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Objective Sample Problems

V/hen you finish thischapter, you will beable to:

1. Read bar graphs, linegraphs, and circlegraphs.

From the bar graph below,

100

90

80

696

Monthly Pa¡nt Jobs at Autobrite

March April

Month

(a) Determine the number of frames assembled by theTuesday day shift.

(b) Calculate the percent decrease in output from theMonday day shift to the Monday night shift.

\lVeekly Frame Assembly

February

Tuesday Wednesday Thursday Friday

Day of Week

From the line graph above,

(c) Determine the maximum number of paintjobs and the month during which it occurred.

(d) Calculate the percent increase in the number ofpaint jobs from January to February.

(e) The average job for ABC Plumbing generates

$227.50. Use the circle graph on the next page tocalculate what portion of this amount is spenton advertising.

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60

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Preview Chapter rz Statistics

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Objective Sample Problems

Percent of Business ExpendituresLabor21"/o

Overhead

Advertising18%

lnsurance Parts16o/"

Transportation

For l.lelpGo to Page

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6%

5"/"

2. Draw bar graphs,

line graphs, and cir-cle graphs from ta-

bles of data.

3. Calculate measures

of central tendency:mean, median, andmode.

4. Calculate measures ofdispersion: range and

standard deviation.

5. Calculate mean and

standard deviation fordata grouped in a fre-quency distribution.

Work Output Day

150 Mon

360 Tues

435 Wed

375 Thurs

180 Fri

For the data in thetable, draw:

(a) A bar graph.(b) A line graph.(c) A circle graph

697104707

Find (a) the mean, (b) the median, and(c) the mode for the following set of lengths.All measurements are in meters.

12.7 16.2 15.5 t3.9 t3.2 17.1 15.5

Find (a) the range and (b) the standard

deviation for the set of lengths in objective 3

(a)(b)(c)

1t8719719

728729(b)

Welding The regulator pressure settings on an

oxygen cylinder used in oxyacetylene welding (OAW)must fall between a range of 110 and 160 pounds per

square inch (psi) when welding material is one inch thick.

The following frequency distribution shows the pressures

for 40 such welding jobs. Use these data to find (a) themean and (b) the standard deviation. (a)

(b)Class Intervals Frequency, F

110-119 psi 4

120-129 psi 8

130-139 psi t2

140-149 psi 10

150-159 psi 6

722IJJ

692 Preview Chapter t2 Statistics

(Ans,wers ts these previw problems are given in the Appendix. Also, workod solutionsto mâny of these problems appe'ar in the ehapter Summary)

If you are certain you cân work øIl these problems conectly, firm to page 7 47 for a set ofpractice problems. If you cannot work one or more of the preview problems, turn to thepage indicated after the problem. Those who wish to master this material with the great-est success chould turn to Section I2-l and begin work there.

Þreview Chapter r: Stati¡tic¡ 695

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Reading and Constructing Graphs

Mea¡ures of Centrel Tendency

Measures of Dispersion

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12-I READING AND CONSTRUCTING GRAPI{S

Working with the flood of information available to us today is a major problem for sci-entists and technicians, including workers in the trades. We need ways to organize and

analyze numerical data in order to make it meaningful and useful. Statistics is a branchof mathematics that provides us with the tools we need to do this. In this chapter, youwill learn the basics of how to prepare and read statistical graphs, calculate some statis-

tical measures, and use these to help analyze data.

Note Þ In discussing statistics, we often use the word "data." Data refers to a collection of mea-

surement numbers that describe some specific characteristic of an object or person or agroup of objects or people. {

Sec. t2-I Reading and Constructing Graphs ó95

Reading Bar Graphs

Note Þ

Electrician HVAC-RTechnician

Machinist AircraftMechanic

Graphs allow us to transform a collection of measurement numbers into a visual form thatis useful, simplified, and brief. Every graph tells a story that you need to be able to read.

A bar graph is used to display and compare the sizes of different but related quantities.

The graph below shows the median hourly wage for workers in six trades occupations.Hourly wages, in dollars per hou¡ are listed along the left side on the vertical axis. The sixtrade occupations being compared are listed, evenly spaced, along the horizontal axis.

2004 Median Hourly Wages

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sõo

$25

$20

$ls

$10

$s$o

Welder DieselMechanic

Occupation

Some information is readily available from the graph: The highest hourly wage (tallestbar) is earned by the aircraft mechanic; the lowest median wage (shortest bar) is earnedby the welder. When the top of a bar is directly opposite a mark on the vertical axis, it iseasy to read its value. Welders earn a median wage of $15 per hour. When the top of abar falls between two marks on the vertical axis, we must estimate its value. Aircraft me-chanics earn a median wage of approximately 522 per hour.

Other calculations may be made from the information in the graph. Using the proportiontechnique from Chapter 4, we can calculate the percent increase of an electrician'shourly wage, $21, over that of a welder, $15.

2l-15 R _ 6x100: _ or R: :4OVo15 r00 15frìI¡¡¡¡ I

lrDoDll¡uoDll.ooEl 15 E 1002t 15 -+ q0.

A bar graph like the one in the previous example, drawn with vertical bars, is called averticul bur gruph.It is sometimes helpful to constrlrct altorízontal bar graph by sim-ply switching the horizontal and vertical axes. {A double bar graph allows us to show side-by-side comparisons of related quantities onthe same graph.

The following double bar graph shows the effect of shielding on computer componentsexposed to radiation.

Notice that in drawing the graph, we arranged the computer components in order of sur-vival time, labeled them from Ato F, and marked them along the vertical axis. This isdesigned to make the graph easy to read.

Vertical axis

Tallest barShortest bar

Horizonlal axisEqual spaces

( )

696 Chap. l2 Statistics

Computer Component vs. Radiation

A

C

D

E

F

5 't0 15 20 25 30

Radiation, Dose to Failure (in hours)35

The graph shows that component C survived about 8.5 hr without shielding and 20 hrwith shielding.

Use the preceding graph to answer the following questions.

(a) V/hich computer component, with shielding, survived the longest?

(b) By what percent do the survival times differ between shielded components B and D?

(c) Find the difference in survival times for component E when it is shielded and un-shielded.

B

ocoCLEoo

(a)

(b)

ComponentA survived the radiation longest at28hr

Set up a proportion:

23 - 17.5 R

17.5 100

5.5 x 100Solve R: x 31.47o

r7.5

Component B survived the radiation about 3l%o longer than did component D

(c) The difference is 15 hr - 7 hr : 8 hr.

Drawing Bar Graphs Usually, abar graph is created not from mathematical theory or abstractions, but is con-structed from a set of measurement numbers.

Suppose you want to display the following data in a bar graph showing sales of DVDplayers and televisions in five stores.

Sec. l2-l Reading and Constructing Graphs 697

I shield E No shield

Quarterly Sales of DVD Players and TVs at Five Stores

Store DVD Players Televisions

Ace 140 65

Wilson's 172 130

Martin's 185 200

XXX 195 285

Shop-Rite 190 375

To draw a bar graph, follow these steps:

Step 1 Decide what type of bar graph to use. Because we are comparing two differentitems, we should use a double bar graph. The bars can be placed either horizon-tally or vertically-you get to choose. In this case let's make the bars horizontal.

Step 2 Choose a suitable spacing forthe vertical (side) axis and a suitable scale forthehorizontal (bottom) axis. Label each axis. When you label the vertical or sideaxis it should read in the normal way.

Name ofstore

Quarterly Sales of DVD Players andTVs at Five Stores

Shop-Rite

Martin's

Wilson's

Ace

0¡

oat

ooEGz

Nameofstofe

XXX

EoU'

ooEoz

50 100 150 200 250

Number Sold

Do this: But never this:

Be sure toinclude a titlefor your graph

..as betweenand 250..

Thickness of eachbar is the samefor all bars

Spacebetweenbarsis thesame

...or anyother 50 units

Same spacebetween 150

range

698

300 350 400

Ghap. 12 Statistics

Notice that the numbers on the bottom axis are evenly spaced and cover the entire range

of numbers in the data, in this case from 65 to 375.

Also notice that the stores were arranged in order of amount of sales. The biggest seller,

Shop-Rite, is at the top, and the smallest seller, Ace, is at the bottom. This is not neces-

sary, but it makes the graph easier to read.

Step 3 Use a straightedge to mark the length of each bar according to the data given.Round the numbers if necessary. Drawing your bar graph on graph paper willmake the process easier.

The final bar graph will look like this:

Sales of DVD Players andTVs at Five Stores

Shop-Rite

Martin's

Wilson's

50 100 1s0 200 250 300

Number Sold350 400

XXXEoU'

ooE(ll2

Ace

Use the following data to make a bar graph.

Average Ultimate Compression Strengthof Common Materials

ru DVD ptayers I TVs

MaterialCompression Strength

(psi)

Hard bricks 12,000

Light red bricks 1,000

Portland cement J 000

Portland concrete 1,000

Granite 19,000

Limestone and sandstone 9,000

Trap rock 20,000

Slate 14,000

Sec. t2-l Reading and Constructing Graphs 699

Averge Ultimate Compression Strengthof Common Materials

Trap Granite Slate Hard Lime- Portland Light Portlandrock bricks stone or cement red concrete

sandstone bricks

Materials

Don't worry if your graph does not look exactly like this one. The only requirement isthat all of the data be displayed clearly and accurately.

A Closer Look Þ Nofice that the vertical scale has units of thou,sond,s of pouncls per square inch. Tt wasdrawn this way to save space. {

Energy Consumption of Gas Appliances

Range Log Refrigeratorlighter

1. Using the bar graph shown, answer the following questions.

(a) Which appliance uses the most energy in an hour?

(b) Which appliance uses the least energy in an hour?

(c) How many BTU/hr does a gas barbecue use?

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050-gal Barbecue Top Clolhes Ovenwater burners dryerheater

Gas Appliances


(d) How many BTUlday (24hr) would a 50-gal water heater use?

(e) Is there any difference between the energy consumption of the range and that ofthe top burners plus oven?

(f) How many BTUs are used by a log lighter in 15 min?

(g) What is the difference in energy consumption between a 50-gal water heater

and a clothes dryer?

2. Printing The bar graph shown compares the stamping rate (in stamps per minute,spm) for several high-speed industrial presses.

(a) Which press has the highest stamping rate?

(b) Which press has the slowest stamping rate?

(c) If your shop upgrades from a Minster Production Press to a Bruderer, what per-cent increase in stamp rate would you expect?

Maximum Stamp Rates for High-Speed Presses

MinsterProduction

Press

Company Name

3. Metalworking Draw the bar graph for the following information. (Hint: Use mul-tiples of 5 on the vertical axis.)

Linear Thermal Expansion Coefficients for Materials

MaterialCoefficient

(in parts per million per "C)

Aluminum 23

Copper 17

Gold I4

Silicon J

Concrete t2

Brass t9

Lead 29

o

.g

=oCLatCLE(ú

(t,

600

500

400

300

200

100

0Niagara Press Oak Model LP Bruderer Press

Press

IIt I II I I II I I II I I I

Sec. tz-r Reading and Constructing Graphs 701

I know I need to be able toread graphs in my job, butwhy do I need to learnto draw them too?

