2.2: Display Data

Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs

One simple graph, the stem-and-leaf graph or stemplot, comes from the field of exploratory data analysis. It is a good choice when the data sets are small. To create the plot, divide each observation of data into a stem and a leaf. The leaf consists of a final significant digit. For example, 23 has stem two and leaf three. The number 432 has stem 43 and leaf two. Likewise, the number 5,432 has stem 543 and leaf two. The decimal 9.3 has stem nine and leaf three. Write the stems in a vertical line from smallest to largest. Draw a vertical line to the right of the stems. Then write the leaves in increasing order next to their corresponding stem.

Example \(\PageIndex{1}\)

For the Spring Kinesiology 101 class, data for the minutes of moderate-to-vigorous physical activity (MVPA) per week reported by 31 students were as follows (smallest to largest):
33; 42; 49; 49; 53; 55; 55; 61; 63; 67; 68; 68; 69; 69; 72; 73; 74; 78; 80; 83; 88; 88; 88; 90; 92; 94; 94; 94; 94; 96; 100

Table \(\PageIndex{1}\): Stem and Leaf Graph

Stem	Leaf
3	3
4	2 9 9
5	3 5 5
6	1 3 7 8 8 9 9
7	2 3 4 8
8	0 3 8 8 8
9	0 2 4 4 4 4 6
10	0

Solution

The stemplot shows that most reported minutes fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 data points or approximately 26% \( \left ( \dfrac{8}{31} \right ) \)were in the 90s or 100.

The stemplot is a quick way to graph data and gives an exact picture of the data. You want to look for an overall pattern and any outliers. An outlier is an observation of data that does not fit the rest of the data. It is sometimes called an extreme value. When you graph an outlier, it will appear not to fit the pattern of the graph. Some outliers are due to mistakes (for example, writing down 50 instead of 500) while others may indicate that something unusual is happening. It takes some background information to explain outliers, so we will cover them in more detail later.

Exercise \(\PageIndex{1}\)

The data are the distances (in kilometers) from a participant's home to the nearest Federally Qualified Health Center (FQHC) or major community clinic. Access to healthcare is a critical public health determinant. Create a stem-and-leaf plot (stemplot) using the data:
1.1; 1.5; 2.3; 2.5; 2.7; 3.2; 3.3; 3.3; 3.5; 3.8; 4.0; 4.2; 4.5; 4.5; 4.7; 4.8; 5.5; 5.6; 6.5; 6.7; 12.3

Problem

Do the data seem to have any concentration of values?

Note The leaves are to the right of the decimal.

Answer

Table \(\PageIndex{2}\)

Stem	Leaf
1	1 5
2	3 5 7
3	2 3 3 5 8
4	0 2 5 5 7 8
5	5 6
6	5 7
7
8
9
10
11
12

Solution

The value 12.3 may be an outlier. Values appear to concentrate at three and four kilometers.

Stem	Leaf
1	1 5
2	3 5 7
3	2 3 3 5 8
4	0 2 5 5 7 8
5	5 6
6	5 7
7
8
9
10
11
12	3

Exercise \(\PageIndex{2}\)

A side-by-side stem-and-leaf plot allows a comparison of the two data sets in two columns. In a side-by-side stem-and-leaf plot, two sets of leaves share the same stem. The leaves are to the left and the right of the stems. Table \(\PageIndex{4}\) and Table \(\PageIndex{5}\) show the ages of presidents at their inauguration and at their death. Construct a side-by-side stem-and-leaf plot using this data.

Table \(\PageIndex{4}\): Age of US Presidents when inaugurated

President	Age	President	Age	President	Age
Washington	57	Lincoln	52	Hoover	54
J. Adams	61	A. Johnson	56	F. Roosevelt	51
Jefferson	57	Grant	46	Truman	60
Madison	57	Hayes	54	Eisenhower	62
Monroe	58	Garfield	49	Kennedy	43
J. Q. Adams	57	Arthur	51	L. Johnson	55
Jackson	61	Cleveland	47	Nixon	56
Van Buren	54	B. Harrison	55	Ford	61
W. H. Harrison	68	Cleveland	55	Carter	52
Tyler	51	McKinley	54	Reagan	69
Polk	49	T. Roosevelt	42	G.H.W. Bush	64
Taylor	64	Taft	51	Clinton	47
Fillmore	50	Wilson	56	G. W. Bush	54
Pierce	48	Harding	55	Obama	47
Buchanan	65	Coolidge	51

