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Looking at Data - DistributionsDisplaying Distributions with GraphsIPS Chapter 1.1 2009 W.H. Freeman and Company

Objectives (IPS Chapter 1.1)Displaying distributions with graphs VariablesTypes of variablesGraphs for categorical variablesBar graphs Pie chartsGraphs for quantitative variablesHistograms Stemplots Stemplots versus histogramsInterpreting histogramsTime plots

VariablesIn a study, we collect informationdatafrom individuals. Individuals can be people, animals, plants, or any object of interest.

A variable is any characteristic of an individual. A variable varies among individuals.

Example: age, height, blood pressure, ethnicity, leaf length, first language

The distribution of a variable tells us what values the variable takes and how often it takes these values.

Two types of variables Variables can be either quantitative

Something that takes numerical values for which arithmetic operations, such as adding and averaging, make sense.Example: How tall you are, your age, your blood cholesterol level, the number of credit cards you own.

or categorical.

Something that falls into one of several categories. What can be counted is the count or proportion of individuals in each category.Example: Your blood type (A, B, AB, O), your hair color, your ethnicity, whether you paid income tax last tax year or not.

How do you know if a variable is categorical or quantitative?Ask: What are the n individuals/units in the sample (of size n)?What is being recorded about those n individuals/units?Is that a number ( quantitative) or a statement ( categorical)?Quantitative Each individual is attributed a numerical value.Categorical Each individual is assigned to one of several categories.

Ways to chart categorical dataBecause the variable is categorical, the data in the graph can be ordered any way we want (alphabetical, by increasing value, by year, by personal preference, etc.)

Bar graphs Each category is represented by a bar.

Pie charts The slices must represent the parts of one whole.

Example: Top 10 causes of death in the United States 2001For each individual who died in the United States in 2001, we record what was the cause of death. The table above is a summary of that information.

Top 10 causes of deaths in the United States 2001Bar graphs

Each category is represented by one bar. The bars height shows the count (or sometimes the percentage) for that particular category.

Chart20

700.142

553.768

163.538

123.013

101.537

71.372

62.034

53.852

39.48

32.238

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Counts (x1000)

Sheet1

Heart disease909,894

All Races, Females Percent* 1. Heart Disease 28.6 2. Cancer 21.6 3. Stroke 8.0 4. Chronic lower respiratory diseases 5.2 5. Alzheimer's disease 3.4 6. Diabetes 3.1 7. Unintentional injuries 3.0 8. Influenza and pneumonia 3.0 9. Kidney disease 1.7

All Races, FemalesPercent*Percent

1. Heart Disease28.6

2. Cancer21.6

3. Stroke8

4. Chronic lower respiratory diseases5.2

5. Alzheimer's disease3.4

6. Diabetes3.1

7. Unintentional injuries3

8. Influenza and pneumonia3

9. Kidney disease1.7

10. Septicemia1.5

All other causes20.9

79.1

Rank1Causes of deathNumberDeaths per

100,000 populationAll causes2,416,425848.51.Diseases of heart700,142245.82.Malignant neoplasms (cancer)553,768194.43.Cerebrovascular diseases 163,53857.44.Chronic lower respiratory diseases123,01343.25.Accidents (unintentional injuries)101,53735.76.Diabe

2,416,425

RankCauses of deathCountsPercentsCounts (x1,000)

1Heart diseases700,14229.0%700.142

2Cancers553,76822.9%553.768

3Cerebrovascular163,5386.8%163.538

4Chronic respiratory123,0135.1%123.013

5Accidents101,5374.2%101.537

6Diabetes mellitus71,3723.0%71.372

7Flu & pneumonia62,0342.6%62.034

8Alzheimer's disease53,8522.2%53.852

9Kidney disorders39,4801.6%39.48

10Septicemia32,2381.3%32.238

All other causes629,96726.1%629.967

http://www.infoplease.com/ipa/A0005110.html

1. Rank based on number of deaths.

Source: U.S. National Center for Health Statistics, National Vital Statistics Report, vol. 52, no. 3, Sept. 18, 2003. Web: www.cdc.gov/nchs .

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Cerebrovasc diseases

Bar graph sorted by rank Easy to analyzeTop 10 causes of deaths in the United States 2001Sorted alphabetically Much less useful

Chart20

700.142

553.768

163.538

123.013

101.537

71.372

62.034

53.852

39.48

32.238

Percents

Counts (x1000)

Sheet1





2. Cancer21.6

3. Stroke8



6. Diabetes3.1




10. Septicemia1.5


79.1



2,416,425


1Heart diseases700,14229.0%700.142

2Cancers553,76822.9%553.768



5Accidents101,5374.2%101.537


7Flu & pneumonia62,0342.6%62.034



10Septicemia32,2381.3%32.238





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Chart21

101.537

53.852

553.768

163.538

123.013

71.372

62.034

700.142

39.48

32.238

Percents

Counts (x1000)

Sheet1





2. Cancer21.6

3. Stroke8



6. Diabetes3.1




10. Septicemia1.5


79.1



2,416,425


5Accidents101,5374.2%101.537


2Cancers553,76822.9%553.768




7Flu & pneumonia62,0342.6%62.034

1Heart diseases700,14229.0%700.142


10Septicemia32,2381.3%32.238





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Percent of people dying fromtop 10 causes of death in the United States in 2000Pie chartsEach slice represents a piece of one whole. The size of a slice depends on what percent of the whole this category represents.