1. (a)

(d)

(e)

2' (a)

Range (b) Refrigerator

1,200,000 BTU/day (e) No

15,000 BTU/hr

Bruderer (b) Minster

Linear Thermal Expansion Coefficients forMaterials

(c) Approximately 837o

(c)

(f)

50,000 BTU/hf

6250 BTU

30

25

20

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Reading Line Graphs A line graph, or broken-line graph, is a display that shows the change in a quantity,usually as it changes over a period of time.

The following broken-line graph shows monthly sales of tires at Treadwell Tire Com-pany. The months of the year are indicated along the horizontal axis, while the numbersof tires sold are shown along the vertical axis. The actual data ranges from 550 to 950,so the vertical axis begins at 500 and ends at 1000, with each interval representing 50tires. Each dot represents the number of tires sold in the month that is directly below thedot, and straight line segments connect the dots to show the monthly fluctuations insales.

T

702 Chap. tz Statistics

Monthly Tire Sales at Treadwell Tlre Company

1000

950

900

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800

750

700

650

600

550

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Month

The following example illustrates how to read information from a brokenJine graph.

To find the number of tires sold in May, find May along the horizontal axis and follow theperpendicular line up to the graph. From this point, look directly across to the vertical axis

and read or estimate the number sold. There were approximately 750 tires sold in May.

{9ö ooN dg\

Other useful information can be gathered from the graph:

The highest monthly total was 950 inAugust.The largest jump in sales occuned from July to August, with sales rising from 800 to

950 tires. To calculate the percent inmease in August, set up the following proportion:

950 - 800 R 150 A

850

800

750

70a

650

600

550

or800 100 800 100

150 x 100R_800

= t8.75%o

The increase in sales from July to August was approximately t9%,

Sec. ru-r Reading and Con¡tructing Graphs

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705

(a)

(b)

(c)

What was the lowest monthly total, and when did this occur?

How many tires were sold during the first three months of the year combined?

During which two consecutive months did the largest drop in sales occur? By whatpercent did sales decrease?

Only 550 tires were sold in December.

We estimate that 580 tires were sold in January, 650 in February, and 700 in March.Adding these, we get a total of 1930 for the three months.

The largest drop in sales occurred from October to November, when sales de-creased from 750 to 600. Calculating the percent decrease, we have

750 - 600 R rsO R

(a)

(b)

(c)

750 100 750 100

150 x 100R_750

: 20Vo

Sales decreased by 207o from October to November

Drawing Line Graphs Constructing a broken-line graph is similar to drawing a bar graph. If possible, beginwith data ananged in a table of pairs. Then follow the steps shown in the example.

The given table shows the average unit production cost for an electronic component dur-ing the years 2001-2006.

YearProduction Costs

(per unit)

2001 $5.16

2002 $s.33

2003 $s.04

2004 $5.s7

2005 $6.ss

2006 $6.94

Step I Draw and label the axes. According to convention time is plotted on the hori-zontal axis. In this case, production cost is placed on the vertical axis. Space theyears equally on the horizontal axis, placing them directly on a graph line andnot between lines. Choose a suitable scale for the verlical axis. In this case, eachgraph line represents an interval of $0.20. Notice that we begin the verlical axisat $5.00 to avoid a large gap at the bottom of the graph. Be sure to title the graph.

Chap. tz Statistics

or

EXAMPLE

701

Step 2 For each pair of numbers, locate the year on the horizontal scale and the costfor that year on the vertical scale. Imagine a line extended up from the year andanother line extended horizontally from the cost. Place a dot where these twolines intersect.

For the table given, at2002 the cost is $5.33.

5.20

2003

Step 3 After all the number pairs have been placed on the graph, connect adjacentpoints with straight-line segments.

Production Cost, 2001-2006

$7.00

$6.80

$6.60

$6.40

$ô.20

$ su.oo

$5.80

$5.60

$5.40

$5.20

2042 2003 2004

Year

2005 2006

A double-line graph is one on which two separate sets of information are plotted. This kindof graph is very useful for comparing two quantities that vary over the same time period.

Plot the following data on the graph in the previous example to create a double-line graph.

Year Shipping Cost (per unit)

2001 $s.30

2002 $s.61

2003 $6.0s

2004 $6.20

2005 $6.40

2006 $6.50

5

2001

/

4- '/

5.33 *

2002

Sec. t2-l Reading and Conrtructing Graphs 7o5

o Shipping costo Production cost

anoo

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'/-'- ,/- Y

2002 2003 2004 2005 2006

A circle graph, or pie chart, is used to show what fraction of the whole of some quan-tity is represented by its separate parts.

The circle graph shown below gives the distribution of questions contained on a com-prehensive welding exam. The area of the circle represents the entire exam, and the

wedge-shaped sectors represent the percentage of questions in each part of the exam.

The percents on the sectors add up to I00Vo.

Welding applications take up 30Vo of the exam, so its sector is307o of the 360" that makeup a circle.

Number of Questions on a ComprehensiveWelding Exam

Occu pational

30%

skills24%

Gouging Quality control16%applications

8"/o

Cuttingapplications

22o/"

Use this graph to answer the following questions.

(a) What percent of the exam dealt with cutting?

(b) Which question topic was asked about least?

(c) If you had only 4 hr to review for this exam, how much of that time should youspend on quality control? (Assume that study time for each section is proportionalto its percent of the test.)

Chap. t2 Statistics

$7.00

$6.80

$6.60

$6.40

$6.20

$6.00

$5.80

$5.60

$5.40

$5.20

Year

Reading CircleGraphs

Weldingapplications

706

(a)

(c)

22Vo (b) Gouging

If p hours are spent studying quality control, and 4 hr or 240 min are spent study-ing total, then set up a proportion:

o16240

o

100

16 x 240

100

: 38.4 min = 38 min

ConstructingCircle Graphs

If the data to be used for a circle graph is given in percent form, determine the number

of degrees for each sector using the quick calculation method explained in Chapter 4.

Convert each percent to a decimal and multiply by 360'.

In a survey of automotive tasks, the following data were found to represent a typicalwork week.

Transmission

Tune-up

Front-end

Diagnostics

Exhaust

657o

I57o

I07o

5Vo

5%o

Extend the table to show the calculations for the angles for each sector

Finally, using a protractor mark off the sectors and complete the circle graph. Notice that

each sector is labeled with a category name and its percent.

Automotive TechniciansWorkWeek

Exhaust, 5%

Diagnostic,5%

Front-end,'lo%

ïune-up, 15%Transmission,

65To

Transmission

Tune-up

Front-end

Diagnostic

Exhaust

234,þ657o of 360"

54 <= I5qo of 360"

36 <r IjVo of360"18+ 57oof360'

657o

I5Vo

rc%57o

5Vo

0.65

0.15

0.10

0.05

0.05

360"

360'360"

360"

360'

234"

54"

36

1go

180

X

X

X

X

X+ 57oof360"

Sum = 100% Round to thenearest

Sum = 360"


If the data for a circle graph are not given in percent form, first convert the data to per-cents and then convert the percents to degrees.

TheZapp Electric Company produces novelty electrical toys. The company has a verytop-heavy compensation plan. The CEO is paid $175,000 yearly; the VP for finance,$70,000; a shop supervisor, $55,000; and 10 hourly workers, $25,000 each. Draw a cir-cle graph of this situation.

It is helpful to organize our information in a table such as this:

First, fincl the sum of the compensations.

$175,000 + 70,000 + 55,000 + 250,000 : $550,000

Second, use this sum as the base to calculate the percents needed. Because all of the com-pensations end in three zeros, these may be dropped when setting up the proportions.

n5c70v55s2508550 100 550 100 550 100 550 100

Cx32o/o VxI3Vo S:107o Ex45Vo

Third, use the percent values to calculate the angles in the last column.

0.32 x 360' æ 115o 0.13 x 360" x 47"

0.10 X 360o : 36o 0.45 X 360" : 162o

The completed table:

Finally, use the angles in the last column to draw the circle graph. Label each sector as

shown.

Employee Compensation Percent Angle

CEO $175,000

VP 70,000

Supervisor 55,000

10 employees 250,0001...

sum = $550,000

Employee Compensation Percent Angle

CEO $175,000 32Vo 5011

VP 70,000 137o 47"

Supervisor 55,000 I07o 36

10 employees 250,000t\

45%/ 162'

Sum = I 00%sum = $550,000 Sum = 360"


cEo32%Supervisor

10% Employee Compensationat Zapp Electric

VP 13%

10 employees45"/"

Jane the plumber keeps a record of the number of trips she makes to answer emergency

calls. A summary of her records looks like this:

Trip Length Number of Trips

Less than 5 miles r525-9 miles 25

l0-19 miles 49

20-49 miles 18

50 or more miles 10

Calculate the percents and the angles for each category and plot Jane's data in a circle graph.

The completed table shows the percents and angles for each category

Trip Length Number of Trips Percent Angle

Less than 5 r52 607o 216

5-9 25 I0Vo 360

10-19 49 19Vo 68"

20-49 18 77o 250

50+ 10 Vo r4"

254 I007o 359"

Because of rounding, the percents will not always sum to exactly I00Vo and the angles

will not always sum to exactly 360". In this case, the angles added up to 359". This willnot noticeably affect the appearance of the graph.

The circle graph is shown below.

Jane's Trips5-9 mi

1Oo/o

Less than 5 mi60%

10-19 mi19lo

20-49 mi

50+ mi4o/o

Sec. t2-t Reading and Constructing Graphs 709

Now turn to Exercises I2-I for more practice in reading and constructing bar graphs,line graphs, and circle graphs.

Exercises Lz-L Reading and Constructing Graphs

A. Answer the questions following each graph.

1. Automotive Trades and Police Science The following bar graph from theU.S. Department of Transportation, National Highway Traffic Safety Administra-tion, shows the 2003 percent rollover occuffence in fatal crashes by vehicle type.

2003 Percent Rollover Occurrence in Fatal Crashes byVehicle Type

Passenger Pickup SUV Van Other l¡ghttruckcar

Vehicle TypeSource: Trffic Safety Facts 2003, Washington DC, January 2005

(a) Which vehicle type had the highest percent of rollover occurrence?

(b) Which vehicle had the lowest?

(c) What was the percent of rollover occuffence for a passenger car?

(d) Calculate the percent increase in rollover occurrences between passenger

cars and SUVs.

Machine Trades The following bar graph shows the recommended cuttingspeeds when turning certain materials.

Recommended Cutting Speeds for TurningCeftain Materials Using a Carbide Tool

ootro

oooLo.9õtr

o(,)

0)o.

40.035.030.025.020.015.0

10.0

5.0

0.0

700

600

500

400

300

200

100

2.

o=,g

=í)CL

0,otl-

II II I

I I III III I I I I0 Carbon Alloy steel Stainless

steel

Material

Manganesebronze

Brasssteel

(a) Which material requires the lowest cutting speed, and what is that speed?

(b) Which material requires the highest cutting speed, and what is that speed?

(c) What is the cutting speed for alloy steel?

(d) What is the difference in cutting speeds between manganese bronze andcarbon steel?

35.7

24.5

9.8


3. Heating and Air Gonditioning The following double-bar graph represents

the energy efficiency ratio (EER) and seasonal energy efficiency ratio (SEER)

for different models of residential air conditioners. The EER is the measure ofthe instantaneous energy efiiciency of the cooling equipment in an air condi-tioner. The SEER is the measure of the energy efficiency of the equipment overthe entire cooling season. Both efficiency ratios are given in British thermalunits per watt-hour (Btu/Wh).

