chapter 4 displaying and summarizing quantitative data

GRAPHICAL METHODS FOR QUANTITATIVE DATA

Chapter 4Displaying and Summarizing Quantitative DataCHAPTER OBJECTIVESAt the conclusion of this chapter you should be able to: 1)Construct graphs that appropriately describe quantitative data 2)Calculate and interpret numerical summaries of quantitative data. 3)Combine numerical methods with graphical methods to analyze a data set. 4)Apply graphical methods of summarizing data to choose appropriate numerical summaries. 5)Apply software and/or calculators to automate graphical and numerical summary procedures.

Displaying Quantitative DataHistogramsStem and Leaf DisplaysRelative Frequency Histogram of Exam Grades0.05.10.15.20.25.30405060708090GradeRelative frequency1006Sample histogram;

Do not confuse with bar chartFrequency Histogram

HistogramsA histogram shows three general types of information:It provides visual indication of where the approximate center of the data is.We can gain an understanding of the degree of spread, or variation, in the data.We can observe the shape of the distribution.Center, spread, shapeAll 200 m Races 20.2 secs or lessHistograms Showing Different Centers

Histograms Showing DifferentCenters(football head coach salaries)

Histograms - Same Center, Different Spread(football head coach salaries)

Excel Example: 2012-13 NFL SalariesStatcrunch Example: 2012-13 NFL Salaries

Grades on a statistics examData:75 66 77 66 64 73 91 65 59 86 61 86 6158 70 77 80 58 94 78 62 79 83 54 52 4582 48 67 55Frequency Distribution of Grades Class Limits Frequency40 up to 5050 up to 6060 up to 7070 up to 8080 up to 9090 up to 100Total 2 6 8 7 5 2304Relative Frequency Distribution of Grades Class Limits Relative Frequency40 up to 5050 up to 6060 up to 7070 up to 8080 up to 9090 up to 100 2/30 = .067 6/30 = .200 8/30 = .267 7/30 = .233 5/30 = .167 2/30 = .0675Relative Frequency Histogram of Grades0.05.10.15.20.25.30405060708090GradeRelative frequency1006Frequency and relative frequency histogram of same data will have the same shape.Based on the histo-gram, about what percent of the values are between 47.5 and 52.5?

50%5%17%30%

Countdown10Stem and leaf displaysHave the following general appearancestemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64

Probably havent seen one before now; heres what one looks like.Stem and Leaf DisplaysPartition each no. in data into a stem and leafConstructing stem and leaf display1) deter. stem and leaf partition (5-20 stems)2) write stems in column with smallest stem at top; include all stems in range of data3) only 1 digit in leaves; drop digits or round off4) record leaf for each no. in corresponding stem row; ordering the leaves in each row helpsExample: employee ages at a small company18 21 22 19 32 33 40 41 56 57 64 28 29 29 38 39; stem: 10s digit; leaf: 1s digit18: stem=1; leaf=8; 18 = 1 | 8stemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64Constructing display;

Order the leaves in each stem rowSuppose a 95 yr. old is hiredstemleaf 18 9 21 2 8 9 9 32 3 8 9 40 1 56 7 64 7 8 95Include all stems in the range of dataNumber of TD passes by NFL teams: 2012-2013 season(stems are 10s digit)stemleaf43032472667778920122223344411346788908Smallest number? Largest number?Pulse Rates n = 138

Ignore the circles showing in the graphic;

*Stem rows have leaves o through 4

Note that leaves in each row are orderedAdvantages/Disadvantages of Stem-and-Leaf DisplaysAdvantages1) each measurement displayed2) ascending order in each stem row3) relatively simple (data set not too large)Disadvantagesdisplay becomes unwieldy for large data setsPopulation of 185 US cities with between 100,000 and 500,000Multiply stems by 100,000

Each leaf should be just 1 digit to keep display simple; sometimes data has to be rounded or truncated

Note four rows for each stem value so display is not too wide.

3|6 = 360,000Back-to-back stem-and-leaf displays. TD passes by NFL teams: 1999-2000, 2012-13multiply stems by 101999-20002012-1324036372324665526677789433222211002012222334449998887666167889421113408Below is a stem-and-leaf display for the pulse rates of 24 women at a health clinic. How many pulses are between 67 and 77?

