Chapter 4: Collecting, Displaying, and Analyzing Data

Chapter 4: Collecting, Displaying, and Analyzing Data

Regular Math

Section 4.1: Samples and Surveys Population – the entire group being studied

Sample – part of the population

Biased Sample – not a good representation

Random Sample – every member has an equal chance

Systematic Sample – according to a rule or formula

Stratified Sample – at random from a randomly subgroup

Example 1: Identifying Biased Samples Identify the population and sample. Give a reason why the sample could

be biased. A radio station manager chooses 1500 people from the local phone book to

survey about their listening habits. Population = people in the local area Sample = up to 1500 people that take the survey Biased = not everyone is in the phone book

An advice columnist asks her readers to write in with their opinions about how to hang the toilet paper on the roll. Population = readers of the column Sample = readers who wrote in Biased = only readers with strong opinions would write in

Surveyors in a mall choose shoppers to ask about their product preferences. Population = all shoppers at the mall Sample = people who are polled Biased = surveyors are more likely to approach people who look agreeable

Try these on your own… Identify the population and sample. Give a reason why

the sample could be biased. A record store manager asks customers who make a

purchase how many hours of music they listen to each day. Population = record store customers Sample = customers who make a purchase Biased = Customers who make a purchase may be more

interested in music than others who are in the store. An eighth-grade student council member polls classmates

about a new school mascot. Population = students in the school Sample = classmates Biased = She polls more eighth graders than students in other

grades.

Sampling Method How Members Are Chosen

Random By chance

Systematic According to a rule or formula

Stratified At random from randomly chosen subgroups

Example 2: Identifying Sampling Methods Identify the sampling method used.

An exit poll is taken of every tenth voter. systematic

In a statewide survey, five counties are randomly chosen and 100 people are randomly chosen from each county. stratified

Students in a class write their names on strips of paper and put them in a hat. The teacher draws five names. random

Try these on your own…

Identify the sampling method used. In a county survey, Democratic Party members whose

names begin with the letter D are chosen. systematic

A telephone company randomly chooses customers to survey about its service. random

A high school randomly chooses three classes from each grade and then draws three random names from each class to poll about lunch menus. stratified

Section 4.2: Organizing Data

Stem-and-Leaf Plot

Back –to-Back Stem-and-Leaf Plot

Example 1: Organizing Data in Tables Use the given data to make

a table. Greg has received job

offers as a mechanic at three airlines. The first has a salary range of $20,000-$34,000, benefits worth $12,000, and 10 days’ vacation. The second has 15 days’ vacation, benefits worth $10,500, and salary range of $18,000-$50,000. The third has benefits worth $11,400, a salary range of $14,000-$40,000, and 12 days’ vacation.

Job 1 Job 2 Job 3

Salary Range

$20,000 -

$34,000

$18,000-

$50,000

$14,000-

$40,000

Benefits

$12,000 $10,500 $11,400

Vacation Days

10 15 12

Try this one on your own…

Day To School To Home

Monday 7 min 9 min

Tuesday 5 min 9 min

Wednesday 8 min 7 min

Use the given data to make a table. Jack timed his bus rides

to and from school. On Monday, it took 7 minutes to get to school and 9 minutes to get home. On Tuesday, it took 5 minutes and 9 minutes respectively, and on Wednesday, it took 8 minutes and 7 minutes.

Example 2: Reading Stem-and-Leaf Plots List the data values in

the stem-and-leaf plot.

0 2 5

1 3 3 7 8

2 0 2 6

3 1 7

Key: 3 I 1 means 31


1 2 5

2 0 1 1

3 2 7 9

12, 15, 40, 41, 41, 52, 57, 59

Example 3: Organizing Data in Stem-and-Leaf PlotsAsh 47 Elm 38 Red Maple 55

Beech 40 Grand Fir 77 Sequoia 84

Black Maple 40 Hemlock 74 Spruce 63

Cedar 67 Hickory 58 Sycamore 40

Cherry 42 Oak 61 Western Pine 48

Douglas Fir 91 Pecan 44 Willow 35

Use the data set on heights of trees in the U.S. (m) to make a stem-and-leaf plot.


Use the data on top speeds of animals (mi/h) to make a stem-and-leaf plot.