Percent of deaths from top 10 causesPercent of deaths from all causesMake sure your labels match the data.

Make sure all percentsadd up to 100.

Child poverty before and after government interventionUNICEF, 1996What does this chart tell you?The United States has the highest rate of child poverty among developed nations (22% of under 18).Its government does the leastthrough taxes and subsidiesto remedy the problem (size of orange bars and percent difference between orange/blue bars).

Could you transform this bar graph to fit in 1 pie chart? In two pie charts? Why?The poverty line is defined as 50% of national median income.

Ways to chart quantitative data

Histograms and stemplotsThese are summary graphs for a single variable. They are very useful to understand the pattern of variability in the data.

Line graphs: time plotsUse when there is a meaningful sequence, like time. The line connecting the points helps emphasize any change over time.

HistogramsThe range of values that a variable can take is divided into equal size intervals.

The histogram shows the number of individual data points that fall in each interval.The first column represents all states with a Hispanic percent in their population between 0% and 4.99%. The height of the column shows how many states (27) have a percent in this range.

The last column represents all states with a Hispanic percent in their population between 40% and 44.99%. There is only one such state: New Mexico, at 42.1% Hispanics.

Stem plotsHow to make a stemplot:Separate each observation into a stem, consisting of all but the final (rightmost) digit, and a leaf, which is that remaining final digit. Stems may have as many digits as needed, but each leaf contains only a single digit.Write the stems in a vertical column with the smallest value at the top, and draw a vertical line at the right of this column.Write each leaf in the row to the right of its stem, in increasing order out from the stem.STEMLEAVES

Percent of Hispanic residents in each of the 50 states

Stem PlotTo compare two related distributions, a back-to-back stem plot with common stems is useful.

Stem plots do not work well for large datasets.

When the observed values have too many digits, trim the numbers before making a stem plot.

When plotting a moderate number of observations, you can split each stem.

Stemplots versus histograms Stemplots are quick and dirty histograms that can easily be done by hand, and therefore are very convenient for back of the envelope calculations. However, they are rarely found in scientific or laymen publications.

Interpreting histogramsWhen describing the distribution of a quantitative variable, we look for the overall pattern and for striking deviations from that pattern. We can describe the overall pattern of a histogram by its shape, center, and spread.Histogram with a line connecting each column too detailedHistogram with a smoothed curve highlighting the overall pattern of the distribution

Most common distribution shapesA distribution is symmetric if the right and left sides of the histogram are approximately mirror images of each other.

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.AlaskaFloridaThe overall pattern is fairly symmetrical except for 2 states that clearly do not belong 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.

How to create a histogramIt is an iterative process try and try again.

What bin size should you use?

Not too many bins with either 0 or 1 countsNot overly summarized that you loose all the informationNot so detailed that it is no longer summary rule of thumb: start with 5 to 10 binsLook at the distribution and refine your bins(There isnt a unique or perfect solution)

Same data set

IMPORTANT NOTE:Your data are the way they are. Do not try to force them into a particular shape.It is a common misconception that if you have a large enough data set, the data will eventually turn out nice and symmetrical.Histogram of Drydays in 1995

Line graphs: time plotsIn a time plot, time always goes on the horizontal, x axis. We describe time series by looking for an overall pattern and for striking deviations from that pattern. In a time series:

Retail price of fresh oranges over timeThis time plot shows a regular pattern of yearly variations. These are seasonal variations in fresh orange pricing most likely due to similar seasonal variations in the production of fresh oranges.

There is also an overall upward trend in pricing over time. It could simply be reflecting inflation trends or a more fundamental change in this industry.Time is on the horizontal, x axis.

The variable of interesthere retail price of fresh oranges goes on the vertical, y axis.

A time plot can be used to compare two or more data sets covering the same time period.The pattern over time for the number of flu diagnoses closely resembles that for the number of deaths from the flu, indicating that about 8% to 10% of the people diagnosed that year died shortly afterward, from complications of the flu.

A picture is worth a thousand words,

BUT There is nothing like hard numbers. Look at the scales.Scales matterHow you stretch the axes and choose your scales can give a different impression.

Why does it matter?What's wrong with these graphs?

Careful reading reveals that:

1. The ranking graph covers an 11-year period, the tuition graph 35 years, yet they are shown comparatively on the cover and without a horizontal time scale.2. Ranking and tuition have very different units, yet both graphs are placed on the same page without a vertical axis to show the units.3. The impression of a recent sharp drop in the ranking graph actually shows that Cornells rank has IMPROVED from 15th to 6th ...Cornells tuition over timeCornells ranking over time

Looking at Data - DistributionsDescribing distributions with numbersIPS Chapter 1.2 2009 W.H. Freeman and Company

Objectives (IPS Chapter 1.2)Describing distributions with numbers

Measures of center: mean, medianMean versus medianMeasures of spread: quartiles, standard deviationFive-number summary and boxplotChoosing among summary statisticsChanging the unit of measurement

The mean or arithmetic average

To calculate the average, or mean, add all values, then divide by the number of individuals. It is the center of mass.