The EER and SEER for 3-ton ResidentialCentral Air Conditioners

T EER

H seen

Base model Mid-range model High-end model

Air Conditioner Model

(a) What is the approximate EER for the base model?

(b) Which model has a SEER of about 13?

(c) Calculate the percent increase in SEER from the base model to the high-end model.

4. Police Science The following bar graph represents crime statistics over athree-year period.

Yearly lncidence of Crimes for Gotham City from2003-2005

I 2oo3

Iu

20042005

Robbery Burglary Grand larceny Grand larceny auto

Grimes Against Property

Over which two years did the number of incidents of grand larceny in-crease?

Which two crimes had the highest number of occurrences in 2004?

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16

14

12

10

I6

4

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450

400

350

300

250

200

150

100

50

6trop.Jgoo.clEJz

0

(a)

(b)

Sec. t2-t Reading and Constructing Graphs 711

5

(c) Which two crimes had the highest number of occurrences in 2003?

(d) In which category are crimes decreasing?

Sales at Corey's Beach Fashions

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$40

$zo

$1 40

$120

20

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Day of Week

(a) On what day are sales at Corey's highest?

(b) Calculate the percent increase in sales from Monday to Thursday.

(c) On what days .are

sales abovc $10,000?

6. Automotive Trades The following line graph reflects the average fuel econ-omy of a seleeted group of automobiles.

Fuel Economy by Speed

35

30

-/H

-À-^ --fI

\

I I

15 20 25 30 35 40 45 s0 55 60 65 70 75

Speed (mph)

(a) What speed has the best fuel economy? The worst?

(b) What is the fuel economy at 65 mph? At 40 mph?

(c) At what speed is the fuel economy 29 mpg?

(d) After which two speeds does fuel oconomy begin to decrease?

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I

7t2

7. Transportation Technology The following double-line graph compares thecondition of paved roads in Mission County in 1989 versus 2005.

Condition of Paved Roads in Mission County

2005,/

1989^/

I I

Failed Poor Fair Good Verygood

(a) In 2005, what percent of the Mission County roads were rated as fair?

(b) In 1989, what percent were rated as very poor?

(c) If there were 1500 miles of county roads in 2005, how many miles of countyroads were rated as very good?

Land Area of the Earthby Continents

Africa

Australia

North America16%

5o/"

Europe America7%

Antarctica10"/"

12%

(a) Which are the largest two continents on the earth?

(b) Which two are the smallest?

(c) Which continent covers l27o of the earth?

School ExpendituresLab fee

B%

Materials fee10%

Books andlab manuals

1B"k

Tuition3ïo/o

Tools26%

(a) 'Which category represents the largest expenditure?

(b) Which category represents the smallest expenditure?

(c) If a student spent a total of $2500, how much of this went toward tools andlab fees combined?

(d) If a student spent $300 on materials, how much would she spend on booksand lab manuals?

7060

-505¿oob30À20

10

0Verypoor

8

20%

9

Sec. l2-t Reading and Constructing Graphs 715

B. From the following data, construct the type of graph indicated.

1. Fire Protection Plot the following data as a bar graph.

Gauses of Fires in District 12

Cause of Fire Number of Fires

Appliance 6

Arson 8

Electrical 18

Flammable materials 7

Gas 2

Lightning 2

Motor vehicle 10

Unknown 9

2. Electrical Engineering Plot the following data as a bar graph.

California's Sources of Electricity

Source Percent Source Percent

Geothermal lVo Gas 237o

Nuclear 67o Hydroelectric 22Vo

Coal 77o oil 41.7o

3. Finance Plot the following data as a bar graph. The following table showsthe revenues of the Hy-Tek Corporation by quarters and billions of dollars.

Quarter Revenue (x $1 billion)

1 $6

2 $+

J $s.s

4 $z

4. Plot the following data as a bar graph.

Radish Tool and Dye Worker Experience

Length of Service Number of Workers

20 years or more 9

r5-19 7

10-14 1.4

5-9 20

1.-4 33

Less than 1 year 5

7r1 Chap. tz Stati¡tics

5. Construction Plot the following data as a double-bar graph.

Sales of Construction Material: CASH lS US

Year Wood (x $1000) Masonry (X $1000)

2000 $289 $131

2001 $32s $33

2002 8296 $106

2003 $288 $e2

2004 $307 $e4

2005 8412 $8e

6. Finance Plot the following data as a double-bar graph.

Earnings and Dividends for Red lnc.

7 , Finance Plot a brokenJine graph for these data.

Earnings per Share (Sl B.l.G. Co.

Year Earnings per Share

1998 $0.7

t999 $0.s

2000 $1.s

2001 $3.0

2002 $3.2

2003 $1.8

2004 $2,s

2005 82.4

2006 $3.s

Year Earnings per Share Dividends per Share

2000 $1.12 $0.28

2001 $1.27 $0.32

2042 $1.75 $0.40

2003 $1.92 $0.40

2004 $2.1"6 $0.4s

2005 $2.8s $0.ss

Sec. lz-l Reading and Constructing Graphs 715

8. Plumbing Plot a broken-line graph for these data.

Households Lacking Gomplete Plumbing

9. Health Care Plot a broken-line graph for these data.

Scooter lnjuries-Year 2000

10. Health Care Plot a broken-line graph for these data.

Percentage of Male Nurses, 1890-2000

Year Percent Year Percent

1890 13Vo 1950 2Vo

1900 l07o 1960 2Vo

1910 77o 1970 3Vo

t920 4Vo 1980 4Vo

1.930 27o 1990 67o

1940 l7o 2000 77o

11. Meteorology Plot a double broken-line graph for these data.

Yearly Occurrences of Tornados, Kansas and Missouri

Year Kansas Missouri

2000 65 28

200r t1.2 42

2002 t02 -t-'t

2003 93 1 1 1

2004 r25 80

2005 140 40

Dates Percent Dates Percent

1940 457o r970 77o

1950 357o 1980 37o

1960 177o 1990 IVo

Month Number Month Numbcr

January 100 June 1200

February 80 July 2300

March 50 August 6100

April 500 September 8500

May 400 October 7700

716 Chap. 12 Statistics

12. Finance Plot a double broken-line graph for these data.

Actualand Projected Sales, IMD Gorp. (x Sf000l

Month Actual Projected

January $10 $4s

February $18 $s0

March $12 547

April 522 $s0

May $40 $6s

June $39 $zt

July $50 $76

August 542 $7s

September $3s $ss

October $37 $60

November $41 $s0

December 944 $87

13. Electrical Trades The primary battery market in the United States in 2004 isshown in the following table. Plot this information as a circle graph.

Duracell

Energizer

Rayovac

Panasonic

Toshiba

All others

28.5%

24.57o

19.2Vo

9.\Vo

8.87o

9.2Vo

14. Finance Plot a circle graph using these data.

Revenue for MEGA-CASH Gonstruct¡on Co., 2006 (S billions¡

First

Quarter5.5

Second

Quarter7.0

ThirdQuarter

6.5

Fourth

Quarter3.0

15. Aviation An aircraft mechanic spends I2.5Vo of a 40-hr week working on air-craft airframes, 37 .5Vo of the week on landing gear, 43.7 5Vo of the week work-ing on power plants and 6.25Vo of the week on avionics. Plot a circle graphusing this information.

Check your answers in the Appendix, then turn to Section 12-2 to learn about measuresof central tendency.


12.2 MEASURES OF CENTRAL TENDENCY

A measure of central tendency is a single number that summarizes an entire set of data.

The phrase "central tendency" implies that these measures represent a central or middlevalue of the set. Those who work in the trades and other fields, as well as consumers, use

these measures to help them make other important calculations, comparisons, and pro-jections of future data. In this chapter, we shall study three measures of central tendency:the mean, the median, and the mode.

Mean As we learned in Chapter 3, the arithmetic mean, also referred to as simply the meanor the average, of a set of data values is given by the following formula:

sum of the data valuesMean :

the number of data values

Note Þ In this chapter, if the mean is not exact, we will round it to one decimal digit more than

the least precise data value in the set. {

An airplane mechanic was asked to prepare an estimate of the cost for an annual in-spection of a small Bonanza aircraft. The mechanic's records showed that eight priorinspections on the same type of plane had required 9.6, 10.8, 10.0, 8.5, 1 1.0, 10.8, 9.2,

and 11.5 hr. To help prepare his estimate, the mechanic first fbund the mean inspectiontime of the past jobs as follows:

Mean inspection time - sum of prior inspection times

number of prior inspections

9.6 + 10.8 + 10.0 + 8.5 + 11.0 + 10.8 + 9.2 + ll.58

81.4: -:

10.175 = 10.18hr

To complete his cost estimate, the mechanic would then multiply this mean inspectiontime by his hourly labor charge.

Machine Trades A machine shop produces seven copies of a steel disk. The thick-nesses of the disks, as measured by a vernier caliper are, in inches, I.738,1.74I,1.738,1.740, I.739, 1.737 , and 1.740. Find the mean thickness of the disks.

1.738 + t.741 + 1.738 + r.740 + 1.739 + 1.737 + 1.740Mean thickness :

7t2.173 : 1.739 in.

7

t.74+ + + + + + --)7

1.739

718

frll¡¡¡¡ll¡oDDlIiEEEJ

1.738 L.741 1.738 1.74 1.739 t.737

Chap. tz Statistics

Median Another commonly used measure of central tendency is the median. The median of a set

of data values is the middle value when all values are arranged in order from smallest tolargest.

To determine the median thickness of the disks in the previous Your Turn, first wearrange the seven measurements in order of magnitude:

r.737 r.738 r.738 L739 L740 1.740 r.74r

The middle value is L739 because there a¡e three values less than this measurement andthree values greatü than it. Therefore, the median thickness is 1.739 in. Notice that forthese data, the median is equal to the mean. This will not ordinarily happen.

In the previous example, the median was easy to find because there was an odd numberof data values. If a set contains an even number of data values, there is no single middlevalue, so the median is defined to be the mean of the two middle values.

Aviation Find the median inspection time for the data in the first example of this section.

First, arrange the eight prior times in order from smallest to largest as follows:

8.5 9.2 9.6 10.0 10.8 10.8 11.0 11.5

Next, because there is an even number of inspection times, there is no single middlevalue. We must locate the two middle times. To help us find these, we begin crossing outleft-end and right-end values alternately until only the two middle values remain.

8ã e2, % 9Æ 1+ß Lre LÆ

The two middle times

Finally, calculate the mean of the two middle values, This is the median of the set.

Median inspection time : 10'0 t 10'8 : 10.42

Notice that the median inspection time of 10.4 differs slightly from the mean time of 10.18

Note Þ The median and the mean are the two most commonly used "averages" in statisticalanalysis. The median tends to be a more representative measure when the set of datacontains relatively few values that are either much larger or much smaller than the restof the numbers. The mean of such a set will be skewed toward these extreme values andwill therefore be a misleading representation of the data. {

Mode The final, and perhaps least used, measure of central tendency that we will consider is themode. The mode of a set of data is the value that occurs most often in the set. If there is novalue that occurs most often, there is no mode for the set of data. If there is more than onevalue that occurs with the greatest frequenc¡ each of these is considered to be a mode.

Sec. tz-z Measures of Central Tendency 719

EXAMPLE

The mode inspection time for the Bonanza aircraft (see the previous example) is 10.8 hrIt is the only value that occurs twice.