Cheetah 64 Elk 45

Wildebeest 61 Coyote 43

Lion 50 Gray Fox 42

Example 4: Organizing Data in Back-to-Back Stem-and-Leaf Plots Use the given data on Super Bowl scores, 1990-

2000, to make a back-to-back stem-and-leaf plot.

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000

Winning 55 20 37 52 30 49 27 35 31 34 23

Losing 10 19 24 17 13 26 17 21 24 19 16


Use the data on US. Representatives for Selected States, 1950 and 2000, to make a back-to-back stem-and-leaf plot.

IL MA MI NY PA

1950 25 14 18 43 31

2000 19 10 15 29 19

Section 4.3: Measures of Central Tendency

Definition Use to Answer

Mean – the sum of the values, divided by the number of values

“What is the average?”

“What single number best represents the data?”

Median – the middle number in an ordered set of data

“What is the halfway point of the data?”

Mode – the value or values that occur most often

“What is the most common value?”

Section 4.4: Variability

Variability – how spread out the data is

Range – largest number minus the smaller number

Quartile – divide data parts into four equal parts

Box-and-Whisker Plot – shows the distribution of data

Example 1: Finding Measures of Variability Find the range and the first and third quartiles

for each data set. 85, 92, 78, 88, 90, 88, 89

78, 85, 88, 88, 89, 90, 92 Range: 92-78 = 14 1st Quartile = 85 3rd Quartile = 90

14, 12, 15, 17, 15, 16, 17, 18, 15, 19, 20, 17 12, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 20 Range: 20-12 = 8 1st Quartile = (15 + 15) / 2 = 15 3rd Quartile = (17 + 18) / 2 = 17.5


Find the range and the first and third quartiles for each data set. 15, 83, 75, 12, 19, 74,

21

Range: 71 1st Quartile: 15 3rd Quartile: 75

75, 61, 88, 79, 79, 99, 62, 77

Range: 38 1st Quartile: 69 3rd Quartile: 83.5

Example 2: Making a Box-and-Whisker Plot Use the given data to make a box-and-

whisker plot. 22, 17, 22, 49, 55, 21, 49, 62, 21, 16, 18, 44, 42,

48, 40, 33, 45 Find the smallest value, first quartile, median, third

quartile, and largest value. Smallest: 16 1st Quartile: (21+21) / 2 = 21 Median: 40 3rd Quartile: (48+49) / 2 = 48.5 Largest: 62


Use the given data to make a box-and-whisker plot. 21, 25, 15, 13, 17, 19, 19, 21

Smallest: 13 1st Quartile: 16 Median: 18 3rd Quartile: 21 Largest: 25

Example 3: Comparing Data Sets Using Box-and-Whisker Plots These box-and-whisker

plots compare the number of home runs Babe Ruth hit during his 15-year career from 1920 to 1934 with the number Mark McGwire hit during the 15 years from 1986 to 2000.

Compare the medians and ranges.

Compare the ranges of the middle half of the data for each.


These box-and-whisker plots compare the ages of the first ten U.S. presidents with the ages of the last ten presidents (through George W. Bush) when they took office.

Compare the medians and ranges.

Compare the differences between the third quartile and first quartile for each.

Section 4.5: Displaying Data

Bar Graph – display data that can be grouped in categories

Frequency Table – use with data that is given in list

Histogram – type of bar graph; groups by using intervals

Line Graph – show trends or to make estimates

Example 1: Displaying Data in a Bar Graph Organize the data into

a frequency table and make a bar graph.

The following are the ages when a randomly chosen group of 20 teenagers received their driver’s licenses: 18, 17, 16, 16, 17, 16, 16, 16, 19, 16, 16, 17, 16, 17, 18, 16, 18, 16, 19, 16

Age License

Received

16 17 18 19

Frequency 11 4 3 2


Organize the data into a frequency table and a bar graph.

The following data set reflects the number of hours of television watched every day by members of a sixth-grade class: 1,1,3,0,2,0,5,3,1,3

Hours Frequency

0 2

1 3

2 1

3 3

4 0

5 1

Example 2: Displaying Data in a Histogram John surveyed 15 people

to find out how many pages were in the last book they read. Use the data to make a histogram. 368, 153, 27, 187, 240, 636,

98, 114, 64, 212, 302, 144, 76, 195, 200

Make a frequency table first.