Sum of heights is 1598.3divided by 25 women = 63.9 inches Measure of center: the mean

3 small ht

heightbinnormalFrequencyoutlierFrequencystemleavesheightheight

58.25757057058258.258.264.0

59.55858058059559.559.564.5

60.759591591607960.760.764.1

60.960601601619960.960.964.8

61.96161261262224961.961.965.2

61.9626226226319961.961.965.7

62.26363463464015862.262.266.2

62.264644644652762.262.266.7

62.465653653662762.462.467.1

62.966662662671862.962.967.8

63.16767267268963.163.968.9

63.96868268269663.963.169.6

63.96969169163.963.9

64.07070170164.0

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64.57272064.5

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birthweight

Frequency

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Histogram

myeloma

lowerlimitupperlimitNpercentdiepercent deadhttp://www.dhs.vic.gov.au/phb/hce/peri/pn/tables/t20.html

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150019997551.2567.42699

2000249924333.8572.342376

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350039991973630.5350.1819701

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total64689

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YrSinceDiagnosisPercentDeadThisYearCumulativeDeadBinForN=30PatientsYears2LiveBinFrequency

129299112215

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Mathematical notation:Learn right away how to get the mean using your calculators.

Your numerical summary must be meaningful.The distribution of womens heights appears coherent and symmetrical. The mean is a good numerical summary.

58 60 62 64 66 68 70 72 74 76 78 80 82 84A single numerical summary here would not make sense.

Chart11

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heightbinclassFrequencystemleavesheightheightwomanheightwomanheight58.264.04 big outliersFrequencyNumberNamePOSFeetInchesTotalBinUCI womenClassUCI mentotal=allflowerscompressNamefeetinchestotalinchesweightMenFrequencyMenAloneFrequencyclass,women, men, no outliersBinFrequencyBinbluepinkred

58.25757058258.258.264.0158.21464.059.564.55701Lisa FaulknerG55655901012Jesse Obrand627418557074158.257059100

59.55858059559.559.564.5259.51564.560.764.15803Ashley BigginsC64766001014Jerry Green637519058075459.558060100

60.759591607960.760.764.1360.71664.160.964.85914Katie SturgeonF62746102028Jeff Gloger637517559076260.759161200

60.960601619960.960.964.8460.91764.861.965.260110Lisa WoznickG511716202028DeVaughn Peace637518060077260.960162200

61.96161262224961.961.965.2561.91865.261.965.761211Jana CiperovaG60726304045Aras Baskauskas637518761078061.961263400

61.9626226319961.961.965.7661.91965.762.266.262212Wendy GabbeF511716404043Mike Hood647619062079161.962264400

62.26363464015862.262.266.2762.22066.262.266.763413Courtney FergusonG56666513047Jeff Hufford647617563080062.263465310

62.264644652762.262.266.7862.22166.762.467.164414Kristen GreenG58686622043Jordan Harris657721764081162.264466220

62.465653662762.462.467.1962.42267.162.967.865320Kimberly MartinF60726702027Ross Schraeder657718565082162.465467200

62.966662671862.962.967.81062.92367.863.968.966221Erin TomlinsonG59696812036Matt Okoro677922266083162.966468210

63.16767268963.163.968.91163.92468.963.169.667222Chanda McLeodF511716911021J.R. Christ698124567084163.167269110

63.96868269663.963.169.61263.12569.663.974.068224Nina HanG56667001012Stanislav Zuzak6108222068085063.968370100

63.96969163.963.91363.9n=2569134Brandy HudsonF61737140042Ryan Codi6118321069086163.969271040

64.07070164.070142Cindy OparahF511717230031Adam Parada7084240700More064.070172030

64.171More064.171044Christina CallawayF607270.4666666667731001Dave Korfman728627578.3333333333710this is for red plants64.171473010

64.57264.572074101272064.572374011

64.87364.873075004473064.873175004

65.27465.274076102374165.274276012

65.77565.775077002275465.775477002

66.27666.276078040476266.276378000

66.77766.777079001177266.777279001

67.17867.178480000078067.178080000

67.87967.8More081001179167.879181001

68.98068.982001180068.980082001

69.663.98169.683001181169.681183001

78.064.5821598.3sum8400118216582184001

78.08363.9average8500008317683185000

78.0848600118417484186001

78.065.98563.985071850

8686172861

More071More0

66

68average69.6

72

69

71

66

73

71

72

74

75

75

75

75

76

76

77

77

79

81

82

83

84

86

Ht,flowers

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height

birthweight

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height + 4

myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency

Height of women on on UCI basketball team

01

01

02

02

04

04

13

22

02

12

11

01

40

30

10

10

00

10

00

04

00

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches

Number of Individuals

Height of Class, Women's and Men's Teams

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

UCI women

Class

Height in inches

Number of Women

Women's Height: Class and Team

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches


Plant Height and Flower Color

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants

Size of Red-flowered Plants

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants

Height of Plants

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

blue

pink

red


Number of Plants



5009993250.517854.77147

100014993840.65113.28333

150019997551.2567.42699

2000249924333.8572.342376

25002999983215.2620.639770

300034992373936.7540.2323685

350039991973630.5350.1819701

4000449963209.8140.226306

4500499910561.650.471051

5000up1090.210.92108

total64689

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

YrSinceDiagnosisPercentDeadThisYearCumulativeDeadBinForN=30PatientsYears2LiveBinFrequency