Stems are 10s digits

4681012

Countdown10Interpreting Graphical Displays: ShapeA distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other.

Symmetric distribution

Complex, multimodal distributionNot all distributions have a simple overall shape, especially when there are few observations.

Skewed distributionA distribution is skewed to the right if the right side of the histogram (side with larger values) extends much farther out than the left side. It is skewed to the left if the left side of the histogram extends much farther out than the right side.Heights of Students in Recent Stats Class

Shape (cont.)Female heart attack patients in New York stateAge: left-skewedCost: right-skewed

AlaskaFloridaShape (cont.): OutliersAn important kind of deviation is an outlier. Outliers are observations that lie outside the overall pattern of a distribution. Always look for outliers and try to explain them.The overall pattern is fairly symmetrical except for 2 states clearly not belonging to the main trend. Alaska and Florida have unusual representation of the elderly in their population.

A large gap in the distribution is typically a sign of an outlier.

30This is from the book. Imagine you are doing a study of health care in the 50 US states, and need to know how they differ in terms of their elderly population.This is a histogram of the number of states grouped by the percentage of their residents that are 65 or over.You can see there is one very small number and one very large number, with a gap between them and the rest of the distribution.Values that fall outside of the overall pattern are called outliers. They might be interesting, they might be mistakes - I get those in my data from typos in entering RNA sequence data into the computer. They might only indicate that you need more samples. Will be paying a lot of attention to them throughout class both for what we can learn about biology and also because they can cause trouble with your statistics.

Guess which states they are (florida and alaska).Center: typical value of frozen personal pizza? ~$2.65

Spread: fuel efficiency 4, 8 cylinders4 cylinders: more spread8 cylinders: less spread

Other Graphical Methods for Economic DataTime plotsplot observations in time order, with time on the horizontal axis and the vari-able on the vertical axis** Time seriesmeasurements are taken at regular intervals (monthly unemployment, quarterly GDP, weather records, electricity demand, etc.)Heat Maps

Unemployment Rate, by Educational Attainment

Water Use During Super Bowl

Winning Times 100 M Dash

Numerical Summaries of Quantitative DataNumerical and More Graphical Methods to Describe Univariate Data2 characteristics of a data set to measurecentermeasures where the middle of the data is locatedvariabilitymeasures how spread out the data isThe median: a measure of centerGiven a set of n measurements arranged in order of magnitude,Median=middle valuen oddmean of 2 middle values,n evenEx. 2, 4, 6, 8, 10; n=5; median=6Ex. 2, 4, 6, 8; n=4; median=(4+6)/2=5 Student Pulse Rates (n=62)38, 59, 60, 60, 62, 62, 63, 63, 64, 64, 65, 67, 68, 70, 70, 70, 70, 70, 70, 70, 71, 71, 72, 72, 73, 74, 74, 75, 75, 75, 75, 76, 77, 77, 77, 77, 78, 78, 79, 79, 80, 80, 80, 84, 84, 85, 85, 87, 90, 90, 91, 92, 93, 94, 94, 95, 96, 96, 96, 98, 98, 103Median = (75+76)/2 = 75.5Medians are used oftenYear 2014 baseball salariesMedian $1,450,000 (max=$28,000,000 Zack Greinke; min=$500,000)Median fan age: MLB 45; NFL 43; NBA 41; NHL 39Median existing home sales price: May 2011 $166,500; May 2010 $174,600Median household income (2008 dollars) 2009 $50,221; 2008 $52,029The median splits the histogram into 2 halves of equal area

ExamplesExample: n = 717.5 2.8 3.2 13.9 14.1 25.3 45.8Example n = 7 (ordered):2.8 3.2 13.9 14.1 17.5 25.3 45.8Example: n = 817.5 2.8 3.2 13.9 14.1 25.3 35.7 45.8Example n =8 (ordered)2.8 3.2 13.9 14.1 17.5 25.3 35.7 45.8

m = 14.1m = (14.1+17.5)/2 = 15.8Below are the annual tuition charges at 7 public universities. What is the median tuition?

442949604960497152455546758652454965.549604971

Countdown10Below are the annual tuition charges at 7 public universities. What is the median tuition?