Then, use intervals of 100 to make a histogram.

Pages Frequency

0-99

100-199

200-299

300-399

400-499

500-599

600-699


Dollars Frequency

0-0.99

1.00 – 1.99

2.00 – 2.99

3.00 – 3.99

Jimmy surveyed 12 children to find out how much money they received from the tooth fairy. Use the data set to make a histogram. 0.35, 2.00, 0.75, 2.50,

1.50, 3.00, 0.25, 1.00, 1.00, 3.50, 0.50, 3.00

Example 3: Displaying Data in a Line Graph Make a line graph of

the given data.

Use the graph to estimate the number of polio cases in 1993.

Year Number of Polio Cases

Worldwide

1975 49,293

1980 52,552

1985 38,637

1990 23,484

1995 7,035

2000 2,880


Year Salary ($)

1985 42,000

1990 49,000

1995 58,000

2000 69,000

Make a line graph of the given data.

Use the graph to estimate Mr. Yi’s salary in 1992.

Section 4.6: Misleading Graphs and StatisticsExplain why each graph is misleading.


Explain why the graphs are misleading.

Example 2: Identifying Misleading Statistics Explain why each statistic is misleading.

A small business has 5 employees with the following salaries: $90,000 (owner); $18,000; $22,000; $20,000; $23,000. The owner places an ad that reads: “Help Wanted – average salary $34,600” Only one person in the company makes more than $23,000 and that is the owner.

It is not likely that a new person’s salary would be close to $34,600. A market researcher randomly selects 8 people to focus-test three

brands, labeled A, B, and C. Of these, 4 chose brand A, 2 chose brand B, and 2 chose brand C. An ad for brand A states: Preferred 2 to 1 over leading brands!” The sample size is too small. The researcher needs to ask more people to get a

true representation. The total revenue at Worthman’s for the three-month period from June 1

to September 1 was $72,000. The total revenue at Meilleure for the three-month period from October 1 to January 1 was $108,000. They are comparing two different times of the year.


Explain why each statistic is misleading. Sam scored 43 goals for his soccer team during

the season, and Jacob scored only 2.

Four out of five dentists surveyed preferred Ultraclean toothpaste.

Shopping at Save-a-Lot can save you up to $100 a month!

Section 4.7: Scatter Plots

Scatter Plots – show relationships between two data sets

Example 1: Making a Scatter Plot of a Data Set

Compound ZIA Average Effects

Zinc Gluconate 100 Reduced cold 7 days

Zinc Gluconate 44 Reduced cold 4.8 days

Zinc Orotate 0 None

Zinc Gluconate 25 Reduced cold 1.6 days

Zinc Gluconate 13.4 None

Zinc Aspartate 0 None

Zinc Acetate-tartarate-glycine -55 Increased cold 4.4 days

Zinc Gluconate -11 Increased cold 1 day

A scientist studying the effects of zinc lozenges on colds has gathered the following data. Zinc ion availability (ZIA) is a measure of the strength of the lozenge. Use the data to make a scatter plot.


Use the given data to make a scatter plot of the weight and height of each member of a basketball team.

Height (in) Weight (lb)

71 170

68 160

70 175

73 180

74 190

Correlations

Example 2: Identifying the Correlation of Data Do the data sets have positive, a negative, or

no correlation? The population of a state and the number of

representatives positive

The number of weeks a movie has been out and weekly attendance negative

A person’s age and the number of siblings they have No correlation


Do the data sets have a positive, negative, or no correlation? The size of a jar of baby food and the number of

jars a baby eats negative

The speed of a runner and the number of races she wins positive

The size of a person and the number of fingers he has No correlation

Example 3: Using a Scatter Plot to Make Predictions

Use the data to predict the exam grade for a student who studies 10 hours per week.

About 95

Hours Studied Exam Grade

5 80

9 95

3 75

12 98

1 70


Use the date to predict how much a worker will earn in tips in 10 hours.

Approximately $24

Hours Tips ($)

4 12

8 20

3 7

2 7

11 26

Chapter 4: Collecting, Displaying, and Analyzing Data

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