Displaying Data

Objectives:

Students should know the typical graphical displays for the different types of variables.

Students should understand how frequency tables are constructed and how to read absolute and cumulative frequencies from the tables.

Tables or graphs that show the number of occurrences of values in a specified category (nominal and ordinal data) or interval (numerical data).

Frequency tables and graphs can show either values (counts), percentages (of total number), or both.

For quantitative data, cumulative frequencies and percentages can also be shown.

Frequency Distributions

Specialty Frequency Percent

Internal Medicine 183 29.8%

Family Practice 137 22.3%

Pediatrics 98 16.0%

Emergency Medicine 44 7.2%

General Surgery 41 6.7%

Obstetrics/Gynecology 39 6.3%

Other 72 11.7%

Total 614 100.0%

Medical specialties chosen by a sample of graduating medical students (n=614) in 2002

A frequency table for a nominal variable

The most common graphical ways of displaying the data are with bar and pie charts

 For Nominal variables, the categories can be listed in any order. Ordinal variables are usually listed in order from lowest to highest category (or vise versa).

Bar charts plot the categories on the x-axis, and the frequencies (or percents) for each category on the y-axis (bars can be horizontal or vertical).

Pie charts display the relative frequencies for the categories as sections of a circle.

Graphic Displays for Categorical Variables

Medical specialties chosen by graduating medical students (n=614) in

2002

Bar graph

0 25 50 75 100 125 150 175 200

Number of Students

Internal Med.

Family Practice

Pediatrics

Emergency Med.

Ob./Gyn.

Other

General Surgery

Medical specialties chosen by graduating medical students (n=614) in

2002

Pie chart

Internal Medicine

29.8%

Family Practice22.3%

Other11.7%

Pediatrics16.0%

Emergency

Medicine

Ob/Gyn6.3%

GeneralSurgery

6.7%

Graphical displays for quantitative/numerical data:

stem and leaf plots histograms frequency polygons box plots dot plots line graphs scatter plots (used for graphs of two

characteristics)

Graphic Displays for Quantitative Variables

Sex HR Sex HR Sex HR Sex HR Sex HR Sex HR Sex HR

F 55 M 66 F 70 M 73 F 77 M 79 F 82

M 57 F 67 F 70 M 73 F 77 M 79 M 82

M 59 F 67 M 70 M 73 F 77 M 79 F 83

F 61 F 68 M 70 M 73 M 77 F 80 M 83

M 61 F 68 F 71 F 74 M 77 F 80 M 83

M 62 F 68 F 71 F 74 F 78 M 80 F 84

M 62 M 68 M 71 F 74 F 78 F 81 F 84

F 63 F 69 M 71 M 74 F 78 F 81 M 85

F 64 M 69 F 72 F 75 F 78 F 81 F 86

M 64 M 69 M 72 F 75 M 78 M 81 F 86

M 64 M 69 F 73 M 75 M 78 F 82 M 89

M 66 F 70 M 73 M 76 M 79 F 82 M 89

Frequency tables for quantitative variablesExample: Resting heart rates (bpm) for 42 males and 42

females collected during a research study

In this format, it’s difficult to draw any conclusions about the heart rates in the sample.

One way to present a summary of the data is to construct intervals of heart rates, and count the number of observations that fall in each interval:

Heart Rate Interval

Absolute Frequency

55-59 3

60-64 8

65-69 12

70-74 21

75-79 19

80-84 16

85-89 5

This is essentially “collapsing” a continuous variable into an ordinal

variable

Sex HR Sex HR Sex HR Sex HR Sex HR Sex HR Sex HR

F 55 M 66 F 70 M 73 F 77 M 79 F 82

M 57 F 67 F 70 M 73 F 77 M 79 M 82

M 59 F 67 M 70 M 73 F 77 M 79 F 83

F 61 F 68 M 70 M 73 M 77 F 80 M 83

M 61 F 68 F 71 F 74 M 77 F 80 M 83

M 62 F 68 F 71 F 74 F 78 M 80 F 84

M 62 M 68 M 71 F 74 F 78 F 81 F 84

F 63 F 69 M 71 M 74 F 78 F 81 M 85

F 64 M 69 F 72 F 75 F 78 F 81 F 86

M 64 M 69 M 72 F 75 M 78 M 81 F 86

M 64 M 69 F 73 M 75 M 78 F 82 M 89

M 66 F 70 M 73 M 76 M 79 F 82 M 89

A complete frequency table might look like this:

Heart Rate Interval

Absolute Frequency

Cumulative Absolute

Frequency

Relative Frequency

Cumulative Relative

Frequency

55-59 3 3 0.036 0.036

60-64 8 11 0.095 0.131

65-69 12 23 0.143 0.274

70-74 21 44 0.250 0.524

75-79 19 63 0.226 0.750

80-84 16 79 0.190 0.940

85-89 5 84 0.060 1.000

Heart Rate Interval

Absolute Frequency

Cumulative Absolute

Frequency

Relative Frequency

Cumulative Relative

Frequency

55-59 3 3 0.036 0.036

60-64 8 11 0.095 0.131

65-69 12 23 0.143 0.274

70-74 21 44 0.250 0.524

75-79 19 63 0.226 0.750

80-84 16 79 0.190 0.940

85-89 5 84 0.060 1.000

We can easily see (for example) that:• a little more than half (52.4%) of the subjects have

HRs <74 bpm• 21/84 subjects (25%) have HRs between 70 and 74

bpm• <5% (3.6%) of the subjects have HRs <60 bpm

A frequency distribution of a quantitative variable displayed in graphic form is called a histogram:

0

5

10

15

20

25

Fre

quen

cy

55 70 80 8560 65 75 90

Heart Rate (BPM)

//

0

5

10

15

20

25

30

Rel

ativ

e F

requ

ency

(%

)

Heart Rate (BPM)

//52 6257 72 77 82 87 9267

Males (n=42)Females (n=42)

Another type of graph is a frequency polygon, useful for displaying 2 or more quantitative distributions on the same graph:

0 10 20 30 40 50

Age (Years)

Hea

rt R

ate

(BP

M)

50

70

60

80

90

100

Scatterplot of two quantitative variables, age and heart rate:

Misleading Graphs

40

50

60

70

80

90

100

1 2 3 4

Variable NumberP

erc

en

t C

han

ge F

rom

Con

trol

Drug Placebo

80

82

84

86

88

90

92

94

96

98

100

1 2 3 4

Variable Number

Perc

en

t C

han

ge F

rom

Con

trol

Drug Placebo

Warning: When interpreting graphs in the literature, make sure to look at the scales of the axes: different scaling can exaggerate or minimize comparisons