442949605245554649715587758652454965.555464971

Countdown10Measures of SpreadThe range and interquartile range

Ways to measure variabilityrange=largest-smallestOK sometimes; in general, too crude; sensitive to one large or small data valueThe range measures spread by examining the ends of the dataA better way to measure spread is to examine the middle portion of the data

m = median = 3.4Q1= first quartile = 2.3Q3= third quartile = 4.2

Quartiles: Measuring spread by examining the middleThe first quartile, Q1, is the value in the sample that has 25% of the data at or below it (Q1 is the median of the lower half of the sorted data).

The third quartile, Q3, is the value in the sample that has 75% of the data at or below it (Q3 is the median of the upper half of the sorted data).49We are going to start out with a very general way to describe the spread that doesnt matter whether it is symmetric or not - quartiles. Just as the word suggests - quartiles is like quarters or quartets, it involves dividing up the distribution into 4 parts.Now, to get the median, we divided it up into two parts. To get the quartiles we do the exact same thing to the two halves.Use same rules as for median if you have even or odd number of observations.Now, what an we do with these that helps us understand the biology of these diseases?Quartiles and median divide data into 4 pieces

Q1 M Q31/41/41/41/4Quartiles are common measures of spreadhttp://oirp.ncsu.edu/ir/admit

http://oirp.ncsu.edu/univ/peer

University of Southern California

Economic Value of College Majors

Rules for Calculating QuartilesStep 1: find the median of all the data (the median divides the data in half)

Step 2a: find the median of the lower half; this median is Q1;Step 2b: find the median of the upper half; this median is Q3.

Important:when n is odd include the overall median in both halves;when n is even do not include the overall median in either half.Example 2 4 6 8 10 12 14 16 18 20 n = 10

Median m = (10+12)/2 = 22/2 = 11

Q1 : median of lower half 2 4 6 8 10Q1 = 6

Q3 : median of upper half 12 14 16 18 20Q3 = 1611Quartile example: odd no. of data valuesHRs hit by Babe Ruth in each season as a Yankee54 59 35 41 46 25 47 60 54 46 49 46 41 34 22Ordered values:22 25 34 35 41 41 46 46 46 47 49 54 54 59 60Median: value in ordered position 8. median = 46

Lower half (including overall median):22 25 34 35 41 41 46 46

Upper half (including overall median):46 46 47 49 54 54 59 60

Pulse Rates n = 138

Median: mean of pulses in locations 69 & 70: median= (70+70)/2=70Q1: median of lower half (lower half = 69 smallest pulses); Q1 = pulse in ordered position 35;Q1 = 63Q3 median of upper half (upper half = 69 largest pulses); Q3= pulse in position 35 from the high end; Q3=78Below are the weights of 31 linemen on the NCSU football team. What is the value of the first quartile Q1?#stemleaf22255423576242672571026257122759(4)28156715293559910303337314553215523361340

287257.5263.5262.5

Countdown10Interquartile rangelower quartile Q1middle quartile: medianupper quartile Q3interquartile range (IQR)IQR = Q3 Q1measures spread of middle 50% of the dataExample: beginning pulse ratesQ3 = 78; Q1 = 63

IQR = 78 63 = 15Below are the weights of 31 linemen on the NCSU football team. The first quartile Q1 is 263.5. What is the value of the IQR?#stemleaf22255423576242672571026257122759(4)28156715293559910303337314553215523361340

23.539.54669.5

Countdown105-number summary of dataMinimum Q1 median Q3 maximum

Pulse data 45 63 70 78 111m = median = 3.4Q3= third quartile = 4.2Q1= first quartile = 2.3

Largest = max = 6.1Smallest = min = 0.6

Five-number summary:min Q1 m Q3 maxBoxplot: display of 5-number summaryBOXPLOT61Add in a one other thing we know - the spread - the largest and smallest values, and make a box plot.Now, why would you want to make one of these? Boxplot: display of 5-number summaryExample: age of 66 crush victims at rock concerts 1999-2000. 5-number summary:13 17 19 22 47

Boxplot construction1) construct box with ends located at Q1 and Q3; in the box mark the location of median (usually with a line or a +)2) fences are determined by moving a distance 1.5(IQR) from each end of the box;2a) upper fence is 1.5*IQR above the upper quartile2b) lower fence is 1.5*IQR below the lower quartileNote: the fences only help with constructing the boxplot; they do not appear in the final boxplot displayBox plot construction (cont.)3) whiskers: draw lines from the ends of the box left and right to the most extreme data values found within the fences; 4) outliers: special symbols represent each data value beyond the fences;4a) sometimes a different symbol is used for far outliers that are more than 3 IQRs from the quartilesQ3= third quartile = 4.2Q1= first quartile = 2.3