129299112215

21645521446

3125745731664

4966341882

587425110101

668026112120

758517114141

838818116161

9391191More0

102930102More0

111940112

121950122

131960132

141971142

151980152

161990163

1711001173

30183

194

204

214

225

235

246

256

267

278

289

2914

3016

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency

Histogram

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Year since diagnosis

Percent dying per year

Percent of people dying each year post-diagnosis from multiple myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Years since diagnosis

Percent of Patients

Percent Dead from Multiple Myeloma

Measure of center: the medianThe median is the midpoint of a distributionthe number such that half of the observations are smaller and half are larger. 1. Sort observations by size.n = number of observations______________________________

Mean and median for skewed distributionsMean and median for a symmetric distributionLeft skewRight skewMeanMedianMean MedianMeanMedian Comparing the mean and the medianThe mean and the median are the same only if the distribution is symmetrical. The median is a measure of center that is resistant to skew and outliers. The mean is not.

Percent of people dying Mean and median of a distribution with outliers

Impact of skewed data

M = median = 3.4Q1= first quartile = 2.2Q3= third quartile = 4.35Measure of spread: the quartilesThe first quartile, Q1, is the value in the sample that has 25% of the data at or below it ( it is the median of the lower half of the sorted data, excluding M).

The third quartile, Q3, is the value in the sample that has 75% of the data at or below it ( it is the median of the upper half of the sorted data, excluding M).

Ht,flowers

heightbinclassFrequencystemleavesheightheightwomanheightwomanheight58.264.04 big outliersFrequencyNumberNamePOSFeetInchesTotalBinUCI womenClassUCI mentotal=allflowerscompressNamefeetinchestotalinchesweightMenFrequencyMenAloneFrequencyclass,women, men, no outliersBinFrequencyBinwhitepinkred














64.07070164.070142Cindy OparahF511717230031Adam Parada7084240mean700More064.070172030


64.57264.5720741012median72064.572374011

64.87364.87307500447773064.873175004

65.27465.274076102374165.274276012

65.77565.775077002275465.775477002

66.27666.276078040476266.276378000

66.77766.777079001177266.777279001

67.1median7867.178480000078067.178080000

67.863.97967.8More081001179167.879181001

68.9mean8068.982001180068.980082001

69.663.98169.683001181169.681183001

78.0821598.3sum8400118216582184001

78.08363.9average8500008317683185000

78.0mean c out848600118417484186001

78.065.98563.985071850

median with out8686172861

64.1More071More0

66

68average69.6

72

69

71

66

73

71

72

74

75

75

75

75

76

76

77

77

79

81

82

83

84

86

Ht,flowers

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height

birthweight

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height + 4

myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency


Sheet1

01

01

02

02

04

04

13

22

02

12

11

01

40

30

10

10

00

10

00

04

00

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches



00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

UCI women

Class

Height in inches

Number of Women


0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants

Height of All Plants

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

white

pink

red


Number of Plants



5009993250.517854.77147

100014993840.65113.28333

150019997551.2567.42699

2000249924333.8572.342376

25002999983215.2620.639770

300034992373936.7540.2323685

350039991973630.5350.1819701

4000449963209.8140.226306

4500499910561.650.471051

5000up1090.210.92108

total64689

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

YrSinceDiagnosisPercentDeadThisYearOf25NMMyelomabinBinFrequencyNMdisease xbinBinFrequencydisease x+outginBinFrequencytables altered110.610.6110.610.6110.6110.2

1298110.2117110.61110.6111to show221.221.2221.221.2221.2220.7

2164220.7225221.22241.2224even and odd331.631.6331.631.6331.6330.8

3123330.8333331.63361.6336441.941.9441.941.9441.9440.9

492440.9443441.94461.9446551.551.5551.551.5551.5551.0

582551552551.55551.5555662.162.1662.162.1662.1661.0

661661662662.16622.1662772.372.3772.372.3772.3771.0

751771771772.3More12.3771882.382.3882.382.3812.3881.1

830881.1880882.32.3880992.592.5992.592.5922.5991.2

931991.2990992.52.599010102.8102.810102.8102.81032.810101.5

102010101.51010110102.8average3.42916666672.81010011112.9112.911112.9112.91142.911111.6

