4.4: Other Summaries

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Learning Objectives

Create and interpret stemplots, line graphs, time series for univariate data.
Create and interpret contingency tables and scatter plots for bivariate data.

Stem-and-Leaf Diagrams or Stemplots

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 Susan Dean's spring pre-calculus class, scores for the first exam 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

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

The stemplot shows that most scores fell in the 60s, 70s, 80s, and 90s. Eight out of the 31 scores or approximately 26% $\left(\frac{8}{31}\right)$ were in the 90s or 100, a fairly high number of As.

Exercise $\PageIndex{1}$

For the Park City basketball team, scores for the last 30 games were as follows (smallest to largest):

32; 32; 33; 34; 38; 40; 42; 42; 43; 44; 46; 47; 47; 48; 48; 48; 49; 50; 50; 51; 52; 52; 52; 53; 54; 56; 57; 57; 60; 61

Construct a stem plot for the data.

Answer

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

Interactive Exercise $\PageIndex{1}$

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.

Example $\PageIndex{2}$

The data are the distances (in kilometers) from a home to local supermarkets. Create a 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

Do the data seem to have any concentration of values?

HINT: The leaves are to the right of the decimal.

Answer

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 $\PageIndex{2}$

The following data show the distances (in miles) from the homes of off-campus statistics students to the college. Create a stem plot using the data and identify any outliers:

0.5; 0.7; 1.1; 1.2; 1.2; 1.3; 1.3; 1.5; 1.5; 1.7; 1.7; 1.8; 1.9; 2.0; 2.2; 2.5; 2.6; 2.8; 2.8; 2.8; 3.5; 3.8; 4.4; 4.8; 4.9; 5.2; 5.5; 5.7; 5.8; 8.0

Answer

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

The value 8.0 may be an outlier. Values appear to concentrate at one and two miles.

Example $\PageIndex{3}$: Side-by-Side Stem-and-Leaf plot

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. Tables $\PageIndex{1}$ and $\PageIndex{2}$ 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 $\PageIndex{1}$: Presidential Ages at Inauguration
President	Ageat Inauguration	President	Age	President	Age
Pierce	48	Harding	55	Obama	47
Polk	49	T. Roosevelt	42	G.H.W. Bush	64
Fillmore	50	Wilson	56	G. W. Bush	54
Tyler	51	McKinley	54	Reagan	69
Van Buren	54	B. Harrison	55	Ford	61
Washington	57	Lincoln	52	Hoover	54
Jefferson	57	Grant	46	Truman	60
Madison	57	Hayes	54	Eisenhower	62
J. Q. Adams	57	Arthur	51	L. Johnson	55
Monroe	58	Garfield	49	Kennedy	43
J. Adams	61	A. Johnson	56	F. Roosevelt	51
Jackson	61	Cleveland	47	Nixon	56
Taylor	64	Taft	51	Clinton	47
Buchanan	65	Coolidge	51	Trump	70
W. H. Harrison	68	Cleveland	55	Carter	52

$\PageIndex{2}$ *Presidential Age 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

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 1 1 1 1 1 0	5	3 6 6 7 7 8
9 5 4 4 2 1 1 1 0	6	0 0 3 3 4 4 5 6 7 7 7 8
	7	0 0 1 1 1 4 7 8 8 9
	8	0 1 3 5 8
	9	0 0 3 3

Exercise $\PageIndex{3}$

The table shows the number of wins and losses the Atlanta Hawks have had in 42 seasons. Create a side-by-side stem-and-leaf plot of these wins and losses.

Losses	Wins	Year	Losses	Wins	Year
34	48	1968–1969	41	41	1989–1990
34	48	1969–1970	39	43	1990–1991
46	36	1970–1971	44	38	1991–1992
46	36	1971–1972	39	43	1992–1993
36	46	1972–1973	25	57	1993–1994
47	35	1973–1974	40	42	1994–1995
51	31	1974–1975	36	46	1995–1996
53	29	1975–1976	26	56	1996–1997
51	31	1976–1977	32	50	1997–1998
41	41	1977–1978	19	31	1998–1999
36	46	1978–1979	54	28	1999–2000
32	50	1979–1980	57	25	2000–2001
51	31	1980–1981	49	33	2001–2002
40	42	1981–1982	47	35	2002–2003
39	43	1982–1983	54	28	2003–2004
42	40	1983–1984	69	13	2004–2005
48	34	1984–1985	56	26	2005–2006
32	50	1985–1986	52	30	2006–2007
25	57	1986–1987	45	37	2007–2008
32	50	1987–1988	35	47	2008–2009
30	52	1988–1989	29	53	2009–2010

Answer

Table $\PageIndex{A}$: Atlanta Hawks Wins and Losses
Number of Wins		Number of Losses
3	1	9
9 8 8 6 5	2	5 5 9
8 7 6 6 5 5 4 3 1 1 1 1 0	3	0 2 2 2 2 4 4 5 6 6 6 9 9 9
8 8 7 6 6 6 3 3 3 2 2 1 1 0	4	0 0 1 1 2 4 5 6 6 7 7 8 9
7 7 6 3 2 0 0 0 0	5	1 1 1 2 3 4 4 6 7
	6	9

Line Graphs or Frequency Polygons

Another type of graph that is useful for specific data values is a line graph or a frequency polygon. In the particular line graph shown in the example below, 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 $\PageIndex{4}$

In a survey, 40 mothers were asked how many times per week a teenager must be reminded to do his or her chores. The results are shown in Table and in Figure.