If possible, find the mode(s) of each set of numbers.

(a) 3,7,6,4,8,6,5,6,3 (b) 22,26,21,30,25,28 (c) 9,12,13,9, 11, 13,8

(a)

(b)

(c)

The value 6 appears three times, the value 3 appears twice, and all others appear

once. The mode is 6.

No value appears more than once in the set. There is no mode.

Both 9 and 13 appear twice each, and no other value repeats. There are two modes,

9 and 13. A set of data containing two modes is often refered to as a bimodal set.

Find the mean, median, and mode of each set of numbers.

1. 22, 25, 23, 26, 23, 2r, 24, 23

2. 6.5, 8.2,7.7, 6.7, 8.9,7.3, 6.9

3. r57, r53, r55, r57, 160, 153, 159, 158, 166,154, 168

4. 1.73, r.l 4, r.J l, 1.78, 1.72, 1.85

5. Construction For quality control purposes, 8 ft-long 2-by-4s are randomly se-

lected to be carefully measured. Find the mean, median, and mode for the followingmeasurements.

8'0000' 8'å" B'# i'#" 1'1" 8'0000' 8'*" 8'*" 7'1" 8'0000'

1. Mean: 23.375 or 23.4, rounded; median: 23;mode:23

2. Mean: 7.457 . . . or 7.46, rounded; median: 7.3; mode: none

3. Mean: 158.18 . . . or 158.2, rounded; median: 157; modes: 153 and 157

4. Mean: 1.755; median: 1.735: mode:none

t'5. Mean:8- or 8.0125'; Median:8'0000"; Mode:8'0000'80

720 Chap. t2 Statistics

Grouped FrequencyDistributions

When a set of data contains alarge number of values, it can be very cumbersome to dealwith. In such cases, we often condense the data into a form known as a frequency dis-tribution. In a frequency distribution, data values are often grouped into intervals ofthesame length, known as classes, and the number of values within each class is tallied. Theresult of each tally is called the frequency of that class.

Consider the following data values, representing the number of hours during one weekthat a group of drafting students spent working on the conception, design, and modelingof a crankshaft.

19 27 56 37 6r33 30 37 16 2546 16 30 32 63

747466

39 53 1324 t3 23

4215

24

464T

30 48 5934 27 56

26 4532 45

39l7

2I31

To construct a grouped frequency distribution, we must first decide on an appropriate in-terval width for the classes. We shall use intervals of t hr beginning at 10 for this data.*We then construct a table with columns representing the "Class Intervals" and the "Fre-quency, F," ot number of students whose hours on the project fell within each interval.To help determine the frequency within each class, use tally marks to count data points.Cross out each data point as it is counted. The final table looks like this:

Class Intervals Tally Frequency, F

r0-19 I.l.t I 6

20-29 }il.I.III 8

30-39 ìi.tt Ìil.t II 12

40-49 tlll il 7

50-59 4

60-69 il 2

70-79 I

Construct a frequency distribution for the following percent scores on a gas tungstenarc welding project. Use class intervals with a width of 14 percentage points beginningat 417o.

81 75 9365 44 837t 92 79

568355

569375

87 9365 49

4680

6742

9l58

81

791686

75 7062 78

t2 7363 54

*Statisticians consider r.nany tactors when deciding on the width of the olass intervals. This skill is beyond the scope ofthis text, and all ploblems will include an appropriate instluction.

Sec. t2-2 Measures of Central Tendency 721

Mean of GroupedData

Class Intervals Tally Frequency, F

41-55 l'il.tl 6

56-70 r1.l.l. li.+l 10

71-85 'ti.t{fl.1.1.il.+.].II t7

86-100 tf+{ I I 7

To calculate the mean of data that are grouped in a frequency distribution, follow these steps:

Step 1 Find the midpoint, M of each interval by calculating the mean of the endpoints

of the interval. List these in a separate column.

Step 2 Multiply each frequency, F, by its corresponding midpoint, M, and list these

products in a separate column headed "Prodrtct", FM."

Step 3 Find the sum of the frequencies and enter it at the bottom of the F column. We

will refer to this sum as n, the total number of data values.

Step 4 Find the sum of the products from Step 2 and eîter it at the bottom of the FMcolumn.

Step 5 Divide the sum of the products from Step 4 by n. This is defined as the mean

of the grouped data.

To find the mean number of hours worked on the crankshaft design (see the previous ex-

ample), extend the table as shown. The tally column has been omitted to save space. The

results of each step in the process are indicated by arrow diagrams.

Step 2Step 1

Produ'ct, FMClass Intervals Frequency, F Midpoint, M

8710-19 6 t4.5

19620-29 8 24.5

41.430-39 T2 34.5

311.54049 7 44.5

21850-59 4 54.5

12960-69 2 64.5

74.5 74.570-79 1

Sum: 1430n:40

Step 4Step 3


Itt¡¡rooo¡ODDITìNN

sumofprodvcts,FM 1430SteP5 Mean:----t :

^:35.75hr

Using a calculator for Steps 2-5, we enter:

6 X 14.5 X t28 24.5

: 35.8 hr, rounded

34.5 @ 7 44.5 + 4 54.5 2 XX+ +

64.5 @ 74.5 E +O Q +,¡,.,i.,..i$,5¡7

Note Þ The mean of grouped data is not equal to the mean of the ungrouped data unless the midpointof every interval is the mean of all the data in that interval. Nevertheless, this method for find-ing the mean from a frequency distribution provides us with a useful approximation whenthere are a large number of data values. In the previous example, the mean of the ungroupeddata is 35.775 hr, which rounds to the same result obtained from the grouped data. {

Find the mean score from the gas tungsten arc welding project (see the previous Your Tüm).

step 5 Meanscore : sumof products'FM :'q.?t : 72.375von40: 12.A%o,rounded

A Closer Look Þ The mean of the ungrouped scores is7L7007o.There is a slight discrepancy with the meanof the grouped scores because the class intervals were relatively wide in this case. {

Now turn to Exercises l2-2 for more practice on measures of central tendency.

Exercises l2-2 Measures of Central Tendency

A. Find the mean, median, and mode for each set of numbers.

1. 9,4, 4,8,7, 6,2,9, 4,7

2. 37, 32, 31, 34, 36, 33, 35

Sec. l2-2 Measures of Central Tendency

X X +

Step 1 Step 2

Class Intervals Frequency, F Midpoint, M PXduct, FM

4I-55 6 48 288

56-:70 10 63 630

71-85 l7 78 t326

86-100 7 93 651

n:40 Sum 2895

Step 3 Step 4

725

3. 98,79, 99,79, 54, 52,98, 58, 73, 62, 54, 54

4. 3.488, 3.358, 3.346, 3.203, 3.307

5. 123, 163, 149, 132, r83, 167 , 105, 192

6. 7.9, 8.t,9.5, 8.1, 9.6,8.7,7.7,9.5,8.r

7. 1188, 1176, 1128, 1126, 1356, 1151, 1313, 1344, 1367, 1396

g. 0.39, 0.39, 0.84, 0.11, 0.78, 0.18, 0.78

B. Construct an extended frequency distribution for each set of numbers and calculate

the mean of the grouped data. Follow the instructions for interval width.

1. Use intervals of 0.09 beginning at 1.00.

r.52 1.68 r.46 1.13 1.89 r.44 r.56 1.09 1.81 t.19 t.23r.46 1.91 1.40 1.30 1.35 t.79 1.61 1.80 t.26 1.29 r.40r.57 1.53 r.28 1.45 r.12 1.35 1.93 1.86 t.95 t.70 1.88

r.t6 1.05

2. Use intervals of 49 beginning at 100.

245 106 rr2 32r 235 209 263 215 168 350 340 401 r79433 286 r4r 358 468 166 498 341 t7r r19 362 264 325

225 39r r33 121

C. Applications

1. Aviation BF Goodrich produces brake pads for commercial airliners. Two pro-

duction teams are working to produce 15-lb brake pads over a six-month period.

Team A works on one furnace deck, while production team B works on a second

furnace deck. A 15-lb carbon brake pad costs $1000 per pad to produce. Use the

information in the table to answer the questions that follow.

Month TeamA Team B

January 37 ,l50 brake pads 40,000 brake pads

February 34,500 41,500

March 35,250 39,000

April 38,750 35,750

May 39,250 32,500

June 36,500 33,250

(a) Calculate the mean monthly production for team A.

(b) Calculate the mean monthly production for team B.

(c) Find the median monthly production for team A.

(d) Find the median monthly production for team B.

(e) Calculate the average cost per month for the two teams combined.

Chap. tz Statistics721

3

2. The following data from the u.s. Department of Labor represent the 2004 me-dian hourly wages for certain trades workers.

Occupation Hourly Wage

Electrician $20.33

HVAC-R technician st7.43

Welder gr4.t2

Machinist $16.33

Aircraft mechanic szt.77

Diesel mechanic sr7.20

(a) Find the mean hourly wage for the six occupations. Round to the nearest cent.

(b) Find the median hourly wage for the six occupations. Round to the nearest cent.

(c) In a 40-hr work week, how much more would an electrician earn comparedwith the mean earnings of these six trades?

Landscaping A landscaping company builds retaining walls and other proj-ects using blocks that are 8 in. tall, 18 in. wide, and 12 in. deep. The number ofblocks used on each of the last eight projects is given in the table.

Project Number of Blocks

Brick planter #1 64

Landscape wall 420

Retaining wall #1 480

Court yard wall 600

Patio and barbeque area 450

Retaining wall#2 1750

Water channel wall 3600

Brick planter #2 450

(a) Find the mean number of blocks used for the eight projects. Round to thenearest block.

(b) Find the median number of blocks used for the projects.

(c) Find the mode number of blocks used for the projects.

(d) If each block costs $3.09, use your answer to part (a) to calculate the meancost of blocks for the eight projects.

4. Wastewater Technology The seven-day mean of settleable solids cannot ex-ceed 0.15 milliliters per liter (mL/L). During the past seven days, the measure-ments of concentration have been 0.13, 0.18, 0.21, 0.14,0.12,0.11, and 0.15mL/L. 'What was the mean concentration? Was it within the limit?

Sec. lz-z Measures of Central Tendency 725

5. Automotive Trades A mechanic has logged the following numbers of hours in

maintaining and repairing a certain Detroit Series 6 diesel engine for 10 differ-ent servicings.

8.5 15.9 20.0 6.5 rr.2 t9.r 18.0 7.4 9.8 22.5

(a) Find the mean for the hours logged.

(b) Find the median for the hours logged.

(c) If the mechanic earns $17.95 per hour, how much does he earn doing the

median servicing on this engine?

6. Forestry A forest ranger wishes to determine the velocity of a stream. She

throws an object into the stream and clocks the time it takes to travel a premea-

sured distance of 200 ft. She then divides the time into 200 to calculate the ve-

locity in feet per second. For greater accuracy, she repeats this process fourtimes and finds the mean of the four velocities. If the four trials result in times

of 21,23,20, and23 sec, calculate the mean velocity to the nearest 0.1 flsec.

7. Hydrology The following table shows the monthly flow through the DarnvilleDam from October through April.

MonthVolume (in Thousand

Acre Feet, kAF)

October 92

November r33

December 239

January 727

February 348

March 499

April 1 03 1

(a) Calculate the mean monthly water flow.