111111111.61111011112.9median3.42.91111012123.3123.3123.3123.31253.312121.8

121112121.81212012123.43.412122133.4133.4133.4133.4133.4132.5

1310132.513130133.43.4More01413.6143.61413.6143.61413.61412.8

14101412.8141411413.63.61523.7153.71523.7153.71523.71522.8

15101522.815More01523.73.7average3.95925925931633.8163.81633.8163.81633.81633.5

16111633.51633.83.8median3.61743.9173.91743.9173.91743.91743.8

17101743.81743.93.91854.1184.11854.1184.11854.11854.0

2518541854.14.11964.2194.21964.2194.21964.21965.0

1965average3.3521964.24.22074.5204.52074.5204.52074.52075.0

2075median2.52074.54.52184.7214.72184.7214.72114.72185.1

2185.12184.74.72294.9224.92294.9224.92224.92295.5

2295.52294.94.923105.3235.323105.3235.32335.323107.0

2310723105.35.324115.6245.624115.6245.62445.6241110.0

24111024115.65.625126.1256.12556.1251214.0

25121425126.16.1

12

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death

Years until death after diagnosis with disease X

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

Frequency

Years until death



0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Years until death



0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Years until death


Years until death after diagnosis with multiple myeloma

M = median = 3.4Q3= third quartile = 4.35Q1= first quartile = 2.2Largest = max = 6.1Smallest = min = 0.6Five-number summary:min Q1 M Q3 maxFive-number summary and boxplotBOXPLOT

Ht,flowers

















64.57264.5720741012median72064.572374011

64.87364.87307500447773064.873175004

65.27465.274076102374165.274276012

65.77565.775077002275465.775477002

66.27666.276078040476266.276378000

66.77766.777079001177266.777279001

67.1median7867.178480000078067.178080000

67.863.97967.8More081001179167.879181001

68.9mean8068.982001180068.980082001

69.663.98169.683001181169.681183001

78.0821598.3sum8400118216582184001

78.08363.9average8500008317683185000

78.0mean c out848600118417484186001

78.065.98563.985071850


64.1More071More0

66

68average69.6

72

69

71

66

73

71

72

74

75

75

75

75

76

76

77

77

79

81

82

83

84

86

Ht,flowers

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height

birthweight

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height + 4

myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency


Sheet1

01

01

02

02

04

04

13

22

02

12

11

01

40

30

10

10

00

10

00

04

00

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches



00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

UCI women

Class

Height in inches

Number of Women


0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

white

pink

red


Number of Plants



5009993250.517854.77147

100014993840.65113.28333

150019997551.2567.42699

2000249924333.8572.342376

25002999983215.2620.639770

300034992373936.7540.2323685

350039991973630.5350.1819701

4000449963209.8140.226306

4500499910561.650.471051

5000up1090.210.92108

total64689

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0


1298110.2117110.61110.6111to show221.221.2221.221.22455.6220.7

2164220.7225221.22241.2224even and odd331.631.6331.631.62345.3330.8

3123330.8333331.63361.6336441.941.9441.941.92234.9440.9

492440.9443441.94461.9446551.551.5551.551.52124.7551.0

582551552551.55551.5555662.162.1662.162.12014.5661.0

661661662662.16622.1662772.372.3772.372.31964.2771.0

751771771772.3More12.3771882.382.3882.382.31854.1881.1

830881.1880882.32.3880992.592.5992.592.51743.9991.2

931991.2990992.52.599010102.8102.810102.8102.81633.810101.5

102010101.51010110102.8average3.42916666672.81010011112.9112.911112.9112.91523.711111.6

111111111.61111011112.9median3.42.91111012123.3123.3123.3123.31413.612121.8

121112121.81212012123.43.412122133.4133.4133.4133.4133.4132.5

1310132.513130133.43.4More01413.6143.61413.6143.61263.31412.8

14101412.8141411413.63.61523.7153.71523.7153.71152.91522.8

15101522.815More01523.73.7average3.95925925931633.8163.81633.8163.81042.81633.5

16111633.51633.83.8median3.61743.9173.91743.9173.9932.51743.8

17101743.81743.93.91854.1184.11854.1184.1822.31854.0

2518541854.14.11964.2194.21964.2194.2712.31965.0

1965average3.3521964.24.22074.5204.52074.5204.5662.12075.0

2075median2.52074.54.52184.7214.72184.7214.7551.52185.1

2185.12184.74.72294.9224.92294.9224.9441.92295.5

2295.52294.94.923105.3235.323105.3235.3331.623107.0

2310723105.35.324115.6245.624115.6245.6221.2241110.0

24111024115.65.625126.1256.1110.6251214.0

25121425126.16.1

12

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

Frequency

Years until death



0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Years until death



0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Years until death



Comparing box plots for a normal and a right-skewed distributionBoxplots for skewed dataBoxplots remain true to the data and depict clearly symmetry or skew.

Chart7

0.2

1

2.5

4.5

14

0.6

2.2

3.4

4.35

6.1

Disease X Multiple Myeloma

Years until death

Ht,flowers

















64.57264.5720741012median72064.572374011

64.87364.87307500447773064.873175004

65.27465.274076102374165.274276012

65.77565.775077002275465.775477002

66.27666.276078040476266.276378000

66.77766.777079001177266.777279001

67.1median7867.178480000078067.178080000

67.863.97967.8More081001179167.879181001

68.9mean8068.982001180068.980082001

69.663.98169.683001181169.681183001

78.0821598.3sum8400118216582184001

78.08363.9average8500008317683185000

78.0mean c out848600118417484186001

78.065.98563.985071850


64.1More071More0

66

68average69.6

72

69

71

66

73

71

72

74

75

75

75

75

76

76

77

77

79

81

82

83

84

86

Ht,flowers

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height

birthweight

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height + 4

myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency


Sheet1

01

01

02

02

04

04

13

22

02

12

11

01

40

30

10

10

00

10

00

04

00

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches



00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

UCI women

Class

Height in inches

Number of Women


0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

white

pink

red


Number of Plants



5009993250.517854.77147

100014993840.65113.28333

150019997551.2567.42699

2000249924333.8572.342376

25002999983215.2620.639770

300034992373936.7540.2323685

350039991973630.5350.1819701

4000449963209.8140.226306

4500499910561.650.471051

5000up1090.210.92108

total64689

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

YrSinceDiagnosisPercentDeadThisYearOf25NMMyelomabinBinFrequencyNMdisease xbinBinFrequencydisease x+outginBinFrequencytables alteredXXXXdisease X QMM