Number of times teenager is reminded	Frequency
0	2
1	5
2	8
3	14
4	7
5	4

A line graph showing the number of times a teenager needs to be reminded to do chores on the x-axis and frequency on the y-axis. — Figure $\PageIndex{1}$

Exercise $\PageIndex{4}$

In a survey, 40 people were asked how many times per year they had their car in the shop for repairs. The results are shown in Table. Construct a line graph.

Number of times in shop	Frequency
0	7
1	10
2	14
3	9

Answer

Interactive Exercise $\PageIndex{4}$

Frequency polygons can also be used for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

Example $\PageIndex{5}$

We will construct an overlay frequency polygon comparing the final exam scores with the students’ final numeric grade.

Frequency Distribution for Calculus Final Test Scores
Range	Midpoint	Frequency	Cumulative Frequency
49.5-59.5	54.5	5	5
59.5-69.5	64.5	10	15
69.5-79.5	74.5	30	45
79.5-89.5	84.5	40	85
89.5-99.5	94.5	15	100

Frequency Distribution for Calculus Final Grades
Range	Midpoint	Frequency	Cumulative Frequency
49.5-59.5	54.5	10	10
59.5-69.5	64.5	10	20
69.5-79.5	74.5	30	50
79.5-89.5	84.5	45	95
89.5-99.5	94.5	5	100

This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency. — Figure $\PageIndex{6}$.

Time Series Graphs

Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.

One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don't have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.

To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.

Example $\PageIndex{6}$

The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.

Year	Jan	Feb	Mar	Apr	May	Jun	Jul
2003	181.7	183.1	184.2	183.8	183.5	183.7	183.9
2004	185.2	186.2	187.4	188.0	189.1	189.7	189.4
2005	190.7	191.8	193.3	194.6	194.4	194.5	195.4
2006	198.3	198.7	199.8	201.5	202.5	202.9	203.5
2007	202.416	203.499	205.352	206.686	207.949	208.352	208.299
2008	211.080	211.693	213.528	214.823	216.632	218.815	219.964
2009	211.143	212.193	212.709	213.240	213.856	215.693	215.351
2010	216.687	216.741	217.631	218.009	218.178	217.965	218.011
2011	220.223	221.309	223.467	224.906	225.964	225.722	225.922
2012	226.665	227.663	229.392	230.085	229.815	229.478	229.104

Year	Aug	Sep	Oct	Nov	Dec	Annual
2003	184.6	185.2	185.0	184.5	184.3	184.0
2004	189.5	189.9	190.9	191.0	190.3	188.9
2005	196.4	198.8	199.2	197.6	196.8	195.3
2006	203.9	202.9	201.8	201.5	201.8	201.6
2007	207.917	208.490	208.936	210.177	210.036	207.342
2008	219.086	218.783	216.573	212.425	210.228	215.303
2009	215.834	215.969	216.177	216.330	215.949	214.537
2010	218.312	218.439	218.711	218.803	219.179	218.056
2011	226.545	226.889	226.421	226.230	225.672	224.939
2012	230.379	231.407	231.317	230.221	229.601	229.594

Answer

This is a times series graph that matches the supplied data. The x-axis shows years from 2003 to 2012, and the y-axis shows the annual CPI. — Figure $\PageIndex{7}$.

Exercise $\PageIndex{6}$

The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO₂ emissions for the United States.

CO₂ Emissions
	Ukraine	United Kingdom	United States
2003	352,259	540,640	5,681,664
2004	343,121	540,409	5,790,761
2005	339,029	541,990	5,826,394
2006	327,797	542,045	5,737,615
2007	328,357	528,631	5,828,697
2008	323,657	522,247	5,656,839
2009	272,176	474,579	5,299,563

Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.

Bivariate Data

Earlier we discussed the methods of summarizing and organizing the data obtained from observing one variable. Such data are called univariate data. When we are interested in studying the relationship between a pair of variables we must collect and organize the data on two variables at the same time. Such data itself is called bivariate data and there are several ways to organize such data depending on the types of the variables.

Univariate Data

Bivariate Data

The mileage of 5 cars

Car	Mileage (mi)
1	78524
2	12574
3	24914
4	65813
5	39824

The ages and mileage of 5 cars

Car	Age (years)	Mileage (mi)
1	7	78524
2	2	12574
3	1	24914
4	5	65813
5	3	39824

Scatter Plots

When both variables are quantitative, we can construct a scatter plot to organize the data. To construct a scatterplot, we use a horizontal axis for the observations of one variable and a vertical axis for the observations of the other. When picking which axis to use for each variable consider whether you suspect that one variable depends on the other. The independent variable will be on the x-axis and dependent will be on the y axis. Each pair of observations is then plotted as a point.