Largest = max = 7.9Boxplot: display of 5-number summaryBOXPLOT

8Interquartile rangeQ3 Q1=4.2 2.3 = 1.9Distance to Q37.9 4.2 = 3.71.5 * IQR = 1.5*1.9=2.85. Individual #25 has a value of 7.9 years, which is 3.7 years above the third quartile. This is more than 2.85 = 1.5*IQR above Q3. Thus, individual #25 is a suspected outlier.65Add in a one other thing we know - the spread - the largest and smallest values, and make a box plot.Now, why would you want to make one of these? ATM Withdrawals by Day, Month, Holidays

Beg. of class pulses (n=138)Q1 = 63, Q3 = 78IQR=78 63=15

1.5(IQR)=1.5(15)=22.5

Q1 - 1.5(IQR): 63 22.5=40.5

Q3 + 1.5(IQR): 78 + 22.5=100.570637840.5100.545Below is a box plot of the yards gained in a recent season by the 136 NFL receivers who gained at least 50 yards. What is the approximate value of Q3 ?

0136273410547684821958109512321369Pass Catching Yards by Receivers450750215545

Countdown10Rock concert deaths: histogram and boxplot

Automating Boxplot ConstructionExcel out of the box does not draw boxplots.Many add-ins are available on the internet that give Excel the capability to draw box plots.Statcrunch (http://statcrunch.stat.ncsu.edu) draws box plots.Q3= third quartile = 4.2Q1= first quartile = 2.3

Largest = max = 7.9Statcrunch Boxplot

72Add in a one other thing we know - the spread - the largest and smallest values, and make a box plot.Now, why would you want to make one of these? Tuition 4-yr Colleges

Statcrunch: 2012-13 NFL Salaries by Position

College Football Head Coach Salaries by Conference

2013 Major League Baseball Salaries by Team

End of General Numerical Summaries.Next: Numerical Summaries of Symmetric Data

#StemLeaves

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105.5556788899

236*00011111122233333344444

236.55556666667777788888888

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FrequencyHeight in InchesNumber of WomenWomen's Height

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FrequencyHeight in InchesNumber of WomenWomen's Height + 4

myeloma

FrequencyBinFrequencyHeight of women on on UCI basketball team

Sheet1

UCI womenClassUCI menHeight in inchesNumber of IndividualsHeight of Class, Women's and Men's Teams

UCI womenClassHeight in inchesNumber of WomenWomen's Height: Class and Team

FrequencyHeight in centimetersNumber of PlantsSize of Red-flowered Plants

FrequencyHeight in centimetersNumber of PlantsHeight of All Plants

whitepinkredHeight in centimetersNumber of PlantsHeight of Plants by Color

lowerlimitupperlimitNpercentdiepercent deadhttp://www.dhs.vic.gov.au/phb/hce/peri/pn/tables/t20.html5009993250.517854.77147100014993840.65113.28333150019997551.2567.426992000249924333.8572.34237625002999983215.2620.639770300034992373936.7540.2323685350039991973630.5350.18197014000449963209.8140.2263064500499910561.650.4710515000up1090.210.92108total64689