1298110.2117110.61110.6111to show110.610.6110.610.610.620.2

2164220.7225221.22241.2224even and odd221.221.2221.221.212.221

3123330.8333331.63361.6336for median551.551.5551.551.513.422.5

492440.9443441.94461.9446331.631.6331.631.614.3524.5

582551552551.55551.5555441.941.9441.941.916.1214

661661662662.16622.1662662.162.1662.162.110.6

751771771772.3More12.3771772.372.3772.372.312.2

830881.1880882.32.3880882.382.3882.382.313.4

931991.2990992.52.5990992.592.5992.592.514.35

102010101.51010110102.8average3.42916666672.81010010102.8102.810102.8102.816.1

111111111.61111011112.9median3.42.91111011112.9112.911112.9112.9

121112121.81212012123.43.41212212123.3123.3123.3123.3

1310132.513130133.43.4More0133.4133.4133.4133.4

14101412.8141411413.63.61413.6143.61413.6143.6

15101522.815More01523.73.7average3.95925925931523.7153.71523.7153.7

16111633.51633.83.8median3.61633.8163.81633.8163.8

17101743.81743.93.91743.9173.91743.9173.9

2518541854.14.11854.1184.11854.1184.1

1965average3.3521964.24.21964.2194.21964.2194.2

2075median2.52074.54.52074.5204.52074.5204.5

2185.12184.74.72184.7214.72184.7214.7

2295.52294.94.92294.9224.92294.9224.9

2310723105.35.323105.3235.323105.3235.3

24111024115.65.624115.6245.624115.6245.6

25121425126.16.125126.1256.1

12

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

Frequency

Years until death



0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Years until death



7

5

3

3

2

2

1

0

0

1

0

0

0

1

0

Frequency

Years until death



0

0

0

0

0

Years until death

Disease X

0

0

0

0

0

0

0

0

0

0

Disease X Multiple Myeloma

Years until death

Comparison Using Box Plots

Suspected outliersOutliers are troublesome data points, and it is important to be able to identify them.

One way to raise the flag for a suspected outlier is to compare the distance from the suspicious data point to the nearest quartile (Q1 or Q3). We then compare this distance to the interquartile range (distance between Q1 and Q3).

We call an observation a suspected outlier if it falls more than 1.5 times the size of the interquartile range (IQR) above the first quartile or below the third quartile. This is called the 1.5 * IQR rule for outliers.

Q3 = 4.35Q1 = 2.2Interquartile rangeQ3 Q14.35 2.2 = 2.15Distance to Q37.9 4.35 = 3.55Individual #25 has a value of 7.9 years, which is 3.55 years above the third quartile. This is more than 3.225 years, 1.5 * IQR. Thus, individual #25 is a suspected outlier.

Ht,flowers

















64.57264.5720741012median72064.572374011

64.87364.87307500447773064.873175004

65.27465.274076102374165.274276012

65.77565.775077002275465.775477002

66.27666.276078040476266.276378000

66.77766.777079001177266.777279001

67.1median7867.178480000078067.178080000

67.863.97967.8More081001179167.879181001

68.9mean8068.982001180068.980082001

69.663.98169.683001181169.681183001

78.0821598.3sum8400118216582184001

78.08363.9average8500008317683185000

78.0mean c out848600118417484186001

78.065.98563.985071850


64.1More071More0

66

68average69.6

72

69

71

66

73

71

72

74

75

75

75

75

76

76

77

77

79

81

82

83

84

86

Ht,flowers

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height

birthweight

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Height in Inches

Number of Women

Women's Height + 4

myeloma

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency

Bin

Frequency


Sheet1

01

01

02

02

04

04

13

22

02

12

11

01

40

30

10

10

00

10

00

04

00

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

UCI women

Class

UCI men

Height in inches



00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

UCI women

Class

Height in inches

Number of Women


0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Frequency


Number of Plants


000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

000

white

pink

red


Number of Plants



5009993250.517854.77147

100014993840.65113.28333

150019997551.2567.42699

2000249924333.8572.342376

25002999983215.2620.639770

300034992373936.7540.2323685

350039991973630.5350.1819701

4000449963209.8140.226306

4500499910561.650.471051

5000up1090.210.92108

total64689

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0


1298110.2117110.61110.6111to show221.221.2221.221.22456.1220.7

2164220.7225221.22241.2224even and odd331.631.6331.631.62345.3330.8

3123330.8333331.63361.6336441.941.9441.941.92234.9440.9

492440.9443441.94461.9446551.551.5551.551.52124.7551.0

582551552551.55551.5555662.162.1662.162.12014.5661.0

661661662662.16622.1662772.372.3772.372.31964.2771.0

751771771772.3More12.3771882.382.3882.382.31854.1881.1

830881.1880882.32.3880992.592.5992.592.51743.9991.2

931991.2990992.52.599010102.8102.810102.8102.81633.810101.5

102010101.51010110102.8average3.42916666672.81010011112.9112.911112.9112.91523.711111.6