(b) Find the median monthly water flow.

(c) By what percent did the flow in March exceed the mean flow?

8. Meteorology The National Weather Service provides data on the number oftornados in each state that occur throughout the year. Use the following statistics

to answer questions (a)-(e).

Year Kansas Oklahoma

2000 70 46

200r 110 70

2002 r04 23

2003 93 88

2004 125 65

2005 r40 28


(a) Find the mean of the number of tornados in Kansas over the 6-yearperiod.

(b) Find the mean of the number of tornados in Oklahoma over the 6-year period.

(c) Find the median of the number of tornados in Kansas overthe 6-yearperiod.

(d) Find the median of the number of tornados in Oklahoma over the 6-yearperiod.

(e) Suppose a tornado causes an average $250,000 worth of damage. Use themean in part (a) to calculate the average cost per year to repair tornadodamage in Kansas.

9. The U.S. Department of Energy's Energy Information Administration recordsthe annual crude oil field production in the United States. Use the data from199I-2005 to answer the questions that follow.

YearAnnual U.S. Crude Oil FieldProduction (billion banels)

I99 1 2.71

r992 2.62

1993 2.50

1994 2.43

r995 2.39

t996 2.37

1997 2.35

1998 2.28

1999 2.t5

2000 2.r3

2001 2.12

2002 2.r0

2003 2.07

2004 1.98

2005 t.81

(a) Find the mean production over the l5-year period.

(b) Find the median production over the 15-year period.

(c) If one barrel costs $7.50 to produce, what was the median production costfor 1 year?

(d) Study the table again. Why would the mean or the median notbe a reliableindicator of future production?

For Problems 10 and 1 1 , construct an extended frequency distribution for eachset of numbers and calculate the mean of the grouped data. Follow the instruc-tion for interval width.

Sec.'t2-2 Measures of Central Tendency 727

10. Automotive Trades To best determine the fuel economy of a 2006 HondaOdyssey, an automotive technician calculates the mean gas mileage computed

for20 full tanks ofgas.

(a) Use intervals of 0.9 beginning at 19,0 to compute the mean fuel economy(in miles per gallon) from the following data.

2r.7 r9.5 20.6 25.6 24.3 20.r 20.7 22.5 20.2 245 22.3

23.7 20.1 23.1 19.9 23.5 24.4 r9.1 25.2 21.8

(b) At $3.00 per gallon, use the mean fuel economy to calculate the average

cost per mile to operate this vehicle.

11. Health Care A registered nurse has been carefully monitoring the blood workof a patient each day over a 24-day period. Each day a blood analysis was per-

formed, and the patient's medication, in milligrams, was titrated (adjusted) ac-

cordingly. Use the values of the medication administered to determine the mean

dose for the last 24 days. Use intervals of 0.49 beginning at 78.50.

Doses in mg

80.63 79.18 78.78 80.79 81.13 79.17 78.66 79.t3 81.08

79.60 79.29 8r.29 81.33 78.1r 8r.47 79.89 80.41 80.60

80.01 79.46 78.84 79.26 79.58 79.47

Check your answers in the Appendix, then turn to Section I2-3 to learn about measures

of dispersion.

12-5 MEASURES OF DISPERSION

Measures of central tendency, such as the mean, give us valuable information about a set

of data, but they do not indicate how much variation exists within the set. To analyze

whether or not the mean, for example, is typical of the numbers in the set, we use a

measure of dispersion. A measure of dispersion is a statistical measure that indicates

the spread or variability of data values in a set. A large value for a measure of dispersion

indicates that the dafa arc spread out, while a small value indicates that the data values

are tightly clustered and vary little from the mean value of the set. In this section, we

shall study two measures of dispersion: range and standard deviation.

Range The range is the simplest measure of dispersion. In any set of data, the range is the dif-ference between the largest and the smallest value.

Range : largest data value - smallest data value

The numbers below are the resistor values from the E12 Series most recently used by an

electronics technician. Determine the range for the following resistors in ohms.

2.2Q 3.9c) 8.2c) 4.7a 3.3C) r.2a 5.6c) 2.7a 6.8c)

Range : largest resistor value - smallest resistor value

: 8.2 f) - I.2 Q : 7.0 'f)

728 Chap. rz Statistics

Automotive Trades The fcrllowing measrìrements represent the distance between theground electrode and the center electrode on a spark plug. This is commonly known asthe spark plug gap. Determine the range for these spark plug gaps.

0.030 in. 0.045 in. 0.028 in. 0.032 in. 0.020 in.0.025 in. 0.035 in. 0.040 in. 0.015 in.

Range : largest gap - smallest gap

: 0.045in. - 0.015in. : 0.030in.

Standard Deviation The range gives a quick indication ofthe overall spread ofthe data, but, ifthere arejustone or two extreme values in a large set of data, this could be a misleading indication ofdispersion. The standard deviation is a more commonly used measure of dispersionbecause it gives statisticians an indication of how widely scattered all the values of thedata are, not just the two extremes.

A key component in the formula for standard deviation is a value called the deviationfrom the mean. For any data value, x, its deviation from the mean,x, is given by the dif-ference x - x.

The following set of data represents the number of yards (cubic yards) of a compost mixused in the last eight projects completed by a landscaping company.

9.4 r2.8 4.5 15.6 2.2 23.9 8.3 6.r

To calculate the deviations from the mean, first we must find the mean.

9.4 + t2.8 + 4.5 + 15.6 + 2.2 + 23.9 + 8.3 + 6.1r\lean i; r :

: 10.35 yd3

Finally, we construct a table showing the data (refened to as -x-values) in one column and thecorresponding deviations from the mean (x - x), for each value of x, in a second column.

Data, x (x-x)

9.4 9.4-10.35:-0.9512.8 I2.8- 10.35: 2.45

4.5 4.5-10.35:-5.8515.6 15.6- 10.35: 5.25

2.2 2.2-10.35:-8.1523.9 23.9- 10.35:13.55

8.3 8.3-10.35:-2.056.1 6.1 -10.35:-4.25

Difference between adata point and the mean

Sec. 12-5 Measures of Dispersion 729

Health Care A certified nursing assistant (CNA) recorded the following heart rates inbeats per minute (bpm) of a patient over a 12-hr period.

62 69 13 76 67 58 65 73 81 76 1r 63

Calculate the mean heart rate, and then prepare a table showing the data (x) in one col-umn and the deviations from the mean (x - x) in a second column.

62 + 69 + 73 + 76 + 61 + 58 + 65 + 73 + 81 + 76 + 7I + 63Mean (x) :

t2

: 69.5 bpm

Data (x-x)

62 62 - 69.5 : -1.5

69 69-69.5: -0.5

t5 73-69.5: 3.5

76 76-69.5: 6.5

61 67-69.5: -2558 58-69.5:-11.565 65-69.5: -4.5

t-) 73-69.5: 3.5

81 81 -69.5: 11.5

76 76-69.5: 6.5

7l ll-69.5: 1.5

63 63-69.5: -6.5

If we find the sum of the deviations from the exact (unrounded) value of the mean ineach of the last two examples, we obtain zero. This will always happen because the

mean is the "center" of the data and the negative and positive deviations from this cen-

ter will cancel each other out. In calculating standard deviation, we use the squares of the

deviations from the mean to eliminate the difficulty with negative deviations. To calcu-

late the standard deviation (s) for a set of data, follow these steps:

1. Find the mean (x).

2. Calculate each deviation from the mean (x - x).

3. Square each of these deviations: (x - Ð2.

4. Find the sum of these squares.

5. Divide this sum by the number of values (n) in the set. (This result is called the

variance of the data.)

6. Take the square root of this quotient.

Chap. tz Statistics750

'We can summarize this six-step procedure with the following formula:

In the previous Your Thrn, we firund the mean heart rate of a patient (Step 1) and the de-viations from the mean (Step 2). To complete the calculation for the standard deviationof this data, we continue as follows:

Step 3 Extend the table to show an (x *.1)2 column, and square each of the devia-tions from the mean in colurrrn 2 to obtain these numbers.

Step 4 Find the sum ofthese squares.

The results of these two steps are shown in the completed table.

Data (x-r) . _-a\x - )cf

62 -7.5 56.25

69 -0.5 a.25

73 3.5 12.25

76 6.5 42.25

67 -2.s 6.25

58 - 11.5 t32.25

65 -4.5 20,25

73 3.5 12.25

8t 11.5 t3t2.25

76 ó.-1 42.25

7t 1.5 2.25

63 -6.5 4225

Sum:501

Step 5 Divide the sum oJthe (¡ - Ð2 byn, which in rhis case is 12.

sumof (x - x)2 501

n 12 : 41.75

Step 6 Take the $quare root of this quotient.

,: {1L75 = 6.46r...

Sec. rz-5 Mea¡ure¡ of Dlçerrion 75r

lrll¡¡rrll¡Dooll¡0Doll¡úool

'Vy'e round the standard deviation to the same number of decimal digits as the mean.

s = 6.5 bpm

Notice that standard deviation has the same units as the mean and the original data.

Once you have calculated the mean, use the following sequence to fill in the ("x - x)2

column directly:

First, store the mean in memory: 6e.5 @

Thenrusethissequence: 62

69

@ + 56.e5 <- Row I@ + A.ê5 <- Row 2

@

@

...andsoon.

After completing the (x - x)2 column, use your calculator to find the sum of this col-

umn. In this case the sum is 50i. Finally, substitute the sum into the standard deviation

formula:

501 t2 AL75 @ -+ 6.Lt6lt1ægge

Learning Help Þ Remember: The first step in finding the standard deviation is always to calculate the

mean. {

Automotive Trades An automotive technician recorded the number of hours of labor

required for each ofthe last 10 engine rebuilds.

20.3 22.1 18.9 2t.7 19.3 20.8 2t.6 r9.9 20.5 2r.3

Calculate the standard deviation of these data. Show all intermediate results in a table as

shown in the previous example.

The following is the completed table.

0.9216(continued)

Step 3: Find the squaresof the deviations.Step 2: Find the

deviations from the mean.

(x - x)z(x-x) (,Data

0.115620.3-20.64:-0.3420.3

2.t3r622.t 22.I - 20.64: I.46

3.027618.9 18.9 - 20.64: -I.74

21.7 -20.64: 1.06 1.12362t.7

r.7956t9.3 19.3-20.64:-1.34

0.025620.8-20.64: 0.1620.8

752

2t.6 21.6-20.64: 0.96

Chap. tz Statistics

Dafa (x-x) lx - x)'

01440 : 1.007174265

: 1.01 hr, rounded

206.4

10

r0.1440

10 : 1.01440

19.9 19.9-2O.64:-0.14 0.5476

20.5 20.5-20.64:-0.14 0.0196

2t.3 2L3 - 20.64: 0.66 0.4356

sun51l06.a Sum : 10.1440

Step 4: Find the sum ofthe squares.

Step 1: Divide the sum of thedata values by n : 10 to findthe mean.

Step 5i Divide the sumof the squares by n.

Step 6: Take the squareroot of this quotient.

VISUALIZING DISPERSION

To visualize dispersion, imagine two archers shooting at a target. Think of the

bulls-eye as the mean. The first archer is very accurate and consistent. Her shots are

clustered around the center of the target, close to the bulls-eye. They have a smalldispersion. The second archer is a beginner; his shots are widely spread out. Oneway to describe his inconsistency is to say that his shots have a large dispersion.

