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SHAPE: VISUALIZING DISTRIBUTIONS

A really good first cut at understanding whether data are normal or skewed is to lay them out graphically. This is easy to do with any of the full-featured statistics programs out there these days. I’ll show you six ways to lay out your data: bar graphs and pie charts for nominal and ordinal variables; stem-and-leaf plots, box-and-whisker plots, histograms, and frequency polygons for interval variables.

Table 20.8 Some Social Indicators for 50 Randomly Selected Countries

Country

LEXMALE

LEXFEM

INFMORT

MFRATIO

TFR

FEMILLIT

PCGDP

Albania

74.2

80.4

14.2

97

1.85

14.7

3,451

Argentina

72.5

80.0

12.0

96

2.16

2.4

7,705

Armenia

71.3

77.7

22.2

87

1.79

1.4

2,651

Austria

78.2

83.2

4.1

95

1.41

44,087

Bahamas

72.3

78.1

7.9

96

1.95

3.02

1,087

Brunei

75.5

80.5

5.3

107

1.95

7.8

38,417

Cameroon

52.0

53.4

79.8

100

4.20

23.1

1,044

Cape Verde

69.6

74.9

21.4

92

2.52

24.7

3,235

Chad

48.7

51.3

129.9

99

5.78

48.5

692

Chile

76.2

82.3

6.5

98

1.89

2.9

7,884

China

72.3

75.9

20.4

108

1.79

14.1

3,915

Congo

53.5

55.5

77.5

100

3.92

14.1

2,666

Ecuador

72.9

78.9

17.6

100

2.38

6.9

3,700

El Salvador

67.5

77.0

17.5

89

2.22

18.6

4,005

Equatorial Guinea

50.9

53.3

90.9

98

5.08

14.3

8917

Gambia

55.7

59.2

72.2

98

4.64

58.4

442

Ghana

57.1

59.0

67.0

103

4.00

24.2

675

Hungary

69.2

77.4

6.0

90

1.42

0.8

13,413

Iran

71.1

74.1

24.2

103

1.74

20.7

4,966

Ireland

78.1

82.9

4.2

100

1.92

46,826

Japan

80.1

87.2

3.1

95

1.27

37,052

Kenya

56.3

57.5

57.2

100

4.54

13.5

994

Laos

65.3

68.4

41.3

100

3.19

36.2

904

Latvia

68.7

78.1

8.3

85

1.48

0.2

11,300

Lesotho

46.9

46.6

60.7

89

3.05

3.6

591

Liberia

58.7

61.5

87.7

99

4.69

52.3

216

Maldives

71.5

74.9

18.1

102

1.94

1.8

4,054

Mongolia

65.1

71.2

37.3

98

1.92

1.1

1,584

Namibia

61.6

62.5

27.2

97

3.07

11.1

3,682

Nepal

67.2

69.0

35.8

99

2.66

64.0

443

Netherlands

78.5

82.6

4.3

98

1.77

44,122

Nicaragua

71.5

77.8

18.1

98

2.55

29.9

1,041

Pakistan

67.6

68.3

57.4

106

3.58

63.0

1,029

Panama

73.8

79.1

15.7

102

2.41

6.6

7,613

Poland

72.3

80.4

6.2

93

1.29

0.3

10,790

Romania

70.3

77.2

12.9

95

1.35

1.5

7,782

Samoa

69.8

76.0

19.5

109

3.55

1.1

3,189

Senegal

55.5

58.8

55.7

98

4.50

61.9

1,003

Slovenia

75.4

82.6

3.6

95

1.47

0.3

24,613

South Africa

51.8

53.8

37.3

97

2.42

11.7

5,014

Spain

78.6

84.7

3.8

97

1.56

2.0

29,898

Sri Lanka

71.3

78.6

14.2

797

2.22

7.9

1,776

Sudan

58.3

61.4

61.9

101

3.70

40.6

1,512

Suriname

66.5

73.7

20.5

100

2.29

7.4

6,546

Switzerland

80.2

84.7

3.9

95

1.49

60,805

Tanzania

57.4

59.1

55.2

99

5.30

20.9

563

Togo

62.3

65.7

65.7

98

3.87

43.0

394

United Arab Emir.

77.3

79.5

9.0

205

1.88

14.4

46,601

Uruguay

73.7

80.7

11.5

93

2.03

1.3

9,320

Zambia

48.7

50.0

78.4

100

5.34

18.8

1,175

LEXMALE and LEXFEM arethe lifeexpectancies, at birth, forchildren born during 2010-2015. SOURCE: United Nations Department of Economic and Social Affairs/Population Division, Population Estimates and Projections Section. http:// esa.un.org/unpp/index.asp?panel = 2.

INFMORT is the number of children who die, per 1,000 born, during the first year of life. SOURCE: United Nations Department of Economic and Social Affairs/Population Division 35 World Population Prospects: The 2008 Revision. Table A18, for 2010-2015. http://www.un.org/esa/population/publications/wpp2008/wpp2008_highlights.pdf.

MFRATIO is the number of males per 100 females in , 2010-2015. SOURCE: United Nations Department of Economic and Social Affairs/Population Division 35 WorldPopulation Prospects: The 2008Revision. TableA1. http://www.un.org/ esa/population/publications/wpp2008/wpp2008_highlights.pdf.

TFR, or total fertility rate, is an estimate of the number of children a girl born during 2010-2015 will bear if current fertility patterns holds and she lives through her entire child-bearing years. SOURCE: United Nations Department of Economic and Social Affairs/Population Division 35 World Population Prospects: The 2008 Revision. Table A15. http:// www.un.org/esa/population/publications/wpp2008/wpp2008_highlights.pdf.

FEMILLIT is the percentages ofwomen, >15 years of age who are illiterate. SOURCE: UNESCO Institute for Statistics, assessment of July 2002, supplemented by UNESCO Statistical Yearbook 1999. http://www.un.org/Depts/unsd/social/ literacy.htm. Data file ''Illiteracy rate and illiterate population, 15 years and older'' at: http://www.uis.unesco.org/ ev.php?ID = 5794_201&ID2 = DO_TOPIC.

PCGDP is the per capita gross domestic product in 2010. SOURCE: International Monetary Fund. World Economic Outlook Database, April 2009. http://www.imf.org/external/pubs/ft/weo/2009/01/weodata/index.aspx.

BOX 20.2

THE OUTLIER PROBLEM

The mean is one of the all-time great statistics, but it has one very important drawback: In small samples, it's heavily influenced by special cases, called outliers, and even in large samples, it's heavily influenced by big gaps in the distribution of cases. The median, however, is not as strongly affected by extreme scores. In table 20.8, for example, the mean per capita GDP is $10,908, while the median is just $3,808. Take out Liberia (with $216 PCGDP) and Switzerland (with $68,805) and the mean drops 7.5% to $10,091, while the median remains at $3,808.

When interval data (like life expectancy) are normally distributed, the mean is the best indicator of central tendency. When interval data are highly skewed (as is often the case with income), the median may be a better indicator of central tendency. You absolutely must get a feel for the shape of distributions to understand what's going on.

 
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