111111111.61111011112.9median3.42.91111012123.3123.3123.3123.31413.612121.8

121112121.81212012123.43.412122133.4133.4133.4133.4133.4132.5

1310132.513130133.43.4More01413.6143.61413.6143.61263.31412.8

14101412.8141411413.63.61523.7153.71523.7153.71152.91522.8

15101522.815More01523.73.7average3.95925925931633.8163.81633.8163.81042.81633.5

16111633.51633.83.8median3.61743.9173.91743.9173.9932.51743.8

17101743.81743.93.91854.1184.11854.1184.1822.31854.0

2518541854.14.11964.2194.21964.2194.2712.31965.0

1965average3.3521964.24.22074.5204.52074.5204.5662.12075.0

2075median2.52074.54.52184.7214.72184.7214.7551.52185.1

2185.12184.74.72294.9224.92294.9224.9441.92295.5

2295.52294.94.923105.3235.323105.3235.3331.623107.0

2310723105.35.324115.6245.624115.6245.6221.2241110.0

24111024115.65.625126.1256.1110.6251214.0

25121425126.16.1

12

12

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Individuals

Years until death


0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

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Individuals

Years until death


0

0

0

0

0

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0

Frequency

Years until death



0

0

0

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0

0

0

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Frequency

Years until death



0

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0

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Frequency

Years until death



The standard deviation s is used to describe the variation around the mean. Like the mean, it is not resistant to skew or outliers.Measure of spread: the standard deviationMean 1 s.d.

Calculations Well never calculate these by hand, so make sure to know how to get the standard deviation using your calculator or software.Mean = 63.4Sum of squared deviations from mean = 85.2Degrees freedom (df) = (n 1) = 13s2 = variance = 85.2/13 = 6.55 inches squareds = standard deviation = 6.55 = 2.56 inchesWomen height (inches)

Properties of Standard Deviations measures spread about the mean and should be used only when the mean is the measure of center.

s = 0 only when all observations have the same value and there is no spread. Otherwise, s > 0.

s is not resistant to outliers.

s has the same units of measurement as the original observations.

Software output for summary statistics:

Excel - From Menu: Tools/Data Analysis/Descriptive Statistics

Give commonstatistics of your sample data.Minitab

Choosing among summary statisticsBecause the mean is not resistant to outliers or skew, use it to describe distributions that are fairly symmetrical and dont have outliers. Plot the mean and use the standard deviation for error bars.

Otherwise use the median in the five number summary which can be plotted as a boxplot. Boxplot Mean SD

What should you use, when, and why?Arithmetic mean or median?

Middletown is considering imposing an income tax on citizens. City hall wants a numerical summary of its citizens income to estimate the total tax base.

In a study of standard of living of typical families in Middletown, a sociologist makes a numerical summary of family income in that city.Mean: Although income is likely to be right-skewed, the city government wants to know about the total tax base.

Median: The sociologist is interested in a typical family and wants to lessen the impact of extreme incomes.

Changing the unit of measurementVariables can be recorded in different units of measurement. Most often, one measurement unit is a linear transformation of another measurement unit: xnew = a + bx.

Temperatures can be expressed in degrees Fahrenheit or degrees Celsius. TemperatureFahrenheit = 32 + (9/5)* TemperatureCelsius a + bx.

Linear transformations do not change the basic shape of a distribution (skew, symmetry, multimodal). But they do change the measures of center and spread: Multiplying each observation by a positive number b multiplies both measures of center (mean, median) and spread (IQR, s) by b. Adding the same number a (positive or negative) to each observation adds a to measures of center and to quartiles but it does not change measures of spread (IQR, s).

Looking at Data - DistributionsDensity Curves and Normal DistributionsIPS Chapter 1.3 2009 W.H. Freeman and Company

Objectives (IPS Chapter 1.3)Density curves and Normal distributions Density curves Measuring center and spread for density curvesNormal distributionsThe 68-95-99.7 ruleStandardizing observationsUsing the standard Normal Table Inverse Normal calculationsNormal quantile plots

Density curvesA density curve is a mathematical model of a distribution.The total area under the curve, by definition, is equal to 1, or 100%.The area under the curve for a range of values is the proportion of all observations for that range.Histogram of a sample with the smoothed, density curve describing theoretically the population.

Density curves come in any imaginable shape.

Some are well known mathematically and others arent.

Median and mean of a density curveThe median of a density curve is the equal-areas point: the point that divides the area under the curve in half.

The mean of a density curve is the balance point, at which the curve would balance if it were made of solid material.The median and mean are the same for a symmetric density curve. The mean of a skewed curve is pulled in the direction of the long tail.