The standard deviation measures the scatter of the data points from the meanvalue of the data. The set of measurement numbers (24,24,24, 24,24) hasmean equal to 24 and s : 0. There is no dispersion. The set of measurements(22, 23, 24, 25, 26) also has a mean of 24 and a standard deviation of approxi-mately 1.4. The dispersion is small. The set of measurements (2, 8,2I,36,53)still has a mean of 24, but the standard deviation is approximately 18.6, indi-cating a large dispersion.

Grouped Data Recall from Section l2-2 that we often use a frequency distribution to group data whena set is large. The original data are displayed in two columns, one showing class inter-vals and one showing frequencies (F) within each interval. To help us calculate the meanof this grouped data, we added two more columns, one showing the midpoints (M) of theintervals and the other showing the products of the frequencies and the midpoints (FM).Now to help us calculate the standard deviation for grouped data, we create the follow-ing three additional columns:

1. (M - x), to list the deviation between each midpoint and the mean.

2. (M - Ð2, to list the square of each difference.

3. F(M - x)2, to list the product of each square times the frequency of the correspon-ding interval.

Sec. l2-3 Measures of Dispersion 753

Notice that we must calculate the mean before we can fill in these last three columns

The standard deviation for grouped data is defined by the following formula:

Standard Deviation for Grouped Data

sum of F(M -)cs

n

For grouped data, the number of values, n, is the same as the sum of the frequencies

The following frequency distribution represents the diameters of 56 finely machinedholes measured to 0.001 in. The diameters are grouped in intervals of 0.004 in. and the

number of measurements in each interval is shown in the frequency column.

Class Intervals Frequency, F

0.360-0.364 4

0.365-0.369 10

0.370-0.374 T4

0.375-0.379 15

0.380-0.384 8

0.385-0.389 5

To calculate the mean and standard deviation for this grouped data, follow these steps:

Step I Extend the table as shown with the five additional columns.

Step 2 Fill in the midpoint(M) and product (FM) columns.

,/ t//,1Class Interval Frequency, F Midpoint, M Product, FM (M-x) (M-Ð2 F(M - Ð2

0.360-0.364 4 0.362 r.448

0.365-0.369 10 0.361 3.670

o.370-0.374 l4 0.372 5.208

0.375-0.379 15 0.377 5.655

0.380-0.384 8 0.382 3.056

0.385-0.389 5 0.387 r.935

n:56 Sum: 20.972


Step 3 Find the sum of the F and FM columns and use these sums to calculate the mean.

sum of FM 20.972Mean (l) : :

56 : 0.3145 in.

tx

Step 4 Fill in the last three columns

Step 5 Use the sum of the F(M - x)2 column andntocalculate the standard deviation.

sum of F(M

Using a calculator, we can filI in the last column directly as follows:

First, store the mean in memory: .3745 @Then, calculate the F(M - x)2 values like this:

-x 0.0026

56 È 0.0068138514in. : 0.0068in.roundeds:n

frllrr¡r I

lrÕDDll¡oDoll¡pppl

First interval -+ .362 @ @ X 4 -+ O.U006e5

l0Second interval -+ .367...andsoon.

@ @ X -+ 0.00056A9

Once the final column is filled in, calculate the sum of the entries in this column:

0.000625 .0005625 @ .OOOOaZS

0.40e6

+ .0000937s .00045+ + .00078125

-+

Finally, substitute into the standard deviation formula:

56 @ -+ u.u0681385t

Construction The following is a grouped frequency distribution showing the numberand the cost ofprojects in the last year that have been overseen by a construction com-pany. Use the fìve-step method shown in the previous example to calculate the mean andthe standard deviation for this data.

Class IntervalFrequency

FMidpoint,

MProduct,

FM (M-x) (M-Ð2 F(M - Ð2

0.360-0.364 4 0.362 1.448 -0.0125 0.00015625 0.00062s

0.365-0.369 10 0.367 3.670 -0.0075 0.00005625 0.0005625

0.370-0.374 I4 0.372 5.208 -0.0025 0.00000625 0.0000875

0.375-0.379 l-5 0.377 5.655 0.0025 0.00000625 0.00009375

0.380-0.384 8 0382 3.056 0.0075 0.00005625 0.00045

0.385-0.389 5 0.387 r.935 0.0125 0.00015625 0.00078125

n 56 Sum: 20.972 Sum: 0.0026

+

Sec. lz-5 Measures of Dispersion 735


$5,000-$14,ggg 12

$15,000-$24,999 8

$25,000-$34,999 4

$35,000-$44,999 2

$45,000-$54,999 8

$55,000-$64,999 6

$65,000-$74,999 10

Step 2: Fill in thesetwo columns.

Step 1: Attach the fiveadditional columns.

, : DT#: $38,7ee.50 s: sumofF(M -n

26,528,000,000

50

È $23,033.89 x $23,034

The standard deviation of $23,034 for the construction projects is relatively large com-

pared with the mean. This implies that the cost data are widely scattered about the mean

and that the mean would not be very usefil in predicting costs of future projects.

Chap. rz Statistics

Step 4: Fill ¡n thesethree columns.

(M-Ð2 F(M - Ð2Midpt., M Prod., FM (M-x)Þ Clus Interval Freq., F

-28,800 829,440,000 9,953,280,0009,999.5 1r9,994$5,000-$14,999 12

353,440,000 2,927,520,000t59,996 - 18,8008 t9,999.5$15,ooo-$24,999

309,760,000-8,800 77,440,0004 29,999.5 119,998$25,ooo-$34,999

1,440,000 2,880,00039,999.5 79,999 I,200$35,000-$44,999 2

1,003,520,000399,996 ,20011 125,440,0008 49,999.5$45,000-$54,999

21,200 449,440,000 2,696,640,00059,999.5 359,997$55,000-$64,999 6

973,440,000 9,734,400,000699,995 31,20010 69,999.5$65,000-$74,999

Sum : 26,528,000,000n: 5O Sum : 1,939,975

Step 5: Use this sum andn to calculate standard deviation.Step 3: Find the mean using these

two sums.

756

Note Þ

On the other hand, the standard deviation of 0.0068 in. for the finely machined holes inthe preceding Example is relatively small. This implies that the data are closely bunchedabout the mean, and the mean measurement of 0.3745 in. is a reliable representation ofthe data.

More advanced statistical analysis involves methods for calculating the probability thata data value will have a given deviation from the mean, {

Now turn to Exercises I2-3 for more practice in calculating measures of dispersion.

Exercises 12-3 Measures of Dispersion

A. calculate the range and the standard deviation for each set of numbers.

1. 5,4,2,5,7,5,2,4,309

2. 19.9,2.5, 19.9, 5.6, I2.4, 17.I,7.6,9.3,20.I

3. 75,72,93, 95, 77,99,79

4. 364, 350, 355, 320, 349, 310, 353, 347

5. 3.59, 3.45, 3.63,3.99, 3.13, 3.45,3.57

6. 49,35,25,33,39, 31, 17,25, 44,32,36, 14

7 . 2643, r5l L, 3133, 2507, 325r, 3456, 2469, 2900, 3264, 2295g. 2.952, 2.993, 2.691., 2.539, 2.507

B. Calculate the standard deviation of the grouped data in each problem. Create anextended table using the five additional columns shown in the text.

1Class Intervals Frequency, F

500-999 sq. ft 13

1,000-1,499 sq. ft 9

1,500-1,999 sq, ft 11

2,000-2,499 sq. ft 7

2,500-2,999 sq. ft 5

2.Class Intervals Frequency, F

0-9 cm 12

10-19 cm T6

20J;9 cm 9

30-39 cm 8

4049 cm 15

50-59 cm 10

Sec. l2-5 Mea¡ure¡ of Dicper:ion 7rl

JClass Intervals Frequency, F

0-1.9 sec 4

2.0-3.9 sec 6

4.0-5.9 sec 9

6.0-7.9 sec 10

8.0-9.9 sec 8

10.0-11.9 sec 7

12.0-13.9 sec 6


50-149 psi 5

150-249 psi 9

250-349 psi t4

350-449 psi 4

450-549 psi 6

550-649 psi 12

650-:749 psi 8

750-849 psi 2

C. Applications

1. Landscaping The following set of data represents the number of yards (cubic

yards) of a bark mulch used in the last 10 projects completed by a landscaping

company. Determine the range for these amounts.

682r0513731552. Automotive Trades An automotive technician has recorded the number hours of

labor required for each of the last ten engine rebuilds . Find the range of these times '

20.3 22.r 18.9 2r.7 t9.3 20.8 2r.6 19.9 205 2r.3

3. Police Science The following table shows the number of citations issued by

the Pine Valley police department for certain traffic violations from 2000-2005.

Speeding Parking

2000 332 6t2

2001 495 510

2002 380 495

2003 472 550

2004 412 483

2005 366 568

4.


(a) Which category had the largest range, and what was this value?

(b) Find the standard deviation for the speeding data.

(c) Find the standard deviation for the parking data.

4. Finance The following table shows the revenue for each quarter of Mega-cashron in billions of dollars. Use this information to answer the followingquestions.

Quarter 2004 2005

First $6.0 $5.5

Second $4.0 $7.0

Third $s.s $6.s

Fourth $7.0 $3.0

(a) During which year was the range of quarterly revenue the largest?

(b) Find the standard deviation of the revenue for all eight quarters.

5. Aviation BF Goodrich produces brake pads for commercial airliners. Twoproduction teams are working to produce 15-lb brake pads over a 6-monthperiod. Team A works on one furnace deck, while production team B workson a second furnace deck. A 15-lb carbon brake pad costs $1000 perpad to produce. The following table shows the total monthly output of eachteam.

Month Team A Team B

January 37,150 brake pads 40,000 brake pads

February 34,500 41,500

March 35,250 39,000

April 38,750 35,750

May 39,2s0 32,500

June 36,500 33,250

(a) Calculate the mean and standard deviation for the data for productionteamA.

(b) Calculate the mean and standard deviation for the data for productionteam B.

(c) As the lead of the two production teams, which team would you preferand why? (Use the mean and the standard deviation of each to help an-swer this question.)

6. Automotive Trades The following chart shows the number of tires sold permonth last year by the Treadwell Tire Company.

Sec. lz-5 Measures of Dispersion 759

Month Number of Tires Sold

January 580

February 650

March 700

April 630

May 750

June 770

July 800

August 950

September 820

October 750

November 600

December 550

(a) Find the range of the number of tires sold per month.

(b) Find the standard deviation of the monthly sales totals. Round to the near-

est whole number.

7. Machine Trades A tool-and-die company machines a precise pin that is sup-

posed to measure 10.25 mm in diameter. For quality control purposes, 10 pins

are selected at random and measured to determine how closely they match the

specified diameter. The results are as follows:

10.28 10.31 10.24 10.23 10.25 10.22 t0.23 10.32 10.25 10.22

(a) Find the range of the measurements.

(b) Find the standard deviation of measurements.

8. Welding The pressure in acetylene cylinders used in oxyacetylene welding(OAW) must be monitored carefully since the pure form of acetylene is ex-

tremely unstable at a pressure greater than 15 pounds per square inch (psi).

Find the standard deviation of the pressure measurements for the eight cylin-ders shown in the table.

CylinderAcetylene Cylinder

Pressure (in psi)

A 12

B 13

C 15

D 10

E 14

F 15

G 11

H 12


9. Meteorology The following table shows the number of tornados that haveoccurred in four different states over a6-year period.