Table \(\PageIndex{4}\): Age of US Presidents at death

President	Age	President	Age	President	Age
Washington	67	Lincoln	56	Hoover	90
J. Adams	90	A. Johnson	66	F. Roosevelt	63
Jefferson	83	Grant	63	Truman	88
Madison	85	Hayes	70	Eisenhower	78
Monroe	73	Garfield	49	Kennedy	46
J. Q. Adams	80	Arthur	56	L. Johnson	64
Jackson	78	Cleveland	71	Nixon	81
Van Buren	79	B. Harrison	67	Ford	93
W. H. Harrison	68	Cleveland	71	Reagan	93
Tyler	71	McKinley	58
Polk	53	T. Roosevelt	60
Taylor	65	Taft	72
Fillmore	74	Wilson	67
Pierce	64	Harding	57
Buchanan	77	Coolidge	60

Answer

Table \(\PageIndex{5}\):

Ages at Inauguration		Ages at Death
9 9 8 7 7 7 6 3 2	4	6 9
8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 2 1 1 1 1 1 0	5	3 6 6 7 7 8
9 8 5 4 4 2 1 1 1 0	6	0 0 3 3 4 4 5 6 7 7 7 8
	7	0 1 1 1 2 3 4 7 8 8 9
	8	0 1 3 5 8
	9	0 0 3 3

Solution

Ages at Inauguration		Ages at Death
9 9 8 7 7 7 6 3 2	4	6 9
8 7 7 7 7 6 6 6 5 5 5 5 4 4 4 4 4 2 2 1 1 1 1 1 0	5	3 6 6 7 7 8
9 8 5 4 4 2 1 1 1 0	6	0 0 3 3 4 4 5 6 7 7 7 8
	7	0 1 1 1 2 3 4 7 8 8 9
	8	0 1 3 5 8

Line Graphs

Another type of graph that is useful for specific data values is a line graph. In the particular line graph shown in Exercise \(\PageIndex{3}\):, the x-axis (horizontal axis) consists of data values and the y-axis (vertical axis) consists of frequency points. The frequency points are connected using line segments.

Example \(\PageIndex{2}\)

In a survey, 40 patients with a chronic condition (e.g., hypertension) were asked how many days per week they missed taking a prescribed medication dose. The resulting frequency data is used to create a histogram to assess medication adherence.The results are shown in Table \(\PageIndex{7}\) and in Figure \(\PageIndex{1}\)

Table \(\PageIndex{7}\)

Number of times medication was missed	Frequency
0	2
1	5
2	8
3	14
4	7
5	4

Figure \(\PageIndex{1}\)

Bar Graphs

Bar graphs consist of bars that are separated from each other. The bars can be rectangles or they can be rectangular boxes (used in three-dimensional plots), and they can be vertical or horizontal. The bar graph shown in Example 2.5 has age groups represented on the x-axis and proportions on the y-axis.

Exercise \(\PageIndex{3}\)

The percentage of residents in a metropolitan area who have received the seasonal flu vaccine, stratified by age groups, is shown in Table \(\PageIndex{9}\). Construct a bar graph using this data.

Table \(\PageIndex{9}\)

Age groups	Proportion (%) of Residents
10–19	32.5%
20–29	29.5%
30–39	16.4%
40–49	13.9%
50+	7.1%

Answer

Figure \(\PageIndex{2}\)

Solution

Figure 2.3

Exercise \(\PageIndex{1}\)

The columns in Table \(\PageIndex{11}\) show the projected data for the year 2030 for the number and percentages of adults diagnosed with Type 2 Diabetes by geographic region in the United States. These regional differences inform targeted public health funding and interventions. Create a bar graph for this data with the geographic region (qualitative data) on the x-axis and the percentage of adults with Type 2 Diabetes (quantitative data) on the y-axis.

Table \(\PageIndex{11}\)

Region	Number of Cases (in thousands)	Percentage of Population
Northeast	517,720	16.1%
Midwest	695,170	21.6%
South	1,253,540	39.0%
West	749,400	23.3%

Answer

Figure \(\PageIndex{3}\):

Solution

Figure 2.4