Normal or Gaussian distributions are a family of symmetrical, bell-shaped density curves defined by a mean m (mu) and a standard deviation s (sigma) : N(m,s). Normal distributionse = 2.71828 The base of the natural logarithm

= pi = 3.14159xx

A family of density curvesHere, means are different (m = 10, 15, and 20) while standard deviations are the same (s = 3)Here, means are the same (m = 15) while standard deviations are different (s = 2, 4, and 6).

Chart1

0.0005140930.00000049560

0.0037986620.00001112360.000000002

0.00874062970.00004461010.0000000142

0.01799698880.00016009020.0000000885

0.03315904630.0005140930.0000004956

0.05467002490.00147728280.000002482

0.08065690820.0037986620.0000111236

0.10648266850.00874062970.0000446101

0.12579440920.01799698880.0001600902

0.13298076010.03315904630.000514093

0.12579440920.05467002490.0014772828

0.10648266850.08065690820.003798662

0.08065690820.10648266850.0087406297

0.05467002490.12579440920.0179969888

0.03315904630.13298076010.0331590463

0.01799698880.12579440920.0546700249

0.00874062970.10648266850.0806569082

0.0037986620.08065690820.1064826685

0.00147728280.05467002490.1257944092

0.0005140930.03315904630.1329807601

0.00016009020.01799698880.1257944092

0.00001112360.0037986620.0806569082

0.00000049560.0005140930.0331590463

0.0000000020.00001112360.003798662

00.00000049560.000514093

Species A

Species B

Species C

PineNeedles

populationmu1015

sigma46

lengthSpecies ASpecies B

10.0079349130.0043703148

20.01349774160.0063587706

30.02156932970.0089984944

40.03237939890.0123851939

50.04566227130.0165795231

60.06049268110.0215862659

70.0752843580.0273350124

80.08801633170.0336664476

90.09666702920.0403284541

100.09973557010.0469853126

110.09666702920.0532413343

120.08801633170.0586775545

130.0752843580.0628972046

140.06049268110.065573286

150.04566227130.0664903801

160.03237939890.065573286

170.02156932970.0628972046

180.01349774160.0586775545

190.0079349130.0532413343

200.00438207510.0469853126

210.00227339060.0403284541

220.00110796210.0336664476

230.0005072620.0273350124

240.00021817070.0215862659

280.00000399590.0063587706

samplinglengthSpecies ASpecies B

distribution10.00000799190.000002482

n=420.00006691510.0000111236

30.00043634130.0000446101

40.00221592420.0001600902

50.00876415020.000514093

60.02699548330.0014772828

70.06475879780.003798662

80.12098536230.0087406297

90.17603266340.0179969888

100.19947114020.0331590463

110.17603266340.0546700249

120.12098536230.0806569082

130.06475879780.1064826685

140.02699548330.1257944092

150.00876415020.1329807601

160.00221592420.1257944092

170.00043634130.1064826685

180.00006691510.0806569082

190.00000799190.0546700249

200.00000074340.0331590463

210.00000005380.0179969888

220.0000000030.0087406297

230.00000000010.003798662

2400.0014772828

2800.0000111236


distribution100

n=25200

300

400

50.00000000160

60.00000185840

70.00044074460.0000000001

80.02191037560.0000000136

90.22831135670.0000012389

100.49867785050.0000564692

110.22831135670.0012852325

120.02191037560.0146069171

130.00044074460.0828976157

140.00000185840.2349265628

150.00000000160.3324519003

1600.2349265628

1700.0828976157

1800.0146069171

1900.0012852325

2000.0000564692

2100.0000012389

2200.0000000136

2300.0000000001

2400

2800

PineNeedles

000

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Species A

Species B

Needle length (inches)

Population distributions

GestationTime

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Species A

Species B

Mean needle length (inches)

Sampling distributions (n=4)

mu and sigma

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Species B



Sheet2

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Species A

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Population distributions

Sheet3

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Species A

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Species A

Species B



populationmu250266

sigma2015

incrementlengthVitamins onlyVitamins and better food

101400.00000000540

1500.00000007430

1600.00000079920

1700.00000669150

1800.00004363410.0000000019

1900.00022159240.0000000709

2000.0008764150.0000016628

2100.00269954830.0000250189

2200.00647587980.0002413624

2300.01209853620.0014929687

2400.01760326630.0059212307

2500.0199471140.0150575218

2600.01760326630.0245513427

2700.01209853620.025667125

2800.00647587980.0172051884

2900.00269954830.0073947223

3000.0008764150.002037814

3100.00022159240.0003600704

3200.00004363410.0000407935

3300.00000669150.0000029633

3400.00000079920.000000138

3500.00000007430.0000000041

3600.00000000540.0000000001

3700.00000000030

38000


distribution100

n=4200

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distribution100

n=25700

800

8.500

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1000

10.500

1100

11.500

1200

12.500

1300

13.500

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14.500

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18.500

2600

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00

00

00

00

00

00

00

00

00

Vitamins only

Vitamins and better food

Gestation time (days)

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

Species A

Species B



00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

Species A

Species B



0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Vitamins only

Gestation time (days)

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

Species A

Species B

Mean needle length (inches

Documents

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