Year Kansas Oklahoma Arkansas Missouri

2000 70 46 41 2I

200r 110 70 52 42

2002 r04 23 39 30

2003 93 88 89 109

2004 125 65 58 80

2005 r40 28 69 40

(a) In which state was the range of these yearly totals the smallest, and whatwas this range?

(b) Find the standard deviation for the numbers of tornados in Kansas.

(c) Find the standard deviation for the numbers of tornados in Arkansas.

(d) comparing the answers to parts (b) and (c), what does this tell us aboutthe relative reliability of the mean for the two states?

10. The Energy Information Administration of the u.s. Department of Energyrecords the annual crude oil field production in the United States. Use the in-formation in the table to find the standard deviation for a recent 7-year period.

YearAnnual U.S. Crude Oil FieldProduction (billion banels)

1999 2.15

2000 2.r3

200t 2.t2

2002 2.10

2003 2.07

2004 1.98

2005 r.87

For Problems l1 and 12, construct an extended frequency distribution for each setof numbers. Calculate the standard deviation of the grouped data in each problem.Create an extended table using the five additional columns shown in the text. Followthe instruction for interval width.

11. Electrical Trades The following values are amperages measured for a col-lection of wires varying in size from 0000 AWG (American wire Gauge) gaugeto 6 AWG gauge. Use intervals of 59 beginning at 100.

Amps

383 168 389 r79 2r7 309 186 340 22t 156 398 254106 282 290 111 r43 367 264 27s 345 r28 173 r6t326 190 379 267 209 124

Sec. tz-5 Measures of Dispersion 711

Objective

Read bar graphs, linegraphs, and circlegraphs.

12. Automotive Trades The fbllowing data are the corporate average fuel

economy (CAFE) numbers from the National Highway Trafïic Safety Ad-

ministration for domestic passerìgel cals in tlie United States from 1981 to

2004. The CAFE is the average fuel economy in miles per gallon (weighted

by sales) for light trucks and passenger cars in the United States with a

gross vehicle weight rating of 8500 pouncls or less. Use intervals of 0.9 be-

ginning at24.0.

Wei ghtetl Jù e I e t: o rutrnlt ( nryt g )

2',1.1 28.0 24.4 26.3 29.1 28.7 21.4 24.2

25.5 27.5 28.1 21.0 28.1 27.8 21.2 29.1

27.3 29.3 26.9 25.0 26.9 21.0 28.6 21.8

Check your alìswers in the Appenclix, then turn to Problem Set l2 on page 747 for more

practice on graphs and statistics. Ifyou need a quick review ofthe topics ofthis chapter,

visit the chapter Summary fìrst.

StatisticsReview

For vertical bar graphs and all line graphs, locate numerical informatiotr along the vertical

axis and categories or time periods along the horizontal axis. The numerical value associated

with a bar or a point is the nurnber on the vertical scale horizontally across fiom it. For hori-

zontal bar graphs, the labeling of the axes is switched.

Example:

(a) Consider the double-bar graph below. To find the number of frames assembled by

the Tuesday day shift, locate "Tuesday" on the horizontal axis. Then hnd the top

ofthe day shift bar, and look horizontally aÇross to the vertical axis. The day shift

on Tuesday assembled approximately 70 frames.

Weekly Frame Assembly

100

Monday Tuesday Wednesday Thursday Friday

Day of Week

90

BOtt,

Ëzo,Ë uo

ô50EooE!soz

20

10

tt

SUMMARY

T n ht

742 Summary Statistics

Objective

Draw bar graphs, linegraphs, and circlegraphs.

25 100

10 X r00R : ------ -- : 407o

25

Review

(b) Study the broken-line graph below. Suppose we wish to calculate the percent in-crease in the number of paint jobs from January to February. From the graph, wcread the Januaryjob total to be 25, and the February total to be 35. The percentincrease is:

35-25 R

25

10R

Monthly Paint Jobs at Autobrite70

OU

8sooìb408soE

2zo10

For circle graphs, or pie charts, each sector is usually labeled with a percent.

Example: Consider the circle graph below Suppose the average plumbing job gen-erates $227.50. To calculate the portion of this revenue spent on advertising, noticethat the advertising sector is labeled 187o. Therefore, the portion of the average jobspent on advertising would be

0.18 x 9227.5O: $40.95

Percent of Business ExpendituresLabor21o/"

Overhead34/"

Advertising1B%

¡aon uary February

I nsurance6%

March AprilMonth

May June

Parts16%

Transportation5"/"

For bar graphs and line graphs, construct a horizontal and vertical axis. Label the horizontalaxis with categories or time periods and space these equally along the axis. Use the verticalaxis to represent the numerical data. (The axes may be switched to make a horizontal bargraph.) Choose a convenient scale that will cover the range of numbers and provide reason-ably precise readings for the graph. Be sure ro title the graph.

,-_

Summary Statistics 743

Obiective Review

Example: The data below were used to create the bar graph shown

WorkOutput Day

150

360435375180

MondayTuesdayVy'ednesday

ThursdayFriday

Work Output

Monday Tucsday Wednesday Thursday Friday

Day

To create a circle graph, we must first convert each numerical value to its percent of the

total. Then convert each percent to its corresponding portion of 360'. Finally, use a pro-

tractor to measure the sectors of the circle graph. Title the graph and label each sector with

its category name and its percent of the total.

EXample: For the work output data fiom the previous example, ltrst we calculate

that the total output for the week is 1500. The percent represented by Monday can then

be determined as follows:

500

400

300

200

100

o.Yo

=

Mondayoutput--+ 150 _ R e _ 150 x 100 _Weekly output---+ 1500 100 1500

Converting this percent to degrees, we have:

0.10x360'=36"

l07o

DailyWork OutputFriday Monday12T" 10%

Wednesday29"/"

Tuesday

Continuing this process for the remaining days of the week, we obtain the results shown

in the table. The completed circle graph is shown to the right of the table.

Day Percent Degrees

MondayTuesdayWednesdayThursdayFriday

l07o24Vo

297o

257o

l27o

36"86"

104"90"44" Thursday

25%24%

744 Summary Statistics

Objective

Calculate measures ofcentral tcndcncy: mean,median, and mode.

Calculate measures ofdispersion: range andstandard deviation.

Review

The following formula is used to find the mean (x)

sum of the data values"Ï- number of values

The median is the middle value when a set of data is arranged in order from smallest to largest.If there is an even number of data values, the median is the mean of the middle two values.

The mode is the data value that occurs most often in a set of data. There may be more thanone mode or no mode at all.

Example: Find (a) rhe mean, (b) the median, and (c) the mode for the following setof lengths. All measurements are in meters.

12.7 t6.2 15.5 13.9 r3.2 t7.l 15.5

(a) The sum of the lengths is 104. l, and there are 7 measurements. The mean is

loj'l : 14.8'11... ry 14.87m

(b) Rearranging the lengths in numerical order, we have

12.7 13.2 13.9 15.5 15.5 16.2 t7.t

The median, or middle value, is 15.5 m.

(c) There are two measurements of 15.5 m-this is the mode.

To find the range, subtract the smallest data value from the largest.

To find the standard deviation, use the following formula, where (_r - x)2 represents thesquare of the deviation from the mean:

Standard deviation (s) : sumof(x -n

Example: Find (a) the range and (b) the standard deviation for the set of lengths inthe previous example.

(a) The range is

l7.l - 12.7 : 4.4 m.

(b) using the mean value of 14.87 from the previous example, we organize our workinto the table below.

Datarr (x - ,ü) (x - r)2

12.7

16.2r5.513.913.217.t1s.5

-2.171.33

0.63

-0.97-1.67

2.230.63

4.70891.76890.39690.94092.78894.97290.3969

Sum : 15.9"143

using n : 7 and the sum of the (x - i)2 column, we substitute into the standarddeviation formula:

ïr9ß,_ry 1.5 lm"Vj

Summary Statistics 745

Obiective

Calculate mean andstandard deviation fordata grouped in a fre-quency distribution.

The mean is:

* : #: l36psi

Review

For the mean, ¡, prepare a table with columns for class intervals, frequency (F), midpoint

(M), and producr (FM). Find rhe sum of the FM column and divide this by the sum of the

frequencies, n.

For standard cleviation, prepare three additional columns: M - x, (M - Ð2, and F(M - Í)2

Use the following formula:

sumof F(M -n

EXample: Wetding The regulator pressure settings on an oxygen cylinder used

in oxyacetylene welding (OAW) must fall between a range of 110 and 160 psi when

welding material is 1 in. thick. The following frequency distribution shows the

pressures for 40 such welding jobs. Use these data to find the mean and the

standard deviation.


I l0-l 19 psi

120-129 psi130-139 psi

140-149 psi

150-159 psi

'I'he completed table is shown below.

.ç

s:

The standard deviation is:

5,710

40

s È 11.94780315: l1.9psi,rounded

F(M - Ð2w -r) (M -Ð2Product, FMMidpoint,MClass Interval1,849.01,058.0

27.0722.5

2,053.5

-21.5- 11.5

- 1.5

8.518.5

4I

t210

6

I 10-l 19 psi

120-129 psi130-139 psi

140-149 psi

150-159 psi

Sum: 5,710Sum : 5,440n:40

746Summary Statistics

Srlnsncs

1$Éu{

uÉss6

pÊs9

Answers are given in the Appendix.

A. Answer the questions based on the graphs pictured.

Questions 1 to 5 refer to Graph I.Graph I ElectricalTrades

Minimum Size of Circuit Wires

15 20 30 40 50

Circuit Rating (amperes)

l. In general, as amps increase, the necessary wire size(increases, decreases)

2. what is the minimum size of wire needed for a circuit rating of 30 amps?3' what is the minimum wire size needed for a circuit rating of 15 amps?4. V/hat circuit ratings could a wire size of 8 be used for?5. What circuit ratings could a size 14 wire be used for?

Questions 6 to 15 refer to Graph II.Graph ll Printing

Employee Production at Miller printing Co.

e

14

o12.N.nßE58!o.Eatr'

=20

110100

t¡,

5e0'6 80

ï9 ,oÊF 60EanEã 50

bg 40€so220

100

Name

Date

Course/Section

Problem Set Chapter lz Statistics

H,V. P.T. M.M. B.C.

lnitials of Plant Workers6. What was H.V.'s output in February?7. What was P.T.'s output for January?

I January

I February

A.G,

717

8. Who produced the most in JanuarY?

10. Who produced the most in February?

11. Who produced the least in February?

12. Which workers produced more in February than January?

13. Which workers produced leçs in February than January?

14. Determine the tdtãT productftrn-forlamarf- -

15. Determine the total production for February'

Problems t6 to 22 refer to Graph III.

Graph lll Automotive Trades

U.S. Regular Conventional Reta¡l Gasoline Prices

$3.00

$2.50Êoã $2.00oE gr.soatL

$ sr.ooo

$0.50

-.. -.. -.-..r"&

*0".-T -.C -.-"Þqis

,Sos¡e

ooN {e\ S .\$ot-Ëi.:.""u'

Mbnth and Year

16. What was the cost per gallon of regular gas in April 2003?

tT. At what othertime was the cOSt per gallon the same as in April 2003?

