association, Vol. 268, pp . 1578-1580). Among other data, the researchers obtained the body temperatures of 93 healthy humans, as provided on the WeissStats site.

In each Exercises 3.82-3.90, use the technology of your choice to determine and interpret the range and sample standard deviation for those data sets to which those concepts apply. If those concepts don't apply, explain why. Note: If an exercise contains more than one data set, perform the aforementioned tasks for each data set.

Bachelor's Completion. As reported by the U.S. Census Bureau in Educational Attainment in the United States, the percentage of adults in each state and the District of Columbia who have completed a bachelor's degree is provided on the WeissStats site. 

Monthly Motorcycle Casualties. This Scottish Executive Analytical Services Division Transport Statics. compiles data on motorcycle casualties .During one year, monthly casualties resulting from motorcycle accidents in Scotland for built up road and non built up road were as follows,

Part (a) Without doing any calculations, make an educated guess at which of the two data sets, built up or non built or non built up has the greater variation.

Pat (b) Find the range and sample standard deviation of each of the two data sets, Compare your results here to the education guess that you made in part (a)

Monthly Motorcycle Casualties. This Scottish Executive Analytical Services Division Transport Statics. compiles data on motorcycle accidents .During one year, the numbers of motorcycle accidents in Scotland were tabulated by day of the week for built up road and non built up road were as follows,

Part (a) Without doing any calculations, make an educated guess at which of the two data sets, built up or non built or non built up has the greater variation.

Pat (b) Find the range and sample standard deviation of each of the two data sets, Compare your results here to the education guess that you made in part (a)

In each of the Exercises 3.91-3.92.

(a) use the technology of your choice to determine the range and sample standard deviation of each of the two data sets.

(b) compare the two data sets by using your results from part (a). 

Treating Psychotic Illness. L. Petersen et al . evaluated the effects of integrated treatment for patients with a first episode of psychotic illness in the paper "A Randomised Multicentre Trial of Integrated Versus Standard Treatment for Patients With a First Episode of Psychotic Illness" (British Medical Journal, Vol. 331. (517):602)

Part of the study included a questionnaire that was designed to measure client satisfaction for both the integrated treatment and a standard treatment. The data on the WeissStats site are based on the results of the client questionnaire.

Weekly Salaries. In the following table, we repeat the salary data in Data Set II from Example 3.1.

(a) Use Definitions 3.4 and 3.6 on pages 99 and 108, respectively, to obtain the sample mean and sample standard deviation of this (ungrouped) data set. 

(b) A frequency distribution for Data Set II, using single-value grouping, is presented in the first two columns of the following table. The third column of the table is for the xf - values, that is, classmark or midpoint (which here is the same as the class) times class frequency. Complete the missing entries in the table and then use the grouped-data formula to obtain the sample mean.

(c) Compare the answers that you obtained for the sample mean in parts (a) and (b). Explain why the grouped-data formula always yields the actual sample mean when the data are grouped by using single-value grouping. (Hint: What does xf represent for each class?) 

(d) Construct a table similar to the one in part (b) but with columns for $x, f, x-\overline{x} ,{\left( x-\overline{x}\right)}^2, and {\left( x-\overline{x}\right)}^{2}f$.Use the table and the grouped-data formula to obtain the sample standard deviation.

(e) Compare your answers for the sample standard deviation in parts (a) and (d). Explain why the grouped-data formula always yields the actual sample standard deviation when the data are grouped by using single-value grouping. 

Outliers. In Exercise 3.54 on pages 104-105, we discussed outliers, or observations that fall well outside the overall pattern of the data. The following table contains two data sets. Data Set II was obtained by removing the outliers from Data Set I.

(a) Compute the sample standard deviation of each of the two data sets. 

(b) Compute the range of each of the two data sets. 

(c) What effect do outliers have on variation? Explain your answer.

Days to Maturity. The first two columns of the following table provide a frequency distribution, using limit grouping, for the days to maturity of 40 short-term investments, as found in BARRON'S. The third column shows the class marks.

(a) Use the grouped-data formulas to estimate the sample mean and sample standard deviation of the days-to-maturity data. Round your final answers to one decimal place.

(b) The following table gives the raw days-to-maturity data.

Using Definitions 3.4 and 3.6 on pages 99 and 108, respectively, gives the true sample mean and sample standard deviation of the days-to-maturity data as 68.3 and 16.7, respectively, rounded to one decimal place. Compare these actual values of $\overline{x} \mathrm{and} s$ to the estimates from part (a). Explain why the grouped-data formulas generally yield only approximations to the sample mean and sample standard deviation for non-single-value grouping.

In each of the Exercises 3.91-3.92.

(a) use the technology of your choice to determine the range and sample standard deviation of each of the two data sets.

(b) compare the two data sets by using your results from part (a).

The Etruscans. Anthropologists are still trying to unravel the mystery of the origins of the Etruscan empire, a highly advanced Italic civilization formed around the eighth century B.C. in central Italy. Were they native to the Italian peninsula or, as many aspects of their civilization suggest, did they migrate from the East by land or sea? The maximum head breadth, in millimeters, of 70 modern Italian male skulls and that of 84 preserved Etruscan male skulls were analyzed to help researchers decide whether the Etruscans were native to Italy. The resulting data can be found on the WeissStats site. [SOURCE: N. Barnicot and D. Brothwell, "The Evaluation of Metrical Data in the Comparison of Ancient and Modern Bones." In Medical Biology and Etruscan Origins, G. Wolstenholme and C. O'Connor, eds., Little, Brown & Co., 1959 ]