YrSinceDiagnosisPercentDeadThisYearOf25NMMyelomabinBinFrequencyNMdisease xbinBinFrequencydisease x+outginBinFrequencytables altered110.610.6110.610.6110.6110.21298110.2117110.61110.6111to show221.221.2221.221.2221.2220.72164220.7225221.22241.2224even and odd331.631.6331.631.6331.6330.83123330.8333331.63361.6336441.941.9441.941.9441.9440.9492440.9443441.94461.9446551.551.5551.551.5551.5551.0582551552551.55551.5555662.162.1662.162.1662.1661.0661661662662.16622.1662772.372.3772.372.3772.3771.0751771771772.3More12.3771882.382.3882.382.3862.3881.1830881.1880882.32.3880992.592.5992.592.5952.5991.2931991.2990992.52.599010102.8102.810102.8102.81042.810101.5102010101.51010110102.8average3.42916666672.81010011112.9112.911112.9112.91132.911111.6111111111.61111011112.9median3.42.91111012123.3123.3123.3123.31223.312121.8121112121.81212012123.43.412122133.4133.4133.4133.41313.4132.51310132.513130133.43.4More01413.6143.61413.6143.61423.61412.814101412.8141411413.63.61523.7153.71523.7153.71533.71522.815101522.815More01523.73.7average3.95925925931633.8163.81633.8163.81643.81633.516111633.51633.83.8median3.61743.9173.91743.9173.91753.91743.817101743.81743.93.91854.1184.11854.1184.11864.11854.02518541854.14.11964.2194.21964.2194.21974.21965.01965average3.3521964.24.22074.5204.52074.5204.52064.52075.02075median2.52074.54.52184.7214.72184.7214.72154.72185.12185.12184.74.72294.9224.92294.9224.92244.92295.52295.52294.94.923105.3235.323105.3235.32335.323107.02310723105.35.324115.6245.624115.6245.62425.6241110.024111024115.65.625126.1256.12516.1251214.025121425126.16.11212

IndividualsYears until deathYears until death after diagnosis with disease X


FrequencyYears until deathNumber of IndividualsYears until death after diagnosis with disease X


FrequencyYears until deathNumber of IndividualsYears until death after diagnosis with multiple myeloma

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&APage &Pgeneralstandard

tpdfxstandard(df+1)/2ln[G((n+1)/2)]G((n+1)/2)df/2ln[G(n/2)]G(n/2)sqrt(np)1+x^2/n(1+x^2/n)^(-(n+1)/2)xstandardt 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&APage &P

tpdf000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

&APage &Pstandardt diststandard normal and t dist

normpdf (2)xgeneralmu0-50.001028186sigma1.5-4.90.0012811986-4.80.0015893921-4.70.0019629781-4.60.0024136242-4.50.0029545656-4.40.0036007041-4.30.0043686881-4.20.0052769677-4.10.0063458184-40.007597324-3.90.0090553128-3.80.0107452387-3.70.0126939997-3.60.0149296869-3.50.0174812594-3.40.0203781398-3.30.0236497286-3.20.0273248374-3.10.0314310445-30.0359939777-2.90.0410365342-2.80.0465780507-2.70.0526334389-2.60.0592123074-2.50.0663180925-2.40.0739472231-2.30.082088348-2.20.0907216549-2.10.0998183104-20.1093400498-1.90.1192389443-1.80.12945737-1.70.1399281974-1.60.1505752183-1.50.1613138163-1.40.1720518839-1.30.1826909783-1.20.1931277018-1.10.2032552846-10.212965337-0.90.2221497353-0.80.2307025955-0.70.2385222861-0.60.2455134269-0.50.2515888185-0.40.2566712497-0.30.2606951293-0.20.2636078939-0.10.265371150900.26596152030.10.26537115090.20.26360789390.30.26069512930.40.25667124970.50.25158881850.60.24551342690.70.23852228610.80.23070259550.90.222149735310.2129653371.10.20325528461.20.19312770181.30.18269097831.40.17205188391.50.16131381631.60.15057521831.70.13992819741.80.129457371.90.119238944320.10934004982.10.09981831042.20.09072165492.30.0820883482.40.07394722312.50.06631809252.60.05921230742.70.05263343892.80.04657805072.90.041036534230.03599397773.10.03143104453.20.02732483743.30.02364972863.40.02037813983.50.01748125943.60.01492968693.70.01269399973.80.01074523873.90.009055312840.0075973244.10.00634581844.20.00527696774.30.00436868814.40.00360070414.50.00295456564.60.00241362424.70.00196297814.80.00158939214.90.001281198650.0010281865.10.0008214794

&APage &P

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&APage &Pgeneral

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#StemLeaves

4*

34.588

95*001233444

105.5556788899

236*00011111122233333344444

236.55556666667777788888888

167*00000112222334444

237.55555666666777888888999

108*0000112224

108.5555667789

49*0012

29.58

410*0223

10.

111*1


Ht,flowers


birthweight


myeloma


Sheet1














Ht,flowers


birthweight


myeloma


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Ht,flowers


birthweight


myeloma


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chapter 4 displaying and summarizing quantitative data

Documents

quantitative data chapter

stem rowsuppose

smallest stem

leafconstructing stem

data set

leaf displayshave

leaf partition

leaf display1