18. During which time period did the cost of gas decrease?

19. During which time period did the largest price increase occur?

20. What was the overall increase in the cost per gallon over the time frame shown?

2t. By what pefcent did the cost increase from April 1998 to April 200¡4?

22. How much more did it cost to filIup a2}-galtank in 2005 than in 1999?

Problems 23 to 30 refer to GraPh IV.

Graph lV Finance

Proiected vs. Actual Sales

60

50

ct840

3*o*zoID

s --rz-

r I

trtlll

718

Month

Problem Set Chapter tz Stati¡tic¡

23. ÌVhat was the actual sales total in October?

24. What was the projected sales total in May?

25. During which month were actual sales highest?

26. During which month were projected sales highest?

27. During which month were actual sales lowest?

28. During which month were projected sales lowest?

29- During which months were actual sales lower than projected sales?

30. During which month was the gap between actual and projected sales the largest?What was the difference?

Problems 31 to 35 refer to Graph V.

Graph V Metalworking

Composition of Marine Bronze

Copper

Nickel33%

Bismuth1%

Tin16./"

31. What percent of marine bronze is composed of tin?

32- what percent of this alloy is made up of quantities other than copper?

33. Without measuring, calculate how many degrees of the circle should have goneto nickel.

34. How many ounces of zinc are there in 50 lb of marine bronze?

35. How many grams of bismuth are there in 73 grams of this alloy?

B. Construct graphs from the data given.

l. Construction Construct a bar graph based on these data.

Fire Resistance Rat¡ngs of Various Plywoods

45%

Type of Wood Fire Resistance Rating

R: ] m. plywood 10

e: j in. plywood 25

C: J in. tongue and groove 20

O: j in. gypsum wallboard 40

ø: lin. gypsum-two layers 90

f: t m. gypsum wallboard 60

Problem Set Chapter tz Statistics 749

750

2. Masonry Construct a double-bar graph based on these data.

Output of Bricklayers for the ABC Construction Co.

Bricks Laid

Initials of Employee March APril

R.M. t725 1550

H.H. 1350 1485

A.C. 890 1620

C.T 1830 1950

w.F rt15 1 150

s.D. 7125 1875

3. Meteorology Construct a broken-line graph based on these monthly average

high temperatures.

Month Average High Temperature

January 42"

February 44"

March 500

April 560

May 65'

June 75"

July 81'

August 840

September 77"

October 69"

November 57"

December 46"

4. Health Care Construct a double broken-line graph based on these recorded

blood pressures of a patient over a 12-hr period'

Time Systolic Diastolic Time Systolic Diastolic

7 n.u. t24 85 1 p.ivt. 118 78

8 128 88 2 124 78

9 120 80 3 115 75

10 l16 75 4 t12 78

11 r20 78 5 116 74

12p.w. rt4 80 6 1.20 80

Þroblem Set Chapter tz Statistics

5. Metalworkingin the table.

Construct a circle graph based on the alloy composition shown

Metal Alloy Composition

Copper 437o

Aluminum 29Vo

Zinc 37o

Tin ZVo

Lead 23Vo

6. Fire Protection Construct a circle graph based on the portion of calls repre-sented by each category during a pafiicular week at a fire station.

Type of Call Number of Calls Type of Call Number of Calls

Fire 4 False alarm 10

Medical 9 Hazardous leak/spill 2

Traffic accident 12 Other 3

C. Find the mean, median, and mode for each set of numbers.

1. gg, gg, g4,g7, gg,94, g4,g3, g5, gg, g4

2. 5.9, 3.2, 4.r, 3.5, 4.3, 5.2

3. 20.220, 20.43 4, 20.39 5, 20.324, 20.345, 20.3 5 6, 20.324, 20.25g

4. 4297, 4394, 4036, 6699, 649 l, 5460, 3192, 6321, 6642

5. 696, 659, 7 56, 7 27, 606, 607, 7 27, 635, 7 33, 637

6. 14.66, 17.70, 13.93, 15.53, r7.35,12.19, 14.90

D. calculate the range and the standard deviation for each set of numbers.

l. 9, 4,3,2,9,9,2, I,3, 5, 4,9, 5,7

2. 105.4,109.2,101.9, 100.9, 102.2, rl4.l, 117.7

3. 32,500;36,000; 38,500; 35,500; 37,000;36,000; 33,000

4. 0.47 2, 0.7 gg, 0.419, 0.207, 0.639, 0.937, 0.7 35, 0.949, 0.7 rl, 0.7 53

E. Calculate the mean and the standard deviation of the grouped data in each problem.Create an extended table using the five additional columns shown in the text.


0-4.9 cu cm T2

5.0-9.9 cu cm 9

10.0-14.9 cu cm 3

15.0-19.9 cu cm 2

20.0-24.9 cu cm 6

25.0-29.9 cu cm 8

30.0-34.9 cu cm 10

Þroblem Set Chapter tz Statistics 751


22-26 fr 4

27-31 ft 7

32-36 fr 11

3741ft 9

4246ft 5

47-51 ft 2

F. Applications

¡' Printing Technica Replica uses high-speed printers for large-volume copying.

The following values represent the number of copies requested, followed by the

number of impressions that were wasted during each run.

(a) For each case, calculate the percent of waste and fill in the charl. Round to

the nearest 0.17o.

Copies Requested Wasted Impressions Percent of Waste

15,500 409

17,500 345

18,000 432

16,750 546

14,000 3r6

845r6,250

18,500 538

(b) Find the median of the percent of waste'

(c) Find the mean of the percent of waste'

(d) Find the range of the percent of waste.

(e) Find the standard deviation of the percent of waste.

(f) There is an upcoming job that requires 17,000 usable copies. The supervi-

sor decides to print 17,500. Based on the median percent of waste, should

this provide enough of a cushion to complete the job?

2. Meteorology The following table shows the average precipitation in Seattle,

Washington, from March through August over the past 75 years.

Month Inches Month Inches

March 3.75 June 1.50

April 2.50 July o.75

May 1.75 August r.25

752 Problem Set Chapter tz Statistics

(a) Find the median monthly rotal for the 6-monrh period.

(b) Find the mean monthly total.

(c) Find the range of the monthly averages.

3. Agriculture The following data from the united States Department of Agri-culture represent the u.S. average price for corn in dollars per bushel.

Year U.S. Average Price ($/bushel)

r990-r99t 82.28

r99r-1992 s2.37

1992-1993 s2.07

1993-1994 $2.50

1994-1995 s2.26

r995-1996 $3.24

1996-1997 52.7t

t997-1998 s2.4s

r998-t999 $1.94

1999-2000 $ 1.82

2000-200r $1.8s

2001-2002 $1.97

2002-2003 s2.32

2003-2004 $2.42

2004-2005 $2.06

(a)

(b)

(c)

(d)

(e)

Determine the median yearly price from 1990-2005.

Calculate the mean yearly price over this time period.

Determine the range of these prices.

Calculate the standard deviation of the prices.

If a corn grower produces 3,000,000 bushels, use the mean price per bushelto calculate his average earnings.

The Energy Information Administration's Annual Energy Review gives the in-dustrial customer natural gas prices in dollars per million cubic feet ($/MCF) forthe following states for 1998 to 2002.

4

State 1998 t999 2000 200r 2002

Illinois 3.96 4.06 5.81 5.55 5.13

Indiana 4.28 4.16 5.00 8.09 5.74

Wisconsin 3.78 4.05 5.4s 6.75 5.1 1

Problem Set Chapter tz Statistics 753

(a) Find the mean and standard deviation for Illinois.

(b) Find the mean and standard deviation for Indiana.

(c) Find the mean and standard deviation for Wisconsin.

(d) Which state had the highest mean price?

(e) V/hich state had the smallest standard deviation?

5. Police Science To determine the speed that a vehicle was traveling at the time

of an accident, police officers carefully measure the lengths of the skid marks

left at the scene. There are many factors that determine the skid length, and a

quick and simple formula for determining the speed is S: V30'L'd'e,where S is the speed in miles per hour, L is the skid length in feet, d is the drag

factor, and e is the braking effîciency. The following table shows the skid lengths

for 10 accidents that occurred on a particular stretch of an interstate during icy

conditions this Past winter.

(a) use the given formula, with d : 0.2 (icy road) and e : 1 .0, to calculate

the speeds for the given skid lengths and fil1 in column 2. Round to the

nearest whole number.

Skicl Length, L (fr") Speed, S (mph)

5t4

294

726

350

216

600

476

384

486

564

(b) Find the median sPeed.

(c) Find the mean speed.

(d) Find the range of the speeds

(e) Find the standard deviation of the speeds.

6. Automotive Trades The following values from the National Highway Traffic

Safety Administration give the average curb weights in pounds for domestic pas-

senger cars in the United States from the years 1980 to 2003'

Curb Weights (lb)

3075 3101 3002 3153 3031 3033 2977 2982

3025 3062 3060 3071 3100 3046 3098 3146

3111 3143 3rl9 3124 3132 3168 3166 3161

Problem Set ChaPter tz Statistics754

(a)

(b)

(c)

construct an extended frequency distribution for the data using intervals of49 beginning at2976.

Calculate the mean and the standard deviation of the grouped data.

The following chart represents the approximate fuel economy for certaincurb weights.

use the curb weight closest to the mean to determine the fuel economy forthe average passenger car during this time period.

Fuel Economy Curb Weight

30 mpg 2,5001b

26 mpg 3,0001b

23 mpg 3,500Ib

20 mpg 4,000Ib

7. Sheet Metal rhades The following data represent the gauges of the sheetmetal used by a sheet metal worker during his last 8 jobs.

28GA TZGA 16GA 24GA 22GA 18GA 3OGA 8GA(a) use the chart to determine the range of the thicknesses used in inches.

(b) Use the chart to find the mean thickness of the gauges used in millimeters.

Gauge Thickness (in.) Thickness (mm)

30 0.0157 in. 0.3988 mm

28 0.0187 in 0.4750 mm

26 0.0217 in. 0.5512 mm

24 0.0276 in. 0.7010 mm

22 0.0336 in. 0.8534 mm

20 0.0396 in 1.0058 mm

18 0.0516 in. 1.3106 mm

t6 0.0635 in. L6129 mm

t4 0.0785 in. 1.9939 mm

t2 0.1084 in. 2.7534 mm

10 0.1382 in. 3.5103 mm

8 0.1681 in. 4.2697 mm

8. sheet Metal rrades The following data represent the number of 4g-in. by96-in. (4-ft by 8-ft) sheets of aluminum used by the sheet metal worker for the8 jobs in problem 7. Note that the number of sheets corespond to the gauges of

Problem Set Chapter rz Statistics 755

sheet metal, respectively, as given in problem 7.(e.8., the first project used five

48-in. by 96-in. sheets of 28 gauqe sheet metal).

57101691234(a) Calculate the total surface area, in square inches, of sheet metal used for

each job. Then calculate the mean amount of surface area used per job.

(b) Find the total volume of aluminurn, in cubic inches, used in each project by

multiplyi-ng-each-surfaee-area,by-the.eonesponding sheet thieknesses.

(c) Find the mean of the volumes from part (b).

(d) If aluminum costs appfoximately $0.236 per cubic inch, what is the cost ofthe average project?

756 Problem Set Chepter rz Stati¡tic¡

smnsrrcs - lomira 12 statistics… · smnsrrcs u.lÉ,f .¡{ É rÉs peÉ{objective sample problems...

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