Example $\PageIndex{7}$

The summary of the ages and the mileages of a sample of 5 cars is shown below:

Table $\PageIndex{A}$: Table showing the ages and the mileages of a sample of 5 cars.
Car	$x$ (age in years)	$y$ (mileage in miles)
1	7	78524
2	2	12574
3	1	24914
4	5	65813
5	3	39824

To construct the **scatterplot**, we use a horizontal axis for the ages of cars and a vertical axis for the mileage.

$clipboard_e5f9535330dbc03f9a07cde0b598da592.png$

Figure $\PageIndex{B}$: Scatter plot showing the ages and the mileages of a sample of 5 cars.

Exercise $\PageIndex{7}$

Amelia plays basketball for her high school. She wants to improve to play at the college level. She notices that the number of points she scores in a game goes up in response to the number of hours she practices her jump shot each week. She records the following data:

$X$ (hours practicing jump shot)	$Y$ (points scored in a game)
5	15
7	22
9	28
10	31
11	33
12	36

Construct a scatter plot and state if what Amelia thinks appears to be true.

Answer

$This is a scatter plot for the data provided. The x-axis is labeled in increments of 2 from 0 - 16. The y-axis is labeled in increments of 5 from 0 - 35.$

Figure $\PageIndex{2}$

Yes, Amelia’s assumption appears to be correct. The number of points Amelia scores per game goes up when she practices her jump shot more.

Interactive Exercise $\PageIndex{7}$

Contingency Tables

When we are interested in studying the relationship between a pair of variables we must collect and organize the data on two variables at the same time. Recall that such data itself is called bivariate data and there are several ways to organize such data depending on the types of the variables. Below are examples of univariate and bivariate qualitative data.

Univariate Data

Bivariate Data

The colors of 5 cars

Car	Color
1	Light
2	Dark
3	Dark
4	Light
5	Dark

The color and condition of 5 cars

Car	Color	Condition
1	Light	Like new
2	Dark	Used
3	Dark	Like New
4	Light	Used
5	Dark	Used

A table called contingency table can be used to organize bivariate data in which one or both variables are qualitative.

Example $\PageIndex{8}$

To construct the table, we arrange the observed frequencies into rows and columns. The intersection of a row and a column of a contingency table is called a cell. Each cell shows the frequency of observations that fit the description in the corresponding row and column.

Color\Condition	Like new	Used
Light	1	1
Dark	1	2

Many times, it makes sense to add the column and row with the totals for each column and row and for the entire table.

Color\Condition	Like new	Used	Total:
Light	1	1	2
Dark	1	2	3
Total:	2	3	5

Alternatively, the contingency table can be referred to as a two-way frequency table. Why? In the example above, cover the second and third columns and what's left can be easily seen as the frequency table for the colors of cars! Cover the second and third rows instead and what’s left can be easily seen as the frequency table for whether car conditions! Also note that every contingency table can be easily broken down into two frequency tables, but the two frequency tables cannot be combined into a contingency table if the original data is lost!

Exercise $\PageIndex{8}$

Destiny surveyed the students in her neighborhood and obtained the following data. Construct the contingency table that summarizes the school enrollment by level and type.

Student	School level	School type
Student 1	Middle School	Public
Student 2	Middle School	Private
Student 3	Elementary School	Public
Student 4	Middle School	Public
Student 5	Elementary School	Public
Student 6	High School	Public
Student 7	High School	Private
Student 8	Middle School	Public
Student 9	High School	Public
Student 10	Elementary School	Public
Student 11	Middle School	Private
Student 12	Elementary School	Private
Student 13	Elementary School	Private
Student 14	Elementary School	Private
Student 15	Elementary School	Private
Student 16	Middle School	Private
Student 17	Middle School	Private
Student 18	High School	Private
Student 19	Elementary School	Private

Answer

The following contingency table summarizes the school enrollment by level and type:

	Public	Private	Total
Elementary School	3	5	8
Middle School	3	4	7
High School	2	2	4
Total	8	11	19

Search

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Exercise \(\PageIndex{1}\)

Interactive Exercise \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Exercise \(\PageIndex{2}\)

Example \(\PageIndex{3}\): Side-by-Side Stem-and-Leaf plot

Exercise \(\PageIndex{3}\)

Example \(\PageIndex{4}\)

Exercise \(\PageIndex{4}\)

Interactive Exercise \(\PageIndex{4}\)

Example \(\PageIndex{5}\)

Example \(\PageIndex{6}\)

Exercise \(\PageIndex{6}\)

Example \(\PageIndex{7}\)

Exercise \(\PageIndex{7}\)

Interactive Exercise \(\PageIndex{7}\)

Example \(\PageIndex{8}\)

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Exercise \(\PageIndex{8}\)